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]p = Hq(R) ®kk<X>p , and E 2p,q = Hq(R) @kHp(k<X>) . This spectral sequence converges to
H(S); by analogy we also call it a Serre spectral
sequence. Observe that if the requirement that
B
$S simply connected is removed then the
E2-term of the topological spectral sequence becomes "cohomology with twisted coefficients". The identical phenomenon occurs if the requirement that placed by
Ho(R ) D ~ . This is another reason for the parallel:
Ho(R) = ~
in (1.8).
Ho(R) = k
is re-
simple connectivity ~-~
The analogy between the two sides is reinforced by the well known
3.9
Proposition.
Let
~: E ~ B
connected, and either the fibre h-DGF,
morphism with
R
be a fibration with F, or
B
of finite
augmented to
~
and
B
simply connected, E
@-type. Let
R ~ S
path
be an
Ho(R) = ~ . The following assertions
are then equivalent: (i)
The Serre spectral sequence collapses at
(ii)
The morphism
(iii)
H*(E;~)
Proof:
H*(E;~) ~ H*(F;~)
is a free
H~(B;~)
(resp.
module
In the topological case denote
H*(E) ~ H~(F)
is identified with
E2 . H(S) ~ TorR(s;~))
(resp.
H(S)
ApL(B) ~ ApL(E)
is surjective.
is a free
also by
H(R)-module).
R ~ S. Then
H(S) ~ TorR(s;k). In either case, this is one edge
homomorphism for the spectral sequence; the other is
H(R) ~ H(S). Now clearly (i)
(ii) and (iii). To show (ii) ~ (i) we need to reduce to the case topology side we may simply replace side we apply 2.18. Let
~
R
stand for
Now choose cycles
~i E R @ ~ < X >
sequences from
E2
(E2)
(H(S),~)
R
o
~ ~). On the
on the topology side. Then on either side R ~<X>
.
which project to a
R @~
~ R @~<X>
K-basis of
H(k<X>); let
gives an isomorphism of spectral
on.
To see that (iii) ~ (ii) we note
H(R)
Tor**
(or
by an appropriate free model; on the algebra ~
the spectral sequence arises from a model
be their span and observe that
R° = ~
that if (iii) holds the bigraded algebra
is concentrated in bidegrees
tral sequence (cf. (i. IO)) collapses and
(0,*). Thus the Eilenberg-Moore spec-
H(S) ~ TorR(s;~)
H(S) ~ TorH(R)(H(S);~), which in this case is surjective.
is identified with
21
A fibration
satisfying
the conclusions
(TNCZ = totally non cohomologous sions precisely when
3.10
H(S)
The dictionary
is
of (3.9) is said to have TNCZ fibre
to zero). A morphism H(R)-flat.
continued.
R ~ S
satisfies the conclu-
Thus these conditions
are analogous.
We summarize § 3 by the following
table in our dic-
tionary: Fibrations
Free extensions
Homotopy pullback Products
M × N
Homotopy pushout of
M
and
N
Homotopy pushouts The
Cohomology algebra of the homotopy pullback
F-algebra
Homotopy fibre
Homotopy fibre
Cohomology of the homotopy fibre
TorR(s;k)
Homotopy fibre has finite formal dimension
S
Serre spectral
sequence
Homotopy fibre is
of A ~R,S
TorR(S,T)
has finite flat dimension over
Serre spectral
TNCZ
A<X> @A
H(S)
is
R
sequence
H(R)-flat.
§ 4. LOOP SPACES
4.1
Topology.
Let
topy equivalent
(M,*)
be a pointed space. The inclusion of
to the path space fibration PM = {f: I ~ Mlf(O ) = *} ;
Its fibre, ~M, space on
is the space of pointed maps
M. By definition, ~ M
*
in
M
is homo-
z: PM ~ M: zf = f(1).
(sl,*) ~ (M,*)
and is called the loop
is the homotopy pullback of
Evidently any continuous map
@: (M,*) ~ (N,*)
determines
~@: ~M ~ ~N
in the
obvious way. In addition to being a topological *)
~M
space (pointed by the constant
admits a continuous multiplication f(2t)
O < t < 1/2
g(2t-l)
1/2 < t < 1
=
,
(f.g)(t) It is homotopy associative,
and
e,
cocommutative
algebra with diagonal More generally,
@ H*(~M;~);
let
E ~ B
~M
C
~M
at
.
the structure of a graded
has finite
q-type,
then
in this case we get a (dual) commutative
H~(~M;~) ~ H*(~M;~)
consider the pullback diagram
e.
identity.
H.(~M;@)
Hopf algebra° On the other hand, if
~) = H*(~M;~)
f,g
acts as a homotopy
If we pass to rational homology we get in
H*(~M ~ M ;
loop
~M × ~M ~ ~M:
@ H*(~M;~)
Hopf
arising from the multiplication.
be any fibration with fibre
F
over
* E B
and
22
F
~E i BPB
e,
~ PB
E ~ B
it gives a homotopy equivalence is a fibration with fibre (4.2)
(up to homotopy)
F-~EXBPB.
a continuous map (x,f)~. x'f .
Up to homotopy this is an action of E
~B
on
F (x'fg ~ (x.f).g
is itself the path space fibration the resulting map
homotopy equivalent Finally,
to the multiplication
H,(F;Q)
of the Hopf algebra
and
x.e, ~ x)
~B x ~B ~ B
is
defined above.
if we pass to rational homology
(4.3)
EXBPB ~ B
~--F x ~ B
F x~B ~ F ;
and when
On the other hand, the composite
F × ~B; thus
F-~E×BPB
defines
;
in (4.2) we get an action
® H,(~B;~) ~ H,(F;~)
H,(~B;~)
on
H,(F;~)
.
This action is the central object of study in the article
[F-T] of Felix-Thomas
in these proceedings.
4.4
Algebra.
Im g
In analogy with (4.1) we consider an augmented
generating
k. The path space fibration corresponds
h-DGF, ~: R ~ ~
with
to a free model
R<X> ~ k for
g
and its fibre is just the
As observed of quisms
h-DGA, k<X> = ~ ®RR<X>
whose homology
is
Tor(~,~).
in (3.12), a second free model leads to a unique homotopy class (rel ~) ~<X>---~<X'>;
4.5
Definition.
for
R . To understand
The
we can therefore make the
h-DGF
, ~<X>, will be written
~R
and called the loop DGA
the analogue of loop space multiplication
and of loop space action
on a fibre, we note that the analogue of the pullback diagram in (4.1) is, ef. (2.15), k ~-R<X>~----R
k ~.... where
R i s On t h e
a free
other
R<X> ~ ~<X>
hand
R<X>
model the
of
~ ~:
inclusion
a n d we n o t e
R R ~ S
and
R<X> = R @RR.
F x ~ B ~ EXBPB
corresponds
to
the
projection
23
(as
~<X> = k @ ~ < X >
DGF 's).
Thus altogether we get (4.6)
~ +Jl R<X>---~k O ~<X>
in analogy with (4.2). Here (cf. (2.14))
~
is the homotopy fibre of
~ . Observe as well that a
second choice of free models leads to a second version of (4.6); the two versions are then connected by unique homotopy classes of quisms (rel R or
~)
making the resul-
ting diagram homotopy conm~ute. Passing to homology in (4.5) yields the standard (4.7)
TorR(s~k) ~ TorR(s,~) @ TorR(~,~);
when
S = k
TorR(~,~)
this is an associative comultiplication in into a
F-Hopf algebra. For general
Hopf algebra on the
F-algebra
Dualizing with respect to for
S
TorR(k,~)
which makes
(4.7) defines a co-action of this
TorR(s,~). ~
converts
TorR(s,~)
into
ExtR(S,~ ) - this works
DGA,'s as well as for rings. In particular, the diagonal above dualizes to a
graded algebra structure in generally, the map (4. 7)
EXtR(~,~)
and this is eaxctly the Yoneda algebra. More
dualizes to make
EXtR(S;k)
(The Yoneda algebra is defined as follows:
into a module over
HomR(R<X> , R<X>)
by composition - passing to homology defines a product in = EXtR(~,k). ) When H(R)
is piecewise noetherian then
TorR(k,~)
in each degree and in this case the multiplication in multiplication in
4.8
Remark.
If
EXtR(k,~).
has a product given
H(HOmR(R<X> , R<X>)) =
will be finite dimensional
TorR(k,k)
dualizes to a co-
EXtR(k,k) - which is then a Hopf algebra.
R
is a supplemented
a unique homotopy (rel k) inverse
k-algebra then the quism
R<X>~k
k ~ R<X>. In this case we get an
has
h-DGF.
morphism (4.9)
k ~ k @ k<X>
whose (rel k) homotopy class is unique. When
S = ~
the resulting diagonal is homo-
topy associative (rel k); in general it is a homotopy co-action (rel ~). In this case, howewer, the bar construction can be used to provide a strictly associative version of (4.9).
4.10
Th_e dictionary, again.
We have
24
Loop space
B
Loop
Homology algebra, H.(~B;~)
~
> Yoneda algebra,
Action of H.(~B;~) on H.(F;~); F the homotopy fibre of E ~ B
~......~ Action of
4.11
Warning.
space) and
(i)
The graded spaces
EXtR(S;~)
H.(M;~)
DGA ~R = ~<X>, where
R<X>-~
k
ExtR(S,~)
.
EXtR(~,k)
EXtR(k,~)
on
~ational homology of a topological
are two of the examples where we grade downstairs in topology
and upstairs in algebra. (ii)
Although
EXtR(k,k )
sponds to
H*(~B;~)
it is important to observe that
H*
is the dual of
corresponds to
H.(~B;~)
where
Ext
TorR(~,~)
is the dual of
correTor
and
H.. This distinction is necessary when these graded spaces do not
have finite type because then products do not dualize to coproducts. H*{~B;Q)
is a Hopf algebra;
EXtR(~,~)
is only an algebra
H*(~B;~)
is only an algebra;
TorR(~,~)
is a
In this case
and
(iii)
If
M
is~,a simply connected topological space and
integer such that and if
kR ~ 1
F-Hopf algebra.
HM(M;~)
# O
then
~M
= kM-l"
is the first integer such that
The observations
If
R
kM ~ 1
is an
HkR(R) # 0
then
is the first
h-DGF, ~R
with
Ho(R)--~k
= kR +I.
4.11 (ii) and 4.11 (iii) are examples in which simple minded
translation between the two sides fails. It is because of this kind of phenomenon that the translator needs to be very careful.
§ 5 THE HOMOTOPY LIE ALGEBRA
In this chapter
M
q-type while
will always denote an
R
with residue field
will always denote a simply connected topological
Now the rational homotopy groups
via the canonical
ture (the Samelson product);
H.(~M;~)
space of finite
is local (cf. (1.9))
subspace (of elements
~i (M) ® 9, equipped with the Whitehead product
In fact, when this product is transferred to
(degree-l) isomorphism it does give a graded Lie strucM.
h-DGF.'s we need two further identifications.
is a graded Hopf algebra. The Cartan-Serre theorem asserts that
the Hurewicz homomorphism
~.(~M) @ ~ ~ H.(~M;~)
~ ~ ~ ~ 1 + 1 @ ~
is an isomorphism onto the primitive
under the diagonal). Almost by definition
this is Lie isomorphism when the primitive subspace [~,B] = ~ -(-I) deg~degB ~ . Under the duality between Q* = H+/H+-H +
H(R)
the result is the r ati0na! homot0py Lie algebra of
To translate this to rings and Recall that
such that
k.
almost define a graded Lie algebra. ~,(~M) ® ~
h-DGF,
H.(~M;~)
of indecomposables.
and
P,
H*(~M;~)
The diagonal in
is given the Lie bracket
P,
H*(~M;@)
is dual to the space yields a Lie comulti-
25
plication in
Q*
dual to the bracket in
To translate this to
R
P,.
we recall (§4) that
which has the natural structure of a connected F-algebra
A
we will let
I(~ p)
H*(fM;~)
corresponds to
TorR(k,~),
F-Hopf algebra. Now for any connected
denote the linear span of the elements in
Im yP
and call QFA = A+/I ; the space of
I = A+'A+ +
F-indecomposables for
~ I(~ p) pZ2
A. In particular we set
n,(R) = QF(TorR(k,k))
;
to corresponds to the space of indecomposables
Q*(H*(~M;~)).
Dualizing we get a graded subspace ~*(R) ~ EXtR(~,k ) corresponding to space
P,
of
w,(~2M) @ ~. Now
EXtR(~,~ )
w*(R)
is always a subspace of the primitive sub-
and is in fact a sub Lie algebra,
diagonally induced Lie comultiplication in have
(with bracket dual to the
w*). However, when
char k > O
we usually
7, # P, . One justification for this choice resides in the following theorem (cf. Milnor-
Moore [M-M], Andr~ [Al],Sjbdin [Sj]):
5.1
Theorem
(i)
Tile Hopf algebra
H,(~M;~)
is the universal enveloping algebra of
~,(~M) ® ~. (ii)
The Hopf algebra
EXtR(~,~ )
For this reason we call then the numbers
5.2 E ~ B
Fibration sequences. (B
w*(R)
gi = dim ~i(R)
Let
is the universal enveloping algebra of
the homotopy Lie algebra of
are called the deviations of
F -~ E
R. When
R = R
~*(R). o
R.
be the homotopy fibre of a continuous map
path connected); then the sequence
~i(B) ÷ ~i(E) ÷ ~i(F)
is exact. Trans-
ferring to loop spaces yields an exact sequence of Lie algebra maps, (5.3)
~,(~B) ÷ W,(~E) ~- W,(fF). Now suppose
residue fields (5.4)
R ~ S ~
is an
~ , and let
h-DGF, S ~ F
morphism in which
H(R), H(S)
are local with
be a homotopy fibre. Then ([Av2])
w*(R) ~k~ ~ w*(S) ~ ~*(F)
is also an exact sequence of graded Lie algebras. For topological spaces one gets the long exact homotopy sequence from (5.3) as follows. Recall the action (4.2) of
fB
on
F. This restricts to a map
fiB ~ F
which
26
turns out to be the homotopy fibre of (5.5)
B ~+-E i F
+ ~B c ~
F ÷ E. Thus we get an infinite sequence ~E -~k
R ~---*S be as above
is a free model of the aug-
is represented by
j: R ~ R<X>
and
the co-action (4.6) yields in an obvious way R<X> ~ %<X> = ~R ® ~ as the homotopy fibre of (5.7)
j. This leads to the analogue
R ~ R ~ R ~ ~R ® ~
~>~S
~ ~F
...
of (5.5) in which each morphism is the homotopy fibre of the one before. The last step of the analogy fails, however. There is a natural surjection ~(~R)
~ ~*-I(R)
which is an isomorphism if
char ~ = O; when
char ~ > 0
the ker-
nel can easily be infinite dimensional. Thus "usually" there is no analogue for (5.6), although (cf. [AVl]) there are important and useful occasions when such an analogue can be established.
REFERENCES
[A 1 ]
M. Andre, Hopf algebras with divided powers, J. Algebra 18 (1971), 19-50.
[A 2]
M. Andre, Homologie des Alg~bres Commutatives, Springer, Berlin 1974.
[A-M] M.F. Atiyah and I.G. Macdonald, Introduction to Con~nutative Algebra, AddisonWesley, 1969. [Av I] L.L. Avramov, Local algebra and rational homotopy, in Homotopie Alg~brique et Alg~bre Locale, Ast~risque 113/114 (1984), 15-43. [Av 2] L.L. Avramov, Homotopy Lie algebras for commutative rings and DG algebras, to appear. [B-G] A.K. Bousfield and V.K A.M. Gugenheim, ~D the PL De Rham Theory and rational homotopy type, Memoirs Amer. Math. Soc. 179 (1976).
[c]
L. Carroll, Through the Looking Glass and What Alice Found There, Macmillan 1871.
[F-T] Y. Felix and J.-C. Thomas, Sur l'op~ration de l'holonomie rationnelle, these proceedings.
27
[G-L]
T.H. Gulliksen and G. Levin, Homology of Local R~Dgs , Queen's papers in Pure and Applied Mathematics - No. 2_~0Queen's University, Kingston, Ontario, 1969.
[G-M]
V.K.A.M. Gugenheim and J.P. May, On the The0ry and Application of Differential Torsion products, Memoirs of the Amer. Math. Soc. 142 (1974).
[Ha]
S~ Halperin, Lectures on Minimal Models, M~n. de la Soc. Math. de France 9/10, 1983.
[Ha-Le] S. Halperin and J.-M. Lemaire, Suites inertes dans les alg~bres de Lie gradu~es, preprint. (To appear in Math. Scand.) [L-Av]
G. Levin and L.L. Avramov, Factoring out the socle of a local Gorenstein ring, J. Algebra 55 (1978), 74-83.
[M]
J.C. Moore, Alggbre homologique et homologie des espaces elassifiants, S~m~naire H. Cartan, ~cole Normale Sup6rieure, 1959-1960, Expos~ 7, Secretariat Math., Paris, 1961.
[Ma]
H. Matsumura, Commutative Algebra 2nd ed., Benjamin/Cummings, Reading, Mass. 1980.
[M-M]
J.W. Milnor and J.Co Moore, On the structure of Hopf algebras, Annals Math. 81 (1965), 211-264.
[Q1] [Q2 ]
D. Quillen, Homotopical Algebra , Lecture Notes in Math. 43 (1967), Springer Verlag. D. Quillen, On the (co)-homology of commutative rings, Proc. Symp. Pure Math. (17), Amer. Math. Soc. 1970, 65-87.
[R]
J.-E. Roos, Relations between the Poincar~-Betti series of loop spaces and of local rings, Lecture Notes in Math. 740, 285-322, Springer Verlag, Berlin 1979.
[Sj]
G. SjSdin, Hopf algebras and derivations, J. Algebra 64 (1980), 218-229.
[Su]
D. Sullivan, Infinitesimal computations in topology, Publ. Math. IHES 47 (1978), 269-331.
[~]
J. Tate, Homology of noetherian rings and local rings, Illinois J. Math. 1 (1957), 14-27.
[w]
G. Whitehead, Elements of Homotopy Theory, Springer Verlag, 1981.
L. Avramov Institute of Mathematics University of Sofia ul. "Akad. G. Bon~ev" BI. 8 1113 Sofia, Bulgaria
S, Halperin Department of Mathematics University of Toronto Toronto Canada M5S IAI
HILBERT SERIES OF FINITELY PRESENTED ALGEBRAS by David A-NICK and Clas LOFWALL
Summary
Let /
denote the collection
of all Hilbert
connected graded algebras over a field This paper addresses
series of finitely presented
k . What can we say about the s e t / ?
itself to that question.
In 1974 Govorov
conjectured that only rational power series b e l o n g e d t o ~ . was first disproved by Shearer
[Go-2]
This conjecture
[Sh] , using methods which we will generalize
and extend in this paper. We will also show that the set
/
is countable
and derive some of its properties.
Definitions
Let direct
k
denote any field. A graded vector space (over
k) is any countable
sum
A = @ A of finite-dimensional k-spaces. A graded vector space n n=0 is a (connected) graded al~ebr a (over k) if there is an isomorphism
a: A 0 for each
~
k
m
and there are associative and
n
and if
~n0 o (I @ s-l): A n @ k
pairings
~mn: A
m
~0n o (~ I 8 I): k 8 A n ~ An
~ A An
~ A
n
A
m+n
and
agree with scalar multiplication.
Viewing
the A ' s as subspaces of A , we say that a non-zero element x~A is n _ degree ~ , written homo6eneous _of Ixl = n ~ if x E A n . The elements of positive
degree are written
A + . The Hilbert
series of a graded space
A = @ A is denoted A(z) and is defined to be the formal power series n=0 n A(z) B rank(An)zn . If A and B are graded vector spaces, then A n=0 is a graded vector space where (A @ B) n = . ¢. A i @ Bj . The Hilbert series l+j=n A 8 B(z) is equal to A(z)B(z) . If A and B are graded algebras, A 8 B is a graded algebra by the rule
(a I 8 bl)(a 2 ~ b 2) = a]a 2 8 blb 2
paper we have no need for the usually the product).
If
~
is a finite subset of
independent homogeneous which is a polynomial the two-sided space and directly
ideal of
A/ from
A .
introduced
elements,
in
z . If A
~(z) A
A
consisting
denotes
is a graded algebra,
~
. In this case,
of
of linearly
span(~)(z) ~=
is also a graded algebra,
generated by
(in this
sign in the definition
Z z lwl , w~ < ~ > denotes is a graded
both of them inheriting their gradations
33
For a finite set
S ,
S . If each element of k< S >
k< S > S
denotes the free associative k-algebra on
is assigned a positive integral degree, then
becomes a graded algebra in a natural way. An algebra
a quotient of such a
k< S >
finitely 6enerated and if
by a two-sided ideal
I = < ~>
and
~
I
A
is said to be
is finite, we say
A
finitely presented (henceforth abbreviated "f.p."). Furthermore, be chosen to consist only of degree-one generators, we say ~enerated and if in addition each relation a one-two algebra. Let ~ ( k ) ~
wee
which is
A
is if
is de6ree-one
~
and
~i2
=
(over
k
will be taken as fixed, and
. Likewise
degree-one generated algebras one-two algebras
(over
}
~I
= ~-I/~k)
k ) and
k ). Lastly,
{A(z)IA~ ~ 2
A
denote the collection of all f.p. algebras
k . Except in theorem I,
be shortened simply to
can
has degree two, we call l
over
S
J/=
~12
\
(~k)
will
is the collection of all = g~(k)_.12 consist~ of all
{A(z)IA~(~}
and
~I
= {A(z)IA~ ~ I }-
are the corresponding collections of Hilbert
series.
Local rings
Let us briefly mention why it is interesting to study
~12
' In the theory
of local rings, the Yoneda Ext-algebra is defined. This is a graded algebra, but in general not finitely generated
FRo]
. But if the local ring is graded,
the subalgebra generated by the one-dimensional
elements is an object in
~2
And if moreover the cube of the maximal ideal of the ring is zero, then this subalgebra determines the whole Ext-algebra is more than an object in ~ 1 2 the Hopf-algebras
in
ILl, lEo3. Now, this subalgebra
' it is also a Hopf algebra. Let ~ 1 2
(~12 ' and let
~3
denote
denote local algebras with the cube
of the maximal ideal equal to zero. The construction above gives a bijective correspondence between map
(~!2
' ~12
~3
and
2~12 . Now Anick CAn -I]
has defined a
which transforms the Hilbert series in a certain
exponential manner (see theorem 5(a) in this paper) and in theorem 6 in this paper we define a map
(~
~12
Hilbert series. The composite map
which in a sense does not change the (~
~3
was used by Jacobsson [Ja] to
disprove a conjecture by Lemaire.
Properties of
As mentioned, we are concerned in this paper with describing the set ~7~ . It seems unlikely that there is any easy analytic way to characterize the elements of ~
, for we shall see in theorem 5 that J
rather complicated operations.
is closed under some
Of special interest, however, are the rates of
"
34
growth of the sequences analytic properties exponentially,
{rank(An)}n20
of the series
, and these rates are reflected in the
A(z)
. Such sequences generally grow
and the radii of convergence
of their Hilbert
series give us
our crudest measure of their rates of growth. A more subtle measure is in the nature of the singularity at the radius of convergence toward examining this singularity proved that
{rank(An)rn}
infinity, as
z
We first prove that that
/
is taken in JAn-2, thm 4] , where it is
cannot approach
approaches ~
r . A first step
zero and that
A(z)
goes to
r , at least as fast as a first order pole. is countable.
In our next four results~ we show
is closed under certain simple operations,
as well as certain
complicated ones ihvolving infinite products.
Theorem
I
The set ~ j ( k )
, the union taken over all fields
k , is
countable. Proof
We construct
a countable
set which maps surjectively o n t o ~
Firstly, the prime fields, Zp , p prime, and the rationals~are many countable fields. Let of
F
F
is either of the form
F[x]/(p)
where
p
F(x)
, x
is an irreducible
and all these fields are countable.
Now, again, let
F
is given by positive
transcendent polynomial.
over
Since
of
F
F , or of the form F[x~
Repeating this we find that the set
be any countable field. A finite presentation integers
n,dl,...,d n
A map u/~
~(k)
, ]til = d i . Since
over
over
F F . Hence
let
A(z)E
in the presentation
a f.p. algebra over
F ; FE~}.
is defined by taking the Hilbert
of
[J~(k)
k0
rankk(A n) = rankko(B n)
where
series of the algebra
In order to prove that this map is A
is f.p. over a field
A , only a finite number of elements in
involved, there is a subfield and
k0c k0
k
such that
is isomorphic
the Hilbert series for
series for a f.p. algebra over surjective.
F
set is countable: = {finite presentations
derived from the finite presentation. surjective,
over
:and a finite set of homogeneous
F
is countable there are countably many finite presentations
~/~
is countable,
of this standard form
in standard form of the prime fields} is countable.
elements of the graded algebra
the following
countably
be any countable field, then a simple extension
there are only countably many simple extensions
= {finite extensions
(k)
F
A = B 8ko k to a field A
k . Since k
are
where F ~ ~
B
is
. Since
is equal to the Hilbert
and hence the map v ~
~ U . ~ (k)
is
35
Theorem 2
Let
A(z)B(z)(~
is replaced Proof finite
, B(z)(~
by
~I
A = k< S >/< ~ >
sets and
S
and
CO ~ k
and
T
are disjoint.
(this follows
given below on page 6). If product
of
A
and
B
if
S
are degree-one
and
T
product space which
For homogeneous {s @ t E A - B
and y
subsets
A-B
(A'B)(z) Hadamard
Theorem
(or
~12
, TCB
3
Let
belong
to
D
has the presentation
then
of the general
E ~ A~
as to
B
Write
method
is the free
~I
generators~
A =
and
~12
'
then C, D, and E If
~
and
8
" @ An n>O
and
and defined by we also define A
B =
~ Bn n>0
(A'B) n
S°T
and
in which multiplication
by ~I
. Then
A'B ~ I
B
is a
An ~ B n
=
as are graded algebras
is defined
as for
A @ B .
" Furthermore,
(or
~ 1 2 ), then
of convergence
A = k<S>/ generators.Assign
of
,
of
if
A(z) = ~ an zn n=0
~ anbn zn n=0 A(z)
(A.B)(z)
is
B = k/< ~ > every element
is an obvious map of graded algebras
is the
" The same result holds
42
). If the radii of convergence
of degree-one
,
A(z) =
A, B ~ (~I
throughout
C:
is given by the formula
C, D, E ( ~ ] 2
A.B
t ~ T}
C = k<SUT>/
are in degree two.
spaces
are
C(z) = i(z) + B(z) - I .
). Finally,
@
S, ~, T, B
~ a z n and B(z) = ~ b z n , then n n n=O n=O = ~ a b z n . I.e., the Hilbert series of a Segre product n n n=O product of the Hilbert series.
then the radius
Proof
and
Isl = itl). If both
is a graded algebra
is replaced ~ b nz n n=0
SeA
t~T,
. If
hence
of degree-one and
is denoted by
; s~S,
~ where
y = {st,ts I s ~ S ,
series
(see ELe]
of two graded
Note that if we write
=
Let
E = k<SUT>/~U~>
are in degree two as well, then The ~
"
e.g. by an application
consist
generated
The same results hold if
t(T}_Ck<S~T>
and its Hilbert
E(Z) -I = A(Z) -] + B(Z) -I - 1 note that
~.
, B = k< T >/< ~ >
6 = {st-ts I s ~ S ,
U @>
A(z) + B ( z ) - 1 ( ~ ,
J12
Cj ~ A. @ B. for j>0 J J D = A ~ B , then D(z) = A(z)B(z)
Letting
then
or by
and
k/ then
A(z)(~
, and
k<S × T>
and
belongs B(z)
and to
are
if
~I
B(z) = ~1
rA
and
rB ,
rAr B .
, where of
S × T
S, T
are finite
sets
to degree one. There
* k<S>.k
.
86
This is an isomorphism of the form by
a ~ b,
its inverse
where
a
and
~(a ~ b) = (s,t)@(a' ~ b')
The surjections
WA: k<S>---+
@ b
where A
is defined inductively on elements are words in s~S,
and
t~T
~B: £
S
and
T
and
laI=Ib[
and
a=sa ', b=tb ' .
~ B
induces a surjection
,
k<S>.k ----+ A-B , this is an algebra map and its kernel is .k
+ k<S>. . This ideal is generated by the finite set
e.WTL) Ws.B
where
k<S>.k and
WT, W S
are homogeneous k-basis for
is freely one-generated, it followa that
B
are concentrated in degree two,
degree two and hence
A.B~ ~12
~-W T
k, k<S> . Since A'B~ ~ I
and
. Also, if
WS'B
are in
"
The remaining claims of theorem 2 are dispensed with easily. The fact that __~I (or--~12) is closed under Hadamard products is an immediate consequence of the closure of
~I
(or
~ 1 2 ) under Segre products. Lastly, it follows
from EAn-2, see lemma I and proof of thm 2] that whenever
lim{cnl/n}
exists and equals rc1 n-~oo C(z) . Application of this fact to deduce
if
~ cnzn~ ~I n=0 is the radius of convergence of
rC
C(z) =
A(z), B(z) and (A'B)(z) I/n -I B = lim(anbn )I/n = ( l i m a n )(lim bnl/n)~ = r A -I r B-I rA. n-~o n-~o n-~o
Remark
It is essential to assume that
= k<s,t> , Is l = 2 , It l = I , then {tsn 8 snt ; n~0}
A'B
allows us to
A, B E (~I " Indeed, if
A = B =
is not finitely generated, since
is a set of indecomposable elements.
To illustrate the idea that a wide range of series operations are encompassed by Hadamard products, we have the following corollary.
Theorem 4
Proof
Let
Writing
A(z)~ ~ I
A(z) =
(or
/ 1 2 ). Then
~ (n+1)z n = B(z) , where n=0
is a free c o m m u t a t i v e p o l y n o m i a l B ~ 2
•
~12).
~ a zn+1 = ~ (n+1)an zn n n=0 n=0
In view of theorem 3, we need only show that ~ 1 2 ). But
(or
~ a zn , we have n=0 n
d/dz(zA(z)) = d/dz
(or
d/dz(zA(z))& ~ I
ring
~ (n+1)z n n=0
belongs to
~I
B = k[x,y] = k<x,y>/<xy-yx>
on two v a r i a b l e s
o f d e g r e e one a n d
'
37
We now want to discuss a general method of eomputing Hilbert series of f.p. algebras, which will he used in the proof of theorem 5. The method applies also to the problem of deciding whether an algebra has a certain presentation. Let GC~]~. We want to find
G(z)
. There are two ways to attack this problem.
The first is to find a spar~ning linearly independent and hence
set
S~ G
and then prove that
G(z) = S(z)
. That
S
spans
S
is
G
is
usually proved by an easy induction. The proof of independence is however normally harder to get directly. The other way to find
G(z)
defining a graded vector space
S(z)) and a
multiplication on
S
S (with a known series
such that
Then one proves that
S
S
consists of
becomes a graded associative algebra.
has the same presentation as
G . This, again,
consists of two parts, one easy and one hard. The easy one is to find generators for
S
and a set of obvious relations, which, hopefully, constitute
all the relations in relations in
G . The hard part is to prove that there are no more
S . We now are very lucky, because we can combine the two
methods and just
do the easy parts of them and yet be done. In fact the ~coeffieJ_ent-wiseJ easy part of the first method g i v e s ~ i n e q u ~ y S(z) ~ G(z) while the easy part of the second method gives
helpful seen
when
the
second
method
as a generalization
Propositionl
Let
of the
A, B
a map of(graded)vector B 8 A
~
A 8 B
above
spaces. The map
T B @ k T
@
. The following propositions are . ..(in~roa3~sition lJ The eons%ruc~lon~0e--------"
out.
semi-tensor
and
product
for
@: B + @ A +
~
,A@B N@I k 8 B-----+ A 8 B q
is the natural embedding.
stands for multiplication):
(p
1 8ll A
1~ A
(~ J, 8 A
8 B
+
¢ DAB 1 .........:"
1@ (A 8 B) 8 (A 8 B) A 8 B
B+ 8 B+ 8 A+
~ B+ 8 A +
]
A 8 B
Then an associative multiplication on
making
be
A@k
B+ 8 A+ 8 A+
8A
A @ B
may be extended to a map
is the natural twisting isomorphism and
8B
algebras.
¢) by means of the maps
Suppose the following diagrams are commutative
A
~
Hopf
1~q *
and
of
be(graded)algebras
T
where
notion
(also called
kSA
S(z) J G(z)
is carried
¢~
A ~ B
8B
A
8B
@B
may be defined as
1 ~ A ~ A ~ B ~ B
to a(graded)algebra.
B+SA
PA 8 UV ~ A 8 B
pB ~ I B + 8 A +
¢ I 8 pB ,ASB
38
Moreover
A
and
B
may be considered
as subalgebras
of
A ® B
natural way and the resulting left A-module and right B-module A @ B
in the structures
of
are the natural ones.
Proof
The last statement
product.
The first
the product.
statement
Firstly,
follows directly from the definition
means that the composition
rows of the following two commutative
18~818181
diagrams
of the maps in the top
are equal.
~AS~B8181
> ASASBSBSA@B
181818¢81
18~81
~ ASBSASB
1818HB8181 ~ ASASBSASB
ASBSASBSASB
of
the diagrams will commute also after we have removed the
plus signs. Associativity
A@B@ASBSA@B
of the
follows if we can prove the associativity
1818¢~1 HA818181/18~AS~81 .....~ ASASASBSB 18(~81
1818~A8#= ASBSASASBSB
1818~A8181 ],-
5' ASBSASB
18,{{181
'+ A S A S B S B
~ ASB
UASUB t ~ ASASBSB UAS!J~ g ASB
181818PB/1818PB81 .....>'
ASBSASBSB
~ASp
' ASA@BSB
ASASBSBSB
PASPB t ~
ASASBSB .
Going by the lower rows of the diagrams instead, we see that the first A and the last B is u n c h a n g e d (except for the last map). Hence it is sufficient to prove that the composition of the maps of the top rows of the two diagrams b e l o w are equal. From the assumptions commutative
it follows that the diagrams
are
and since the lower rows are equal the proof is finished.
BSASBSA
> ASBSBSA
18]JB81 , ASBSA
1®¢~1
18(~ PA@I , A@A@B , ASB
1818]-1Bt
ASBSASB
'
t ~pA ~ 1 B@ASB@A
~ B@A@A®B
(~ 1
"* B@A@B
t~(~®1 ASB@ASB
UA~PB//¢
ASASBSB
1~PB>
-~ ASB@B
PA81~1 t '
/
A@B
D~ASPB/
AeA@B@B
39
It may be hard in general to check that the conditions of Proposition
I
are satisfied. The next proposition considers a case when this could be done by a machine (if the algebras involved are finitely presented).
Before stating
the theorem we introduce some conventions. If
S
and
given by
T
a
r
~
e
D(x 8 y) = xy
its image under
B
k-linear map
~: k<S> 8 k
is injective and we will identify
---+ k<S U T> k<S> 8 k
as a subspace of k<S U T>. In the same way
identified with a subspace of
k<S U T>. Also
k<S>
and
with
k 8 k<S>
k
is
are identified
with subspaees of k<S> 8 k.
Proposition 2
Suppose
S
and
T
d
%9: T x S
-+
k<S> 8 k
is a map such that im(%9)
c
1 8 k
im(~)
c
k<S> 8 1
@
span(S) 8 k
(1) or
k<S> 8 span(T)
(span(S) means the k-space spanned by @: k (as
@ k<S>
in Proposition
homogeneous let
--+
I).
elements
A = k<S>/
induces
Suppose
making also
elements
B = k/
8 k
~ c k<S> + of
: A 8 B
--+
S
and
. The
and T
to
an
having ~:
extension
associative
B c k +
imbedding
G = k<SUT>/
a unique
degree k<S>
are one)
algebra sets
of
and
@ k
--~
k<S U T>
map
U ~
@ k
(the and
a k-linear
The map
k<S>
has
S). Then
and
¢(t @ a)
5
a U B U {ts - ~(t,s); s 6 S , t 6 T} > .
is injective if and o~ly if
are contained in (2)
@ k
+
k<S> 8
If this is satisfied, then
@
for all
induces a map
to an algebra (as in Proposition
a 6 a, b 6 8, s 6 S, t 6 T .
B @ A
I), and the map
morphism.
The proof makes use of the following two lemmas.
~
--*
A @ B
making
A 8 B
above to an algebra iso-
40
Lemma
I
Suppose
satisfies map
condition
@: k<S U T>
Moreover,
and
T
are disjoint
(I) of Proposition k<S U T>
(a)
~x)
(b)
~ts)
(C)
~(XyZ) = ~ x ~ ( y ) z )
~
sets and
which satisfies
= x
if
>
k<S> 8 k
the following properties
x 6 k<S> 8 k
= ~(t,s)
is
~: T x S
2. Then there is a unique k-linear
--+
the image of
+
k
S
for
s 6 S , t 6 T for all
k<S> 8 k
in
x,y,z
, the restriction
k<S U T>
of
@
.
to
+
@ k<S>
hence
@
Proof
k<S> 8 k
satisfies the conditions
defines an associative
This is an application
im(~) E <SUT>
--*
algebra
of Bergman's
k<S> @ (k<S> ~ Span(T)), ( = the set of monomials
may be factored uniquely as Assign to
w
structure
Diamond lemma
on
SUT
) as follows.
w = O U l U 2 . . . u q , where
+ length(Ul).~
The induced partial order on
k<S> ~
k
I, and
.
[Be]. In the case
we may define a semigroup partial order on
a degree as the ordinal number length(o)
on
of Proposition
<S U T>
Any monomial o C<S>
and
on
SUT
u i 6 T<S>
.
(~ means the first infinite ordinal
+ length(u2)'~ 2 + ... + l e n g t h ( u q ) . ~ q is easily seen to be compatible
D
with
the semigroup structure and also compatible with the reduction rules defined by
~
, since the degree of
of monomials
ts
is
2.~
while
each of which has a degree of at most
There are no inclusion or overlap ambiguities so by [Be]
~(t,s)
~
among
(a) - (c) follow from
algebra im(~) E
a + ~
where
a
is finite.
{ts; (t,s) g T x S} ,
extends uniquely to a reduction rule on the entire free algebra
k<S U T> . The reduced words are obviously precisely
diagrams
is a linear combination
[Be]
of Proposition structure on
k<S> @ k
. At the same time property
I commute,
proving that
~
(c) shows that the
defines an associative
k<S> 8 k, which also follows from [Be]
k ~ (Span(S)
~ k)
may be h a n d l e d
, so properties
similarly.
. The case
41 + Lemma 2
Suppose
S, T, ~
and
@
are as in Lemmal and suppose
a c k<S>
+ and
~ ~ k
are sets of h o m o g e n e o u s elements
h a v i n g degree one) a n d let as a map
k < T > ~ k<S>
B 8 A
--*
--*
A ~ B
A = k<S>/
and
(the elements of
B = k/
k<S> ~ k . Then
~
if and only if condition
S
and
. Consider
T
@
induces a map (2) of P r o p o s i t i o n 2 is
satisfied.
Proof
The "only if" part is obvious.
only the case w h e r e
~
case is similar). Let
gives that
I
(2) is satisfied. We c o n s i d e r
satisfies the first part of condition (I) (the other p
be the natural p r o j e c t i o n
and c o n s i d e r the set l = ( x can prove that
Suppose
C k<SDT>;
k<S> ~ k < T >
--*
A ~ B
p o ¢(x) = 0} . The l e m m a follows if we
is a t w o - s i d e d ideal, since
a c I
and
S c I
then also
k 8 + ~ k<S> c I, which is a r e f o r m u l a t i o n of the claim.
First observe that
x E I
~
sx E I
and
xt E I
for
s E S
and
t E T ,
also by (2) we h a v e
BS c I , T a c I . Next we c l a i m that it is enough to prove
that
S c I . Because,
T c I
and
by induction on the length of
x C
if this is true, then it is p r o v e d that
x < a > ~ I . Indeed, suppose
x c I , then @(tx) = @ ( t ~ x < ~ > ) ) c k + k < S > < ~ >
~
~t(k
+ k<S>)) ~
.
By a similar argument it is p r o v e d that
N o w c o n s i d e r the f i l t r a t i o n of Fr<SUT> The m a p Claim:
@
@(t)k + @(tk<S>)
= (w E < S U T > ;
k<S> c I
k<S U T>
provided
S c I .
induced by
w contains at most
r
elements from
p r e s e r v e s this filtration.
I n FIk<s U T>
Proof of claim: Suppose
is a left
k < T > - module.
x C I n FIk<s U T>
and suppose
t E T . Then
S} .
42
@(x)
6 FIk<sU
since
~
T> 0 ( 8 k < T >
consists
~(x)
where Now,
a. 6 i
5
~(tx)
Now
= ~(tai)b
we a r e a b l e
to p r o v e with p o
= ~a. ~ b. 1 1
by
that
s 6 S,
The next
Fr
Suppose
b 6 k
and
short
~
for
and
ha 6 I
step
T So s u p p o s e
T<e>
is to p r o v e ~
N F r-1 c I
a i,a'i 6 < ~ > N F r-1 ¢(ta) By
= Z ¢ ( @ ( t s i ) a i)
(2)i,
p o
¢(ta")
= 0
8
,
.
claim
then
r
= 0
since
: 0 .
above
also
6 FIk<sUT>
and
for any
it is e n o u g h
btS c I
. But,
, hence
p o @(bS)
= 0
by assumption.
,
A Fr c
+ k<S>
where
I
a 6 N F r @ )
+
since,
that
f o r all
T and
+ = 0
, then @ k
+ Frk<S>
, b. 6 k < T > 1
8
•
, a! 6 k < S > i
,
p
and
= Fr
a. 6 < 5 > n F r 1
Z@(tai)b i
to z e r o b y
A F r-1 c I
bS c I
for all
t 6 T
= @(t¢(ba))=
is m a p p e d
last
+ Za~b! iI for
FIk<s>
F r k < S U T>).
= Za.b. ii . Then
+
hence,
p o ~(tx)
I . By t h e
@(ts)
k
@(ba) b~ 6 < $ > 1
and
that
b! 6 < ~ > i
= p o @(ta~).p(bl)
and
p o @()
I
~ k
F1 c 5
and hence
S c
, t 6 T
6 F r A ( 8 k < T >
and this
where
that
¢(b@(ts))
since
But
and
p(¢(ta~)b~)
¢(ba)
@(tba)
The
=
N Fr c
(we u s e
Hence
~(ta~)b~
is t o p r o v e
T as
+
b 6
= 0
step
i
= FI
~a~ 8 b! 1 1
, a~ 6 k < S > i
(2) a n d
@(bts)
@(b@(ts))
+
to p r o v e
if
8 )
homogeneous e l e m e n t s .
of
, b. 6 k < T > 1
p o @(ta i) = 0
+ k<S>
A Fr c
Z@(ta~)b~ by assumption,
b[ 6 < 8 > 1
•
I .
si,s I 6 S
. Then
and
ZqS(@(ta~)s~) if
and
r
a 6 N F r
and
ta. 6 1 1
a = Zs.a. + a's'.. + a" 1 I 1 1
a" 6 S p a n ( 5 ) . +
qb(ta")
@(ts i) = Zxjbj
Then
with
t 6 T
E k<S>
, bj
[ k
. , xj
,
43
then
@(@(tsi)a i) = Exj@(bjai)
step
p o ¢(bja i) = 0 . Also
and hence
since we have proved that
ta~ 6 1
by assumption
from which it follows that
S c I .
proof is completed by observing that
T =
and from the previous
@(ta~)s~ 6 1 , since
@(ta~) 6 ~ k + k<S> 8
@(ta~)s~ 6 1
Be
and by assumption
T N F 0 = 0
and
U T N F r . r=1
Proof of Proposition Lemma
2.
I proves the first part of the proposition.
it follows that
x = @(x)
in
Consider t h e k-linear map projection
k<S> ~ k
G
for all
p o @: k < S U T > --*
@(x) = x
. Hence in
of each other, b
and
s
~
ker(p o @)
but
~ ( p o ~(hs)) = bs = 0
G
p
is the
(2) is satisfied.
-+
~
A ~ B
(resp.
and
In
ideal
and since
are mutually
a
G
is surjective.
(2) is false.
(or there is
in
is a two-sided
induces a k-linear map
p o @(bs) # 0
~
A ~ B , where
condition
Suppose on the other hand that
inverses
Then there is t
such that
~ ( p o ~(ta)) = ta = 0 )
is not in~ective.
The last assertion
Remark
--*
it is obvious that this map and
such that
p o ~(ta) ~ 0) so
G
p o ~
ts =@(ts)
and hence
A ~ B . Suppose
the proof of Lemma 2 it is proved that in k < S U T >
x
Since
follows from Lemma
I
and
2 .
The proof is valid under the weakened assumption
consists of homogeneous condition
elements
(I) is satisfied.
if the first
(resp.
that
second)
~
(resp0
row of
B )
44
The following corollary to the proposition will be useful in the proof of theorem 6.
Corollary
Suppose
L, R
trivial multiplication ~: R + @ L +
~
A
of the zero map of a map and on R
L+
Moreover,
@ A
if
L @ A ~
and
L ~ A
L @ A @ R
L @ A @ R
@ ~ I ~A ~ A @ A-----~ A c
A
L
and
R
have
is an algebra by means
is an algebra by means
defined as zero on
R + @ (k @ A +)
as the composition
GL, G R and G A
respectively and
are algebras such that
of graded vector spaces. Then
R + ~ (L @ A) +
~
A
(L+) 2 = (R+) 2 = 0). Suppose also given a map
A + @ L +---+
R + @ (L + ~ A) +
and
(i.e.
~
L @ A ~ R .
are minimal generating sets for
has the presentation
k/ , then
L, R and
A
L @ A ~ R
has
the presentation k < G L U G A Y GR>/.
has the presentation
k/< rr'
may be restricted and lifted to a map
~ :
; r,r' E G R >. The map G R x G[j - +
k
which
defines a map :
G R x (G L U G A)
--+
k
by sending
GR x G A
t~ the map
~ . The claim of the corollary follows from the fact that the
extension
~
of
~
to zero and using
satisfies condition
~
on
G R x G L . We apply Proposition 2
(2) of Proposition 2. This again
follows by an explicit computation in a few eases.
All the operations on
~
we have discussed so far have the property that
if we start with rational Hilbert series,we end up with a rational series. In theorem 5 we discuss some operations for which this is not the case. In each of the constructions of theorem 5 , we obtain a Hilbert series which equals a rational function times a transcendental power of the original series.
infinite product, possibly times a
45
Theorem 5 and
Let
G~ ~
and write
~ = {Ul,...,UM } . Let
There exist
A, B, C ( ~
G = k/ , where
gn = rank(Gn) and if
, so that
G( ~
T = {tl,...,t N}
G(z) = n=~0g n z n 6 ~ "
there exist
D, E E ~ I
such that
oo
(a)
A(z):~
if char(k) = 2
~(z)n%11(1 - zn)-gn }~[f(z)K(1+z2n-1)g2n-I/(1-z2n) g2n n=1
where
f(z) = (I - T(z) - z-IT(z)2)-1(1
if char(k) # 2
- z - T(z)) -I
is rational-
oa
(b )
B ( ~ ) = ( I -- Z 2 )- I ( i I z )- I G ( z )2 ~
( ] + zns(z) )
n= I (e)
C(z)
= (t
-
z2)-1(1
- z)-lG(z)2"~(1 n=l
- znG(z)) -I
(d)
D(z) = (O'G)(z2)G(z)fi(1 n=O
+ gn zn+1)
(e)
E(z) = ( G ' G ) ( z 2 ) G ( z ) ~ ( I n=O
- gnzn+]) -I
\ ay Proof N'See [An-l, prop. 8.4] above. In this case and
~
A
. It is also possible to use the general method
is the enveloping algebra of a graded Lie algebra
may be seen as an abelian extension of basic Lie algebras.
(b). This is a generalization proof" 2 . Let
of Shearer's example
G' = k
be a copy of
T' = {t~ .... ,t~}
is a set of generators
i.e.,
for
It~l =]ti[
set of relations as
disjoint from but identical to
i = I ..... N . Likewise, ~ , but among the
[Sh, see "note added in
G . This means that
{t~}
ix' = {u~ ..... u~} instead of the
{t.} . The
1
algebras
G'
and
G
The desired algebra
T ,
is the same 1
are obviously isomorphic. B
has a presentation
as
B = k where
lal = Ibl = 1
and
Icl = 2
and
8 = {ae-ca, bc-aba, b 2} U
U {ati-tia , at~-t~a, eti-tic , ct[-t~c . t.t.-t.t., . . . .bt.-t.b . -
We c o m p u t e t h e k-basis A
for
spanning
a where
series
G set
with for
B(z)
1 G W and let B
is
1
j
l, J
l-
by means o f t h e g e n e r a l W'
be t h e
f o u n d by i n d u c t i o n
1
1
, I < i' < N}.
,U~7
methodf~-Le~
corresponding t o be a l l
W be a
k-basis
for
words of the form
qo p qr c w'w. baqlw, baq2w . . . . ba w. 10 11 12 ir
q1>q2>...>qr~>O
and
P'qO
are arbitrary, w'~ W' , w.lo,...,WirE W .
It is easy to see that this set has the series we look for. We now apply Proposition
2 to define an algebra
is a quotient of
B . Put
L ~ R
with the right series and which
G'
46
R = kI = lwl
where
+
i
i~1}>/ ,
L = k/
and put ~ G ~ G'
laI = I , IcI = 2 . The algebra structure
where
tensor product of algebras denote the element
on
L
(defined in "Definitions").
ZX.w.(i) J 0
in
R
where
is defined as the g EG
For
g = Z~.w. OJ
with
let
X. E k J
g(i)
, w:~ W . J
A map {w(i); w 6 W ,
--*
i_>]} × ( { a , c } U T U T ' )
k
is a finite set and the elements of t
Let
is
. From the proof of theorem 3 we have (W is a k-basis for G with I~W)
G.G °p ~ k.k/ where
G °p
(T x T) (2)
(rep.
T x T
are of degree one.
T x T
(resp. ~
%
~2)
) denote the set
elements of degree two. Let also
b
) with the
be a variable of degree one. Put
2) D=~(/
op Since G'G Op m k/< B > , there is a m a p ( ) ! G'G ~re~.~ algebras which doubles t h e ~ A spannlng set for D
D
((~)is a map of
as a k-space consists
of all elements of the form
(wj1
•w. ~?~w. bw. bwi2.., J2 lO 11
bwi
r
where w.j1,w.j2, Wio ..... Wir ~ W , lw1! = !w21 , lwill > ... > lwir -> 0 and r>_O. This is easily seen by induction. Also, this set of elements has a series which is less than or equal to the indicated series. For the second step of the "general method" define
R = k< ~/
is a minimal
indeed we have ~ (i - 2)rank(Tot ~,i(k, k )) . i=3 if and only if
B E(]L12 . ~ u s
complexity
is a measure of how much an algebra deviates from being a one-two algebra and theorem 6(b) is trivially true if Suppose now in
~I
co(B) = N > 0
whose complexity
co(B) = 0 .
and that theorem 6(b) is true for algebras
is smaller than
N . We will show that there is an
50
algebra
D~(~I
with
integer coefficients
co(D) < N
and a polynomial
~(z) i D(z) 2 coefficient-wise. an algebra
Q1(z)
with non-negative
such that
Since our inductive
C @ (~12
QI (z)B(z)
assumption
applied to
D
gives us
with D(z) ~ C(z) ~ Q(z)D(z)
for a suitable polynomial
Q(z)
, we obtain
B(z) ~ C(z) ~
(Q(z)Q1(z))B(z)
as desired. Let
B = k/< B >
Itil = i
be a minimal
and with co(B) = c o ( B )
and let
B = k/< B - {y]>
presentation
and let
~
D
of degree two and a single
Define algebras LI
is
L, R
{uij ; 1~i,j
. Since,
D is the image of
in =
y'
~l
G/~B
G
= I
= ~I and let
is closed r
53
be the r.c. of to
~ a zn( J. Consider an algebra B which is identical n=O n except that all generators (and hence all relations) have their degrees
A
A(z) :
th d . The n - - graded component
increased uniformly by a factor of be zero unless
d
divides
n , and when it does, B n ~ An/d
B(z) =
That
~
is countable
B
will
. It follows that
Z n = ~ a z dn = A(z d) d|nan/d z n= 0 n
which has radius of convergence
that
of
r
follows
I/d
immediately
from theorem
I. For density,
note
6~
tion
contains every e , where a is the smallest positive root of an equa2 d 1-elz-e2z -...-e,z = 0 with e.>0 integers. To see this choose A =
= k
with
2-P/q~ ~ dense in
~eiz i=I
T(z) =
i , then
for any positive
l-- = 1 / ( 1 - T ( z ) ) . A(z)
rational
p/q
For
T(z) =
2Pz q
we get
and these points are themselves
(0, I]
As to how Hilbert
series behave near their smallest
singularity,
we have the
following~
Theorem 9
Let
the r.c. of
Proof
rE~
C(z)
Since
Shearer's
and
r ~ ~
has r.c. equal to shown that,
• Then there is an algebra
if
= ~I
is an essential
, we may choose
r . In JAn-2, see lemma
A(z) =
algebra,
r
a zn n=0 n
as described
, then
a
CE ~2
singularity
some algebra
such that
of
r
C(z).
A~ ~I
such that
I and proof of t h e o r e m ~
> r -n n --
for each
in [Sh, see "note added"~
is
n . Let . This
it is H
H
A(z)
be has the
property that its r.c. is unity but hence the singularity a degree-one
B~(~I
Z gn zn ~ we have n=O
for each
n
I
generated algebra
By t h e o r e m 3, =
at
lim (I - z)dH(z) = ~ for any d , z÷1is essential. Using theorem 6(a) we construct
•
For
G
with this same property.
and the r.e. of B(z) =
z ~ [O,r)
Z angnZ n=O
B(z)
is also
B(z) > _
B = A.G
r . Writing
and inequalities
it follows that
Let
.
G(z) =
ang n ~ gn r
~ gn (zr-1) n , so n=O
54
substituting
X = zr
lim (r z÷r-
, we obtain
z)dB(z) > lim rd(1 ~÷I-
for any fixed the r.c. of
-I
l)d ~ gn n=0
rdlim (l - I)dG(x) ~+1-
d . Using theorem 6(b) we get an algebra C(z)
also equal to
r
and
C (~12
' with
lim (r - z)dc(z) = ~
for any
z+r
fixed
d . Thus
C(z)
has an essential
singularity at
r .
The reverse question to theorem 9, whether or not for every is an
A(z)~ J
converging
for
Izl < r
r ~ ~
and with a simple pole at
there r ,
remains open. If true, it would follow from theorem 4 that there are series with poles of any desired order at We close by mentioning set
~
one more open question about the set
~
contain any algebraic number which is not the reciprocal
algebraic
integer?1~n
particular,
motivated by the observation form
P(z)/Q(z)
Consequently Q(z)
r .
, where
, is the reciprocal
2/3
that when
Q(z)
the r.c. of
does
A(z)
A(z)
belong to ~ ?
. Does the of an
This question
is rational,
is
it always has the
has constant coefficient unity and
P, Q ~ Z [ ~
.
, which coincides with the smallest root of
of an algebraic
mean an algebra in which the sequence
integer. A r.c. of
{rank(An)}
2/3
grows like
It would be of interest to see how closely the coefficients
would
{(3/2) n} .
could approximate
such a sequence. REFERENCES [An-l]
ANICK, D., A counterexample to a conjecture of Serre, Ann. Math. 1-33. Correction: Ann. Math., 116, 1983, 661.
[An-2]
ANICK, D., The smallest 1982, 35-44.
[Be]
BERGMAN, G.M., The diamond lemma for ring theory, Advances 1978, 178-218.
[Go-l]
GOVOROV, V.E., Graded algebras, Math. Notes of the Acad. Sc. of the USSR, 12, 1972, 552-556.
[Go-2]
GOVOROV, V.E., On the dimension of ~raded al~ebras, Math. Notes of the Acad. Sc. of the USSR, 14, 1973, 678-682.
singularity of a Hilbert
1)The answer is now known to be yes (added in proof).
115, 1982
series, Math. Scand., 51, in Math., 29,
55
[Ja]
JACOBSSON, C., On the double Poincar$ series of the envelopin~ al~ebras of certain ~raded Lie al~ebras, Math. Scand. 51, 1982, 45-58.
[Le]
LEMAIRE, J.-M., Al~bres connexes et homolo~ie des es~aces de lacets, Lecture Notes in Mathematics, 422, 1974, Springer-Verlag, Berlin, Heidelberg, New York.
[L6]
L~FWALL, C., 0n the subal~ebra ~enerated by the one-dimensional elements in the Yoneda Ext-al~ebra, these proceedings.
[Ro]
ROOS, J.-E., Relations between the Poincar@-Betti series of loop spaces and of local rin~s, Lecture Notes in Mathematics, 740, 1979, 285-322, SpringerVerlag, Berlin, Heidelberg, New York.
[Sh]
SHEARER, J.B., A ~raded al~ebra with non-rational Hilbert series, Journ. of Algebra, 62, 1980, 228-231.
David ANICK Department of Mathematics Mass. Institute of Technology Cambridge, Mass. 02139
(USA)
Clas L~FWALL Department of Mathematics University of Stockholm Box 6701 s-113 85 STOCKHOLM
(SWSnSN)
ON E N D O M O R P H I S M RINGS OF C A N O N I C A L MODULES (joint work with Shiro @oto) Yoichi Aoyama Department
of M a t h e m a t i c s
Faculty of Science Ehime U n i v e r s i t y Matsuyama,
The purpose
of this note
790 Japan
is to show the main result
of the paper
[3]. A ring will always mean a commutative n o e t h e r i a n ring with unit. Le$ R be a ring, a an ideal cf R and T an R-module. injective envelope
ER(T) denotes an
of T and ~Ij(T) is the i-th local cohomology module
of T with respect to a. We denote by ^ the J a c o b s o n radical adic completion over a semi-local ring.
Q(R) denotes the total quotient ring of
R and we define dimR0 to be -~. First we recall the d e f i n i t i o n of the canonical module. Definition
1([5, D e f i n i t i o n 5.6]).
with m a x i m a l ideal n. An R-module if C ® R ~
Let R be an n - d i m e n s i o n a l
local ring
C is called the canonical module of R
HomR(H~(R),ER(R/~)).
When R is complete, module which represents
the canonical module the functor HomR(H~(~
(H~(M),ER(R/n)) ~ HomR(M,C)
(functorial)
2]~. For e l e m e n t a r y p r o p e r t i e s reader to[4,
C of R exists and is the ),ER(R/n)) , that is, HornR
f o ~ any R-module M ([5, Satz 5.
of the canonical module, we refer the
§6], [5, 5 und 6 Vortr~ge]
and [2, ~I]. If R is a h o m o m o r -
phio image of a G o r e n s t e i n ring, R has the canonical module C and it is well known that Cp is the canonical module of Rp for every p in SuppR(C) ([5, Korollar 5.25]).
On the other hand,
as was--shown by Ogoma [7,
~6],
there exists a local ring with canonical module and n o n - G o r e n s t e i n formal fibre, hence not a homomorphic general,
image of a G o r e n s t e i n ring. But,
in
the following fact holds.
Theorem 2([2, Corollary
4.3]).
Let R be a local ring with canonical
module C and let E be in SuppR(C).
Then C
is the canonical module
of R .
Let R be a rings M a finitely g e n e r a t e d R-module and t an integer. We say that M is (S t ) if depth M (M). maximal
Throughout
~ min ( t , dim M E ) for every ~ in Supp R
this note A denotes
a d-dimensional
ideal m and canonical module K, H = EndA(K)
ural map from A to H. We put U A ~ q
local ring with
and let h be the nat-
where ~ runs through all the primary
57
components
of the
zero ideal
in A such that
dim A / ~ = d. We have a n n A ( K )
= u i (cf. [2, (1.8)~. Lemma
3([7,
Lemma
4.1]
and
[3]).
If A is
($2) , t h e n d i m A / E = d for
every p in Ass(A). (Proof)
We p r o c e e d
obvious.
Let d ~ 3 and let
the zero
ideal
s and b = q s + l ~ . . . ~ q t
in S u D P A ( K ). T h e n U A
the c a n o n i c a l
module
= 0 by the
of A ~.p Since
b. S u p p o s e
that
tradiction
f r o m the exact
Proposition
on d. If d ~ 2, then the a s s e r t i o n
(0) = ql A - - - ~ q _ t
in A such that d i m A/qi
We put a = ~ l ~ . . . a q deal
by i n d u c t i o n
be a p r i m a r y
= d if and only
UA
hypothesis
= (UA) p by [2,
s < t. T h e n a + b is a ~ m - p r i m a r y
4([1],
sequence
[7, P r o p o s i t i o n
4.2] and
prime
because
(1.9)],
[31).
i-
K
is
we h a v e _P
ideal and we have
0 + A ~ A/a@A/~
of
if i s s (i s s ~ t).
. Let p be a n o n - m a x i m a l induction
is
decomposition
a con-
÷ A / a + b ÷ 0. The f o l l o w i n g
(q.e.d.) are
equivalent: a) The map h is an i s o m o r p h i s m .
b) ~ is (S2). c) A is (S2). Proof) We show
(a)~(b)
follows
(c)~(a)
and the a s s e r t i o n hypothesis
is k n o w n
and T h e o r e m
m" By L e m m a
from
by i n d u c t i o n
3, we h a v e
Assume
(S2)~.locus
6([3]).
0 ~ A + H ÷ Coker(h)
prime
÷ 0.
ideal
(q.e.d.)
that d i m A / R = d for e v e r y m i n i m a l
( p ~ S p e c ( A ) I A E is
Let R be an A - a l g e b r a
T h e n the f o l l o w i n g
Let d > 2. By the i n d u c t i o n
Ker(h) = a n n A ( K ) =--UA = 0. H e n c e we have C o k e r ( h )
Corollary
Theorem
b)----~(c) is well known.
If d ~ 2, then A is C o h e n - M a c a u l a y
([5~ 6 V o r t r a g ] ) .
sequence
p. Then the
(i. I0)] and
2, we have C o k e r ( h p ) = 0 for e v e r y
= 0 from the exact 5([3]).
[2, on d.
($2)}
with
prime
ideal
is open in Spec(A).
structure
homomorphism
f.
are e q u i v a l e n t :
(a) R ~ H as A - a l g e b r a s . (b) R s a t i s f i e s ( i ) R As
the f o l l o w i n g
conditions
(S 2) and a f i n i t e l y
(ii) For e w ~ r y m a x i m a l
generated
A-module,
ideal n of R, d i m R n = d,
(i~) d i m A C o k e r ( f ) S d - 2 and d i m A K e r ( f ) ~ d---l. (Proof)
(a)~(b):([2,
K is the c a n o n i c a l
Theorem
module
3.2])
Since
of A/U A by
[2,
H= EndA(K) =EndA/UA(K)
(1.8)],
we m a y a s s u m e
and
that
A
is u n m i x e d . (It is o b v i o u s that d i m A U A < d.) Let Ass(A) = { ~ l , . . . , ~ t ) and t S = A\i~_~l~i . Since K is t o r s i o n free ([2, (1.7)]), H is a l s o t o r s i o n free and the n a t u r a l m a p EA(A/Pi)
by
= Q(A).
Hence
[2, L e m m a
f r o m H to S-IH is i n j e c t i v e .
3.1],
we have
H is c o m m u t a t i v e .
Since
S-IH~ HomA(S'IK,s'IK)~
Since
K is a f i n i t e l y
S - I K ~ .$, l=i I i~iAp. ~ S- A
generated--~S 2) A-
58
module,
the c o n d i t ! o n
(i) is satisfied.
integral e x t e n s i o n of A contained proven by virtue of [6,
(34.6)].
sume that A is complete,
Since A !s u n i m i x e d and H is an
in Q(A), the condition
To show dimACoker(h) ~ d - 2, we may as-
Let p be a prime ideal of heght
is the canonical module of A
(ii) can be
([5, Satz 5.22]) and A
one. Then K
is Cohen-Macaulay.
P R Hence we have Coker(hp) = 0, ~hat is, dimACoker(h) ~ d - 2. (b)~--~(a):([3, TheoremS]
By the conditions, we can prove Ker(f) = U A. We
may assume that U A = 0 and f is injective because K is the c a n o n i c a l m o d ule of A/U A and H = EndA/UA(K). nonical module of R Since d i m A R / A S d -
We put L = H o m A ( R , K )_ Then L n is the ca-
for every maximal
ideal n of R by [5,--Satz 5.12]. n 2~ H o m A ( R / A , K ) = 0 and Ext~(R/A,K) = 0 by [2, (I.I0)].
Hence we have an i s o m o r p h i s m L = H o m A ( R , K ) ~ + H O m A ( A , K ) ~ K from the exact sequence 0 + A ÷ R ÷ R/A ÷ 0. From this isomorphism, gebra i s o m o r p h i s m from H to EndA(L).
we obtain an A-al-
It is obvious that E n d A ( L ) = EndR(
L). Since R is ($2) , R ~ EndR(L) by P r o p o s i t i o n 4. Hence we have R ~ H as A-algebras.
(q.e.d.)
For a r e l a t i o n between H and idea~ transforms, layness of H, we refer the reader to the paper Finally we note the following facts.
and the C o h e n - M a c a u -
[3].
They cad be proven by using
our T h e o r e m 2. Theorem 7-
Assume that Hi(A)
homomorphic
image of a G o F e n s t e i n ring.
is of finite length for i # d. Then A is a
m
Corollary 8.
If a B u c h s n a u m local ring has the canonical module,
it is a homomorphic P r o p o s i t i o n 9.
then
image of a G o r e n s t e i n ring.
If d = 2 and dim A / p = 2 for every minimal prime
then A is a homomorphic Acknowledgement.
ideal p,
image of a G o r e n s t e i n ring. The author was p a r t i a l l y
supported by G r a n t - i n -
Aid for C o - o p e r a t i v e Research.
References [I] Y. Aoyama, On the depth and the p r o j e c t i v e d i m e n s i o n of the canonical module, Japan. J. Math., 6(1980), 6 1 ~ 66. [2] Y. Aoy~ma, Some basic results on canonical modules, Univ., 23(1983), 8 5 - 94. [3] Y. A o y ~ m a and S. Goto, Preprint. [4] A. Grothendieck, Verlag, 1967.
J. Math.
Kyoto
On e n d o m o r p h i s m rings of canonical modules,
Local cohomology,
Lect. Notes in Math.
41, Springer
[5] J. Herzog, E. Kunz et al., Der k a n o n i s c h e Modul eines C o h e n - M a c a u l a y -Rings, Lect. Notes in Math. 238, Springer Verlag, 1971. [6] M. Nagata,
Local rings,
Interscience,
1962.
[7] T. Ogoma, Existence of d u a l i z i n g complexes, 24(1984), 27 - 48.
J. Math.
Kyoto Univ.,
GOLOD H O ~ M O R P H I S M S
by Luchezar L. Avramov (*)
There is a growing understanding homotopy
category
that theorems and constructions
reflect and are reflected by the homological
perties of local noetherian rings. Accordingly, have been perceived as the topological a closer inspection reveals
from the rational
and homotopical
pro-
maps which fibre as wedges of spheres
ghosts of local Golod homomorphisms.
a rather unsatisfactory
materialization
However,
on the algebraic
side of the looking-glass. Indeed,
in topology such a fibre
racterized by either its homotopy, of its loop space. According
is up to rational
to anybody's dictionary,
perties displayed by the homotopy, generally,
F
or its cohomology,
homology,
DG algebra with divided powers),
one should look for similar pro-
imposing such conditions
f: R ~ S
(2.3),
criteria
of
F . In this paper we
involving only the fibre. In
(3.4) and (4.6), one has:
be a local homomorphism,
and let
X
be a DG algebra with di-
vided powers~ which also is a free resolution of the residue field that the homology
H(F) = TorR(S,k)
on the fibre,
assumption on the map it-
self, and often this is harder to verify than the properties make the extra work unnecessary, by establishing
Theorem. Let
(or, more
which arises from a local homomorphism.
to make some additional
fact, as a particular case of theorems
ring structure
or Yoneda algebra of some ring,
The trouble comes from the fact that besides algebraists have found it necessary
equivalence uniquely cha-
or the Pontrjagin
of the fibre
F = S ~R X
k
o__ff R . Suppose
has length S # 2.
Then
the following are equivalent: (I)
TorR(S,k)
(2)
the homotopy Lie algebra
(3)
the Poincar~ P
s
(t)
has trivial Massey products of all orders
series of
~*(F) R
and PR (t)
> 2 ;
is free; S
are connected by:
=
+t - I lengthsTor~(S,k)ti+iDl i>0
As an immediate corollary one obtains: For a local ring
dimension
< I
R , the universal
if and only if
R
enveloping algebra
U
of
~2(R)
has global
is a Golod ring.
(~) During the preparation of this paper the author was a G.A. Miller Visiting Scholar at the University of Illinois (Urbana), partially supported by the National Science Foundation of the United States; and a Visiting Professor at the University of Toronto, supported by the National Science and Engineering Research Council of Canada.
60 This represents
a first step in answering ~ question of Roos
III]: he proved that if the completion through a sequence of
n
of
R
[Ro, §10, P r o b l e m
can be reached from a regular ring
surjective Golod maps,
then
gl dim U < n
[ibid, T h e o r e m
5], and has asked whether the converse holds. The first two sections needed for Golod's
contain mostly
construction;
definitions
and the yoga of Massey products,
for the reader's and the author's
messy part has not been skipped.
convenience,
Section 3 puts the Golod conditions
the
in the perspective
of the h o m o t o p y Lie Algebra theory of [Av4]. The fourth section deals with Golod h o m o morphisms remarks.
proper,
and this is followed by a last section containing
The reader should be warned
different
from that found in previous
included as Remarks
(2.5),
some miscellaneous
that our use of the "Golod lexicon" publications.
At the referee's
(4.7). and (5.4) detailed
comparisons
is somewhat
request,
I have
of the different
notions. I should like to thank Steve Halperin, interest
in this paper.
by the Universities
I.
SO~
of Illinois
(Urbana)
We shall need to manipulate
are either non-negative
case the standard
rules
n +~
"series"
integers,
that only a finite number of negative
with
and of Toronto,
Coefficientwise ~ < ~
comparison,
and
(1.1.1)
If
V~ = 0 = V'f
for
= ~ dim~Vitl
n < ~ V'
or the symbol
i
(1.1.2)
denoted
V"
acknowledged.
V'
More generally,
for the one involving (1.2)
n > 0
of "series"
and
In this
m > I, along
in the usual fashion. with the understanding
small
then the Hilbert = Hilbv,(t)
is a subfactor Hilbert
"series"
of
"series"
~ , such that
Hilbv,(t)
=
+ Hilbv,,(t), Hilbv,~v,,(t)
V", then
=
Hilbv,(t) ~ Hilbv,,(t) .
can be defined by means of length func-
ring. The formulas
above
still hold, except
tensor products.
Differential
non-negatively
for
coefficients.
n C IN.
and Hilbv,@v,,(t)
. If
restricted by the condition
~ , can also be attempted,
for graded modules over an arbitrary
algebra
provided
t , whose coeffi-
are graded vector spaces over some field
sufficiently
is defined,
=,
degrees occur with n o n - z e r o
= = + ~ = m .~ = ~ -~ = ~
for any
and
= Hilbv,(t)'Hilbv,,(t)
tions,
are gratefully
in an indeter~inate
0 . = = 0 , allow addition and m u l t i p l i c a t i o n
that
working conditions,
DEFINITIONS
(1.1) cients
Gerson Levin, and Jan-Erik Roos for their
The financial help and excellent
graded,
graded
(= DG)
skew-commutative,
algebras will be, unless with differentials
specified,
of degree
considered
-I. A graded
is said to be piecewise noetherian, if F is a noetherian ring, and for o F. is a finitely-generated F -module. A DG algebra F is augmented, if a l o surjective h o m o m o r p h i s m e to a field % is fixed, such that e d I = 0 ; we set
each
F
i
IF = Ker e . D G modules the sense ef Eilenberg
are non-negatively and Moore
graded,
and torsion products
[Mo]. In particular,
if
X
is a
are taken in
F-module,
such that
61
X#
(= X
of
F-modules
with trivial differential)
is
(i.e. the induced map
F#-free, and
H(X) -~ H(M)
X ~ M
is a quasi-isomorphism
is an isomorphism), then for any
N, H(X @F N) ~ TorF(M,N) canonically. Observe that, if (1.2.~) and only if
H(F)
is piecewise noetherian, then:
The Hilbert "series" H (F)
lilenghtHo(F)Hi(F)t i
has integer coefficients if
is an artinian ring; and
O
(I .2.2)
If
g: F -+ % •
is an augmentation, F
~
the Poincar~ series
"
PF(t) = ~ dlm~T°ri( ,~)t I has integer coefficients. (I .3) let
If
~ C X°
ex: X -~ ~
is a quasi-isomorphism of
be an element such that
augmentation zero, then
F-modules, with
gX(R) = I C ~. If
z E IZ(F)
z ]i 6 IZ(X) , hence there exists an
X#
F#-free,
is a cycle with
y C IX
such that
dy = z.
The assignment [z] ~
[y ~ I] c x ~F~
gives rise to a degree one map of
Ho(F)-modules
oF: IH(F) -~ TorF(k,k) , called the suspension. It is well-defined, and natural: for details cf. e.g. [GM, (3.6)]. (1.4.1) If there exist
h I .... ,hn
aij• 6 IF
are classes in
(I < i < j < n)
with
IH(F) , their Massey product is defined if J -dai, i = 0 , [ai, i] = hi' daij =v=1 I a.iv a vj•
_
(where
n
a = (-I) deg(a)+1
and (i,j) # (1,n)) . Then 5-a ' v = I Ivavn and all classes defined by such cycles form the Massey product (1.4.2)
Let
B = {h }
is a cycle in
IF
' c IH(F) .
be a subset (finite of infinite) of
IH(F) . It is said
e
to admit a trivial Massey operation disjoint union
=o
T/i=IB
i
to
(t.M.o.), if there exists a function
Y
from the
IF , such that
y(h e) = z e 6 IZ(F) , with n
[z e] = he;
and
dy(hel .... 'han ) = v=15-Y(h~1, .... hav) " Y(hev+1 '''''h~n ) ' (1.4.3) in
Ordinary Massey products have been generalized to operations on matrices
[Ma], to which we refer for definitions. We shall only need to know the set of all
matric Massey products in a
H (F)-submodule in
IH(F) , denoted by
MH(F) , and the
O
following result: (1.4.4)
[GM, (5.12)]. For a piecewise noetherian
MH(F) = Ker o (I .5)
F , there is equality:
F
The DG algebras
F
and
F'
are said to be (homology) equivalent, if there
exists a chain of quasi-isomorphisms F ~ G (I) ~ F (I) ~ G (2) ~
...
~ F (n) ~ G (n)
-+
F'
°
In the augmented case, one furthermore requires all these algebras to map to the
62
same field
2.
£ , and the quasl-isomorphisms
GOLOD ALGEBRAS
For an augmented
I +t Z (2.0) algebra,
F , denote by
~ .
GF(t)
its Golod
"series":
[ length= (IF)iti) -I i>0 ~o side denotes the "series" (IF) i t i + It ~ length
length
In this section,
such that
(2.1)
on
= (I - t
the right-hand
where
the identity
VIA HOMOLOGY
algebra
GF(t)
to induce
Lepta.
H (F) o
g: F ~
contains
For any such
%
(iF)iti] 2 + . . . .
denotes
a piecewise
noetherian
augmented
DG
some field.
F ,
the inequality
PF (t) < GH(F)(t) holds. Proof.
~ %) does not change the left-hand side, and can only o decrease the right-hand one, hence we can assume F local with maximal ideal m . o Completing in the m-adic topology leaves both sides unchanged, hence we shall moreover
assume
Localizing
F
at
Ker(s:
complete.
o
(2.1.1)
F
In this context
[Av4].
then it is equivalent
If
F
we have:
is an augmented
to a supplemented
DG
DG algebra
k-algebra
F'
with (i.e.
~': F' ~ ~ is the identity on %) , with F' complete and local. o o Both sides of the inequality being preserved by equivalences, supplemented, construction (2.1.2) Filtering sequence,
and replace of Eilenberg TorF(%,~)
B(F)
length Ho(F) and MacLane
by
TorF(%,%)
one obtains
EI = (BH(F)) = (siN(F) ®p) P,q P,q q
(2.1.4)
E2 = TorH(F)(%,%) P,q P,q
(2.2) upper bound
=
is reached
(For a comparison section.)
inequality
of "series"
• ~ i < ~ ( ~ dlm%Ep,q)t i>0 p+q=i
Definition.
we can assume
denoting
F
the reduced bar-
An algebra
F
in the previous with Levin's
an (Eilenberg-Moore)
spectral
, such that
(2.1.3)
PF(t)
B
= HB(F) .
to
Now the required
With
complete and local o % c F' , and o
[Ca], one has
by "the number of bars",
converging
dim~ .
F
follows
from:
I i = GH(F) (t) . ~ ( ~ dim Ep,q)t i>O p+q=i
as in (2.0)
is called
a Golod algebra,
lemma,
if
= GH(F)(t)
definition
i.e.
PF(t)
of a Golod algebra,
if the
.
cf. the end of this
63 (2.3)
Theorem.
Let
F
satisfy
the conditions
of (2.0). Then the following
are
equivalent: (I)
F
is Golod;
(2)
Ker o F = 0 ;
(3)
MH(F)
= 0 ;
(4)
H(F)
has trivial Masse y products,
hl,...,h n
(h.l 6 IH(F)),
i.e. for every
the Massey product
and every set
n ~ 2
is defined and contains
o n l y zero; (4 ~) there exists a set of generators
(5)
B = {h } a
every Nassey product
every set of elements
of
(5') some set of generators (6)(i)
(Ker g)-IH(F)
(ii) IH(F)
of
IH(F)
is defined
H (F) , such that o
over
for all
n > 0 ;
admits a trivial Massey operation;
IH(F)
over
H (F) o
admits a
t.M.o.;
= 0 ;
has an
k-basis
(iii) taking a free zero map
IH(F)
of --
{h }a6 A
F ° -module ~
V * IH(F)
V
w h i c h admits a t.M.o. Y;
w i t h basis
{va}a6 A
and the surjective
degree
(v~ * h a) , set:
d(1 @ SVal @ ... @ sv n) = and extend this map to
~ (hal,...,h) @ sv @ ... @ sv i=I i ai+1 an X = F @ T(sV) by r e q u i r i n g additivity and
= df @ v - ~ @ dv
v = SVal ~ ... @ sv~
d(f @ v) =
Fo then
d
for
is a differential
(iv) the natural
augmentation
T(sV) ;
i n the tensor algebra n
on X ;
e(f ~ v) = g(f)e(v)
induces an isomorphism
H(E): H(X) ~ ~; (6') there exists a DG (i)
X ~ ~F #
F-module
T(sV)
jective
X
with the following
V
is a free graded
properties:
F -module equipped with a suro ....... v * IH(F) , which induces an isomorphism V @F L ~ IH(F);
where
@Fo Fo-linear ms p
(ii) dX c (IF)X ;
o
(iii) the augmentation (7)
(Ker g) IH(F) = 0 , and
(7') F Note.
is equivalent If
W
W 2 = 0 , and Proof.
(I) ~
so is (2) by F
E(f @ v) = e(f)E(v)
contains
is an
F
is equivalent
to the trivial
£
induces an isomorphism
vector space,
to the trivial
extension
of
£
£ ~ W = % @ W
H(X) ~ % .
extension
by some
£ ~ IH(F) ;
~-vector space.
as vector spaces,
~
is a subring,
dW = 0 . (2). Condition [Ma, Theorem
(I) clearly
since
oF
under homology
equivalences,
1.5]. Hence we can, as in the proof of Lemma
L . Now the equality
(2.1.3). However, F this implies o
is invariant
(I) implies
that
is the map induced
is injective.
E I = E°°
(2.1), assume
in the spectral
in homology by
and
sequence
IF 9 x ~ x 6 BI,,(F) ,
64
(2) ~=~ (3) is Gugenheim (3)
~
(4). ~ s s e y
hence are always
defined,
assume any product this holds
for
Definition
1.2],
contains
and May's
products
of
of
theorem,
quoted
2 elements
and by the assumption
< n
elements
for
in (1.4.4).
are (up to sign) are trivial.
Inductively,
is defined and contains
only zero.
I J j - i J n - 2 . According
is defined,
ordinary
and the equality
to [ ~ , MH(F)
products, one can
In particular,
Lemma
= 0
1.3 and
implies
it
only zero.
(4) ~ (5). Let for an arbitrary
B = {h a}
be a set of classes
argument
to
shows
of the cycle
z
in
IH(F) , and set
y(h e) = z e
h . By induction, one assumes e n - 1IB i . Then by (1.4.1) the class y already defined on Ei= z = l y ( h I ,...,h a1'''''en ~i x y(h e ,...,h a ) belongs to < h ...,h a >, hence by the assumption it is aboundary. i+I n el' n It follows that for each sequence (el,...,e n) one can choose Y(h~1'''''he. n ) 6 IF , which bounds
choice
in its class
z~ I, • .. e ' hence one can extend the t.M.o. Y to H i=I n B l . The same 'n that (4') implies (5'), while (4') and (5') follow trivially from (4)
and (5) respectively. (5') ~ (6). Let defined,
and denote by
is a maximal
ideal
i = 1,...,m
and all
basis
of
choose
B = {h }
IM(F):
that map
be the system of generators
h 1,...,h m
~
of
those of degree zero.
Ho(F) , the fact that
a , shows
that
the restriction
Since
¥(hi)Y(h
¥
still
is a t.M.o.
d2 = 0
on
{v } . e is verified through
is
IH(F) ° = Ker(Ho(F) ~ ~)
) = dY(hi,h e)
n IH (F) = 0 . One can now cut
of
y
B
for down to a
It is now clear how to
V ~ IM(F) , and the basis
The fact that
X
makes use of the definition
(1.4.2)
the inclusion
leads
of
on which a t.M.o,
F
in
X
of a t.M.o.
to an exact
an instant
To show that
computation,
which
H(X) = ~ , note that
sequence
0 ~ F ~ X ~ X @F sV ~ 0 o where
the tensor product
is of DG modules,
with
d(sV)
= 0 . It yields
an exact
se-
quence . . . . i+j=n+1~ Hi(X) Obviously, n . However,
Bn+1 -->
®Fo Vj-I Ho(X)
working
= ~ , and
Hn(F)
~ Hn(X) ~i+j=n @ Hi(X)
3n+i(I
@ v e) = h a , hence
®Fo Vj-I . . . . . ~n
is surjective
for all
back from the sequence
0 ~ HI(X) ~ Ho(X)
@ V°
61> HI(F) ~ 0
I @ H I (F) one easily
sees that this implies
(6) = (6') is trivial. (cf.
(1.2)),
whence
H.(X) = 0 for i > I o I To deduce (I) from (6'), note that
while the assumptions
on
X
imply:
H(X @F ~) = X @F ~ = T~(sV @F %) = T~(sIH(F)) o the equality of power series.
,
TorF(%,~)
= H(X @F ~)
65
(7) ~ (7') needs no proof, while (7') implies (I) because the equality to be established is invariant under homology equivalences, and one has isomorphisms of vector spaces: Tor £ ~W(£,~) = H(B(~ ~W)) = B ( £ ~ W )
~ T(sW)
.
In order to complete the proof we show: (4) ~ (7). By the preceding, one can assume piecewise finite-dimensional Furthermore,
let
(sW) v
graded Lie algebra on on f
be the graded (sW) y
and let
s-IL(sw)V; G
= @ G n where r r=q_ n q (sL(sw)V) ®n for which
on
Gn q
h-dual of G
E(a)
sW , let
L(sW) v
be the free
denote the algebra of alternating cochains
is the set of degree
f(sv I ~... ® s v n) = s(~)f(sv (I) .... ,sv (n)), with
~ c F , and one knows IH(F) is a 2 (IH(F)) = 0 ; set W = IH(F) .
R-vector space with
v i C L(sW) v,
q-n
linear functionals
Z deg sv i = -q ,
standing for the usual ("Koszul") sign involved in a permutation of homo-
geneous symbols. Note that, by construction, on the vector space
G
is a free skew-commutative
(s~ (sw)V) v = s-1@L(sW)V) v. Furthermore,
G
R-algebra
has a differential
defined by df(SVl .... 'SVn) =i<jZ ±f(s[vi,v j], sv 1,...,svi,.^ ..,s~j,...,sv n) . (For the correct signs and a more detailed description, we refer the reader to the excellent discussion in
[Ta, Chapter I].)
Clearly, the construction of algebra, and its homology
H~(G)
G
can be performed starting from any (graded) Lie
is, by a definition (appropriately stretched for
the occasion), the (bi-) graded cohomology of L = L (sW) v, H~(G) = £,
L . In particular, for the free algebra
H~(G) ~ Der£0L(sW)V,%) ~ (sW) vv = sW , and
Hi(G), = 0
for
i > I . It follows the total homology to tion
H,(G) = @ Hn(G) is isomorphic as an algebra grp=, q £ ~W . Furthermore, this isomorphism is easlly seen to be induced by the projecG ~ £~ W , which extends the surjection
dualizing the canonical inclusion
(sL(sw)V) v ~ (s(sw)V) v= W
On the other hand dualizing the inclusion of surjection
algebra u tion
L(sW) v
into
j: T(sW) = (T(sW)V) v ~ (L(sw)V) v. Thus, a basis
system of generators G.
{u~1"''~n = s-lj(sw~| '''''swan )}
of
By the remarks above, the map which sends (n > 2)
obtained by
(sw)Vc-~L(sW) v. T(sW) v, one gets a
{w }
of
W
defines a
(~(sw)V) v, hence of the
u
C G
to
w
C W,
and
to zero, is a quasi-isomorphism. Also, a straightforward computa-
[Ta, (1.4.2)], shows the differential
d
of
G
can be expressed by the formula:
n =-~u u du~1...~ n i=i ~i...~i ai+1...~n Assuming (4), it is now easy to construct a quasi-isomorphism will establish (7). In fact, choose for each
e
a cycle
z
in
g: G ~ F , which IF
which maps to
66
w~
under
IZ(F) ~ IH(F) ~ ~ W , and set G , hence
""
~i u
~i+I "''~n
C F , so that
Y~l"''~n
[Gu],
g
u i...~ n. For any one among them, the for-
(I < i < j < n) ---
) = = dY~1
g(u 1...~n) =
(2.4). Remarks. In the l i t e r a t u r e [Go],
one can assume
{u 1.1.~mlm < n} . Choose
"''~n
extends
Y~I"
g
G 0, --
to denote a DG algebra
local ring with maximal F.
i
(7)
I am grateful
it out to me, with a somewhat different proof.
The term "Golod algebra" has been employed in the notes
context,
is a noetherian
(ii) for every
in algebra, but well known to topologists.
for pointing
of the
is a free
F
[Lev 2]
such that:
ideal
m ;
F -module of finite rank; o
dF c m F ; every set of elements of
IH(F)
admits a t.M.o, with values
in
mF.
H (F) contains a field, (2.3.5) shows our definition (2.2) is much more o general than Levin's. However, it is not for generality's sake that we have adopted
Thus,
if
the new framework.
The point is that the equicharacteristic
assumption will come in
for free for the DG algebras which arise in the main applications (ef. Sections 4 and 5), while in the same context to verify the last part of condition (2.5.2). A Golod algebra to be of the form
~
W
F
of Theorem (2.3)
it will be particularly
difficult
(iv) above.
which is a ring (i.e.
for some finite-dimensional
F = F ) has, by (2.3.7), o k-vector space. This is a very
67
particular approach
3.
instance of the "Golod rings" of local algebra: how they fit in the present
is made explicit
in (5.4).
GOLOD ALGEBRAS VIA HOMOTOPY
An algebra
F
is called a
defined on elements e.g.
x
F-algebra,
if it has a system of divided powers
of even positive degree, and satisfying
[Ca, Expos~ 7]; if moreover
F
is a DG algebra whose differential
dyi(x) = (dx)yi-1(x) , it is called a DG
Next we give a very brief account of the results of
(3.1). For any piecewise noetherian
with the Hopf diagonal, (cf.
augmented DG
P-algebra.
F-algebra structure on
[Av 4] (partly announced
F-algebra
If moreover,
HB(F) , where
I (2)
F ~ £ , TorF(~,%)
% c F , then this coincides
B(F)
for the ideal generated by
is of even positive degree and
i > 2.
Then the
in
in this section.
has the shuffle product,
and divided powers, defined in the work of Eilenberg-~cLane
[Ca]) . Write
satisfies
F-algebra.
[Av3]) , which are needed in order to go on with the exposition
has a natural structure of Hopf
{yi}i> 0 ,
the usual axioms:
12
and all
and Cartan
yi(x) , when
x
~-vector space
(ITorF(k,k)/l (2) TorF(k,k)) v is a graded Lie algebra, If F
F ~ G ~ F' and
F'
are DG
to note that
called the homotopy Lie algebra of
are quasi-~somorphisms
G
F-algebras,
then
need not be a
of DG algebras ~*(F) ~ ~*(F')
H (F) o Recall that for a graded Lie algebra
by all
F-algebra,
[a,b]
where
K
of
(cf.
L
such that
(a, b C L)
in characteristic
is denoted by
[Av3],
operator
~*(F) .
~) , and if both
in a natural way (it is important
denotes a piecewise notherian
contains a field. L , the graded Lie subalgebra, [L,L]
generated
(and called the commutator of
[L,L]
also all
ideal);
m(a) ,
L 2i+I ~ L 4i+2 , which is part of the structure
[Av 4] for details).
(3.3). Lemma. The suspension
graded
c: F ~ %
2, we include among the generators
is the quadratic
to
F-algebra).
(3.2). For the rest of this section, augmented DG
F , and denoted
(augmented
o
F
of (|.3) defines a natural degree zero map of
%-vector spaces: TF: z*(F)/[~*(F),
Moreover,
~*(F)] ~ Hom F (slH(F),~) . o
Im T F = Im(oF) v .
Proof. Compose
the degree
IH(F) ~ ITorF(£,£)
-I
isomorphism
(deg o F = I)
slH(F) ~ IH(F)
with the suspension
and follow this by the projection
F o :
68
ITorF(~,~) ~ i Tor F (~,~)/I (2) Tor F (~,~) . Since
HOmFo(,%)
= Hom~(,%)
yields a degree zero map
for vector spaces, dualization of this composition
~: ~*(F) ~ Hom F (sIH(F),%). It remains to show •
o
m [~*(F),~*(F)], which can also be wrltten as
[~*(F),~*(F)].Im o
F
Ker T m
= 0 . Since commu-
tators and images of the quadratic operator are decomposable in the universal envelope
(TorF(~,~)) v
of
~*(F) , it suffices to show (ITorF(%,~)v)2.Im o F = 0 . This F Im o in the ~-dual of the indecomposables of
is equivalent to the inclusion of the graded algebra of
TorF(~,~) v, which is canonically
TorF(~,~) . We shall show
oF[z]
identified with the primitives
is primitive for any
z C IH(F) . As at the
beginning of the proof of (2.~), one can for this purpose replace mented DG
k-algebra
F', with
,
~
H(F ) = H(F) @F Fo ' and •
Tor
F'
F
by a supple, F
(%,~) = Tor (%,~) ,
O
•
•
both equalities being provided by serles of DG algebra maps. However, w~th the IdenF' F' tification Tor (%,%) = HB(F') , and for z' C IZ,(F') , o [z'] = Ix] , where x is the element shows
x
z'
in
BI,,(F) ~ IF . The definition of the diagonal
~
of
B(F')
is primitive, whence the claim.
For the final statement, it suffices to note that since
~*(F)¢-~ (TorF(%,~)) v
is the inclusion of the Lie algebra into its universal envelope, it induces a canonical isomorphism of graded vector spaces ~*(F)/[~*(F),~*(F)] ~ (ITorF(~,~)v)/(ITorF(~,%)v) 2 . (3.4). Theorem.
For a DG
F-algebra
F , which satisfies the conditions of (3.2),
the following are equialent: (I) (2)
F
is a Golod alg.ebra;
IH(F)
is a vector space (through the augmentation
~: F ° ~ ~) , and there is
an isomorphism of graded Lie algebras ; ~: ~*(F) ~ L(slH(F)) v (= L) where
L
denotes the free Lie algebra functor; furthermore
cal map in the following commutative diasram:
~ * (F) / [ ~ * ( F ) , ~ * ( F ) ]
(sIH(F)) v
L/[L!] in which (3)
TF
(4)
~*(F)
p
is the canonical isomorphism;
is surjeetive; is a free Lie algebra.
~
induces the verti-
69
Proof. F
(I) ~ (2). It is easily seen that
to its completion
equivalence
F
in the
(2.1.1), which links
by using only homomorphisms
~*(F)
F
of DG
to a supplemented F-algebras
remarks at the beginning of this section, r-algebra
F'
over
F-algebra
(with
the homology
F', can be achieved
[Av4]). Hence, by the
can be replaced by a supplemented DG
y1(x) = 0
i > 2) , one can further replace
Furthermore,
%-algebra
(for details cf.
F
Z . Since the trivial extension
a natural way a DG and every
does not change when one passes from
Ker (F ° ~ %)-adic topology.
% ~W
(cf.
for every
F'
by
x
k~W
(2.3), Note))
is in
of even positive degree
. But now all the claims of
(2) are obvious. It is clear that (2) implies both (3) and (4). Noting that (3) is equivalent Ker o F = 0 ,
(3) implies
Now we assume
(I) by Theorem
to
(2.3)•
(4) and shall prove that
H(F)
is a Golod algebra.
Consider first
the homomorphism
f: F ~ F = H (F) of DG F-algebras, and the induced homomorphism o f*: ~*(F) ~ ~*(F) of graded Lie ~Igebras over L . Since ~I(F) is naturally isomorphic
to
(~/~2)v, where
~ = Ker(F ~ Z)
(cf.
[Av4]) ,
fl
is an isomorphism.
Subalgebras of free Lie algebras being free [Lem, Proposition A 1.10], it follows that I (F) generates a free Lie subalgebra in ~*(F) , hence I ( ~ ) generates a free subalgebra
L
of
The ring
~*(F) . ~
being equicharacteristic,
we can after localization
and completion
assume it is the homomorphic an ideal
a
there exist
minimally
image of the formal power series ring %[XI,...,Xn] by 2 generated ~y a 1,...,a r with a.l 6 n , n = (XI,..,X n) . Hence
~ • 6 Z , such that lj
ah -
_h n3 I a•.X.X. 6 i q
C , its elements
i.e. they are in the image of the map v:
( I H B ( F ) , ) v -~ E I'*~ ~
Since composed
with the isomorphism
the dual of the suspension to elements
of
free Lie subalgebra the tensor algebra
L' T
of
choose
the smallest
guarantees
r
integer
q
that
E 1'q = r drE1'qr # 0 .
r , for which
, d e f i n e d by t h e c o n d i t i o n s
are permanent
cycles
in the spectral
sequence,
El'* . E II,* ~ (IH(F),)v , v (3.3)
form a basis
By our assumption
d
(= edge homomorphism)
o F , Lemma
z*(F) , which
this Lie algebra.
inequality,
of the action of
and
for the generators
[Lem, Proposition
~*(F) , hence
on the graded vector
gives up to a degree
shows that a basis
of
of degree
A 1.10],
(TorF(%,Z)) v = Uz*(F) space associated
C
to
shift,
can be lifted ~ q +I
of
they generate D UL' , which
C . Moreover,
a is
the
71
inclusion
is an equality
in degrees
dim Hq+2(B(F))
= dim E l 'q+1 + dim T q+2 .
Let now that
denote
the bigraded
dE l ' q c D . T h i s i s r q' < q , and by the
with If
D
r = 1
and
< q +I , while
clear
subalgebra
when
definition
a E E l'q
in degree
of
q +2
it yields
E** , generated by r in this case dE l ' q
r > 2 , since -of D one has
E
= D
C . We claim ~ Ep ' ' q '
in these
dimensions.
write
da = I c~ ~ c? + b i i •
>2
wzth
b E El-- '
linearly
*
1,0
independent.
dc~ = 0 , 1
Since
-~2 = 0 ,
Comparing
filtration
degrees,
c'~ E K e r 1,q = cl,q 1 dl " With T as above,
the
formula
same degrees
of some
among the
one sees that
a
r is
the
map
) # O.
r+l
which c o n t r a d i c t s Having forms
On t h e
of
the
r+1
equality
of
EI
established
= E
other
hand,
it
guarantees
the
exis-
q +2
'
generators
of
+ dim
Tq+2
earlier.
, it follows the
immediately
k-algebra
that
E I = E~
E 1. Now we h a v e ,
since
in view of
the
(H i B ( F ) ) t i = K ( K dim EP'q)t i = (I - K d i m IH.(F)ti+1) -I i p+q=i ~ i 1 (2.2)
of a Goiod algebra.
(3.5).
Remark.
Theorem
Theorem
(1.4) and Corollary
the results
(3.4)
(and most of the arguments)
is the basic assumption
(see below)
dc'~ = 0 , i.e. i
H.(F): I
definition
F-algebra
hence
s,t El,q+1 K dim Er+ I < dim ~ s+t=q+2 s>2
particular:
DG
shows
that
= dim E I'q+I~ +
that
a system
K dim i
context,
B(F)
d (a) # 0 , which produces a relation of degree r < q + I of E*'* hence in degree q +2
This means
the
proved
g-finiteness
is
of
ci
T ~ E*'* , defined by the inclusion r+1 E*'*r = E*'*~ in total degree _< q + I , and
shows that
a E E 1'q with r of degree
dim H q + 2 B ( F )
which
differential
,
the
a:
an isomorphism.
generators
(T ~ E
El'*
the
C'. @dc': = 0, i i
--
Ker
for
choosing
= db C ~ 3 , *
consider
C c Rr+ I . The choice of
tence
v
= (n) , and furthermore
hence
c! @ dc'~ = d(da +b) i i
in the
--
, c~ E E~ 'q , e~ E E,
resolution
of
looks
[Av2,
(and is) similar (1.6)].
of the previous (1.3)]
F ° . Indeed,
that we want to deduce
to results
The crucial
that
F
[Av2;
in
which makes
paper unusable
in the present
is a subalgebra
of the miniraal
it is precisely
from the properties
in
difference,
of
this condition TorR(s,K)
•
on
F = S~R X
72
4.
LOCAL HOMOMORPHISMS
In this section
f: (R,m,k) ~ (S,n,~)
denotes a homomorphism of local (noetherian
and commutative) rings, such that
f(m) c n . Furthermore,
F-algebra over
has finite
R , such that
by a natural augmentation The DG
F-algebra
Xi
e(= gX ) , commuting with that of
F = S ®R X , augmented to
the (homotopy) fibre of
X
denotes a free DG
R-rank for any
%
by
f , and the canonical inclusion
i, and
H,(X) ~ k
R(~ R/m = k) .
(S ~ S/~ = %) @ s , is called S ~ S ®R X
is denoted by
g .
The fundamental importance of the fibre in the study of the homology of the map f
is given by the next result proved in [Av4]: (4.1) There is a natural exact sequence of graded Lie algebras over (4.1.1)
~ @k~*(R)
f+---*~*(S)
g+-~*~*(F) +~ C(Coker f*) - - 0
where for any piecewise finite-dimensional graded vector space with
F~
denoting the free
Moreover,
Im 6
F-algebra of
is central in
Wv
over
W , C_*(W) = ~*(F%wv),
~ .
~*(F) .
If furthermore the flat dimension in odd degrees, its dimension is
~:
fdRS
is finite, then Coker f*
j fdRS + edim (S/mS) , and
(Here and below we use the notation
is concentrated
~i(Coker f) ~ Coker(fi-1).
edim R = dimk(m/m2)) .
(4.2). Lemma. For any local homomorphism, there is an inequality of power "series": Ps(t) < PR(t)'C~(t) where
G~(t) = (I - I lengths(ITor~(S,k))ti+1)-1
Proof. Setting
ei( ) = dim i (
)
is the Golod "series" of
a. = dim (Coker fl)
H(F) .
b. = dim Cl(Coker f*)
the exact sequence (4.1) yields (4.2.1)
e.(S) + a. + b. = e.(R) + e.(F) . I l i I l
Recall that for the universal envelope of a graded Lie algebra
L , the
PBW
theorem
gives the equality of formal power series
dim (UL)it i =
dim L I 3 dim L 3 (I +t) (I + t ) ... t2) dim L2 4 dim L 4 (I -
Write
A(t)
)
. . .
for the Hilbert series of the free
for that of the universal envelope of (4.2.2).
(I - t
Ps(t) < Ps(t)A(t)B(t)
F-algebra on Coker f*, and
C(Coker f*) . One now has: because
A(t) ~ 0,
= PR(t)PF(t)
by (4.2.1);
< PR(f)G~(t)
by (2.1) .
B(t) ~ 0 ;
B(t)
73
It should be emphasized quasi-isomorphic is uniquely property
DG
defined
to
(4.3). Definition. is a Golod algebra
f , i.e.
if
homomorphisms.
ITorR(S,k)
is necessarily
terion of flatness,
products
Golod.
is relegated
TorR(S,k)
~n(F)
is
Ker g* # 0 ;
(b)
Coker f* # 0 ;
(c)
Coker fl # 0 .
(4.4.2) Moreover,
hence
. Moreover,
for
become
0 # Ker (f1:
£ @k m / m 2
sequence
= I . This means
to
(trivially)
of
Tor I
~.
trivial
Tor-s,
implies,
hence
i = 0 , or
i = I.
is a quasi-isomorphism, ~*(S/mS)
I (these are very I
by the local cri-
in (4.4.1)
S/mS ~ ~ ~ % , hence
is the free
special
~ ~ ~ 72 . According
cases of (3.4)).
to (4.1) the
in the easily checked
form:
~ ~/22) "
I J dim£Ker
from
one-dimensional
sequence
are
the last inequality
fl = dim%Coker
hold throughout. (4.1)
Asstmle both conditions
(abelian)
is an exact
F
equivalent:
= I , hence equalities
(4.4.3).
if
products).
i and isomorphic
F = S @RX ~ S/mS
n # I , 2 , with
note that in this case
dim%~2(F)
the next
F .
it presents some deviations from theusualpat-
of all higher
R-flat,
£ , one can express
= 0 , the homotopy
denote
= 0
conditions
(a)
over
F
look at this simple situation.
Lie algebra on a single generator of degree
Dualizing
of
to (4.7) below.)
lengths(ITorR(s,%))
that since the vanishing
~*(F) ~ ~*(S/mS)
following
produce
. In view of this,
has trivial Massey
in degree
of
However,
the triviality
i = 0 , S
Accordingly,
f
structure
is called a Golod homomorphism
Suppose
is concentrated
First of all note,
so that
the homology
i , such that
tern, hence we have to give a closer
When
f
TorR(S,k)
with earlier definitions
In this case the Massey such a map
= TorR(S,k))
A local homomorphism
(4.4). Exceptional
l
H(F)
in particular
it does not depend on the choice of
(equivalently:
(A comparison
(4.4)..
so that
(note also that
is intrinsic
there exists an
that any two constructions of a fibre for
F-algebras,
reduces
(4.4) and
Lie algebras,
fl = dim~C2(Coker
C1(coker f*)
Since by construction
to the following (4.4.2)
hold,
concentrated
f,)
... > b s. Then S
S-I
Proof by induction on (If S ~ 2 then ^(rest = ^(rest
S and by the modular
law (34).
of factors) ^ (aS_iVb s) ^ a s =
of factors) a (a6_ I V (bsAas))
= ( ^(rest of f a c t o r s ) ^ a s _ I )
v (%Ab s)
.
)
C0ROLLARYI - Assume t h a t t > 1, t h a t a 1 ..... a2t,b 1 ..... b2t E L and t h a t a I b2t • Then
(39) (yl(a2U_IAb2u--1))A
(a2uAb2u)
=
= V (auAbu+I)
=
U= 1
For, ( ~ (a2u_IAb2u_I)) A ( ~ (a2uAb2~)) = t-1 t-] = blA(u61= ( a 2 u - l V b 2 u + l ) ) A a 2 t - l A b 2 A ( A ( a 2 u V b 2 u + 2 ) ) A a 2 t
=
12t-2 = b2Atug~= I ( a u v b ~ + 2 ) ) A a 2 t - 1
=
~(a~Abu+1)
•
5, PROOF OF (33), By (30) and by the shearing property
(35),
,__m_l ),i ,f h,d,e +tF ,_m+l,f,d,e~ (IPm)~'d'eN (pmI)~'d'en (tiF )i,h ' = ,h+l
= (zPm)fi'd'en(Fmz)f'd'en
"~*
'i,h+l
~
~
~'i,h
-
Fp--->Fp_ i--->...--->F0--->B--> 0 , where each F
is generated (as a graded K-module) by elements of degrees < P < pq0(2,B). Tensoring with k and taking homology, we get
Cp,q
0
for q>pq0(2,B)
,
whence by (66) rate(B) < q0(2,B) . Thus, by Anick's argument rate(R) < q0(2,A)
(and actually rate(R) O
is a rational function of Z.
More ~recisely, i_~f(3) holds, then there is a finite set of algebra ~enerator { {~i}in1.=
of (2) with bidegree(~i) =(mi,n i) (contrary to the convention in [26] we let both
102
degrees take non-negative values) and n m. n. PR(Z) = H'(I-(-I) iZ l)/po](Z) i=I 2
where
2
.
means that we take the product over those i:s such that m.+n. Is even,char k#2,
and where pol(Z) is a ~olynomial in Z.[ In particular, algebra (2) are of bidegree
if the generators of the
(1,1), it follows that PR(Z) = (1+z)n/pol(Z).]
(ii) EXtR(k,k) is finitely ~enerated as an al~ebra. (iii) When we present~ usin~ (ii), EXtR(k,k) as a ~uotient of a finitely ~enerated free ~raded alsebra, it is also true that the (twosided) kernel ideal is ~enerated (as a twosided ideal) by a finite set of ~enerators.
[We say that ExtR(k,k)
is a
finitely presentad algebra.] (iv) Furthermore, there are only a finite number of relations between the relations in (iii) etc. The
first example where (ii) is false is given in [24], p.314 , using ideas from
J.-M. Lemaire's thesis [15]. Using similar ideas, one can construct examples (R,~), where (ii) is true, but (iii) is violated etc. Later Anick [I] gave many examples (R,m), where PR(Z) is non rational, thus violating
(i).
Thus there are many examples where (3) is not satisfied. However, in [23], one of us proved that if (R,m) is a Golod ring (examples: R = ~/~s
, s~1, where (R,~) is
a regular local noetherian ring, or R = R/x.a , where x6~ and ~ is a proper ideal of R...), then (3) holds, and so does even the following more precise assertion [(3) follows from (4), if we take M = R]: If M is a finitely senerated R-module, then the (left) Ext*
-(~)
(k,k)-module
Ext~(k,k)
(*)
Ext*
(Ext~(M,k),k)
Ext~(k,k) is n o e t h e r i a n . Note that here the Yoneda product Ext~(k,k) ~ Ext~(M,k)
> Ext~+J(M,k)
gives Ext~(M,k) the structure of a left Ext~(k,k)-module, (~) becomes a left Ext* (k,k)-module. Ext~(k,k)
and that in a similar way
Using a reasoning, similar to that in
[23], it follows that if R comes from a local complete intersection S by a Golod map S
> R, then (4) and (~ fortiori)
this, cf.
§ 2
intersection,
(3) are true. For a complete proof of
below. In particular,
(3) and (4) are true if R is a local complete
a fact which can also be deduced from [12].
In [6],one of us proved among other things, that if k is a field and R is the quotient
103
of a formal power series ring: R = k[[X1, .... Xn]]/(M I .... ,Mr) , where the Mi:s are monomials in the X.:s, then PR(Z) is rational of the form (1+z)n/pol(Z). It was 0 therefore very natural to ask whether the stronger statement (3) could hold for these rings with monomial relations. This is indeed the case, and that is one of the main results of the present paper (Theorem 5 of § 3). The proof is a combination of the ideas from our papers [6] and [23]. It follows from the theory of § i~ below, combined with results of C. Jacobsson [14], that the stronger statement hold in general for these rings with monomial relations.
(4) can not
The fact that (3) does hold
for these rings has already been used by Anick [2] in his study of the homology of certain loop spaces. Indeed he proves in [2] that H,(O~X,Q) is finitely presented if X is a generalized fat wedge. The paper by us, cited as [4] in [2], has been incorporated in the present paper. We study commutative and not anti-commutative R:s here, but the methods are the same. The plan of the present paper is as follows: In § I we deduce general consequences of the validity of (3) and (4) for a local ring. In § 2 and § 3 we prove the main theorems, using general results about the (co)homology spectral sequences of extension~ of Hopf algebras. Finally, in § 4 some open problems are mentioned.
§ I. THE DOUBLE Ext-ALGEBRA AND THE DOUBLE Ext-MODULES.
THEOREM I.- Let (R,m) be a local, commutative noetherian ring such that Ext*
(k,k) is a (bigraded) noetherian ring. Then:
Ext~ (k ,k ) (a) EXt*xt~(k,k)(k'k)E has a finite set of algebra generators
{~i)i=~n with
bidegree(~i )= (mi'ni) ' and we have the following formula for the corresponding double Hilbert series : (5)
~ dia_(ExtP, q ~ \ (k,k)).Xpyq = p, q~0 Ext R(k ,k )
PoI(X,Y) H ~ ( I-xmiy ni ) 10
Pol (- I ,~)
PROOF: We will need the general observation
"
[15], Chapitre I, that if A is a graded
connected algebra over k, then a minimal set of graded generators of A corresponds to a basis of the graded vector space Tor~(k,k) = I(A)/I(A) 2 , where I(A) is the augmentation
ideal of A. In the same way, a minimal set of graded relations for A A corresponds to a basis of the graded vector space Tor2(k,k) , etc. Furthermore, i the graded vector spaces EXtA(k,k) are dual to the Tor~(k,k). We will use this below for A = Ext~(k,k). But let us first recall that we have supposed that the bigraded algebra B = Ext~(k,k) is noetherian.
~us
I(B)/I(B) 2 is finite-dimensional
and
therefore B has a finite set of generators{~.} n as in (a) of Theorem I. Since B i i=I is the Ext-algebra of the Hopf algebra A, it follows from [26], Th@or~me I, p.15-13, m-m-+n.n. that ~$~j = (-I) I $ i ~ j ~ i " We will now use the following bigraded variant of Theorem 11.1 in [3]: LEMMA.- Let B be a bigraded connected algebra over a field k. Suppose that B ha___~s a finite set of algebra generators (*)
{~i}1 Ext
C
, it follows that Ext~(V,k) is also a
In the spectral sequence (15) (or (13)...) the E2-term
is also a left Ext~(k,k)-module by means of the Yoneda product.
How is the spectral sequence (15) (or (]3)...) related to these Ext~(k,k)-module structures? Here is the answer (Ming [21] Theorem 2.7, p. 238): There are structure maps: Ext~(k,k) @ EP'qr
@ r
>
EP+S'qr
108
such that: i)
02
is exactly the Yoneda product Ext~(k,k) 8 Ext~(Tor~(k,V)~k) .....
> Ext~+S(Tor~(k,V),k)
ii) The differential dr of (15) is a left graded Ext~(k,k)-module homomorphism, and @r+1 is induced from Qr by passing to cohomology. iii) Let { FPExt~(V,k) }p~O
be the decreasing filtration of Ext~(V,k), corresponding
to the spectral sequence (15), and let pp be the natural quotient map defined by: 0p 0 > FP+IExt~(V,k) > FPExt~(V,k) > Ep'n-p~ > O With these notations the following diagram is commutative: yFP+SExt~+S(v,k
Ext~(k,k) ~ FPExt~(V,k)
i inclusion
Id 8 inclusion
.> E~+s,n-P
Extc(k,k) ~ Ep'n-p
Here Y" is induced by the Yoneda product and the natural algebra map
j*
Ext~(k,k)
> Ext~(k,k) , so that the following diagram is commutative: .8
J ~ Id Ext~(k,k) ~ E x t ~ ( V , k ) - - ~
Yoneda > Ext~+S(v,k)
Ext~(k,k) ~ Ext~(V,k)
Id ~ inclusion
inclusion
y" Ext~(k,k) ~ FPExt~(V,k)
>
FP+S~ _n+s~ ~ ~x~ B Iv,K)
Summing up, one can say that the spectral sequence (15) is compatible with all the *
left Extc(k,k)-module structures in sight. We will use this in some special cases: THEOREM 3.- Let k
>A
>B
>C
>k
be an extensi0n of cocommutative 6raded connected Hopf al~ebras such that A is is a free al~ebra (i.e. gldim A = I). Then, for each ~raded left B-module V, we have an exact sequence of left Ext~(k,k)-modules: B * .-I(TorB(c,V),k ) --> .. (16)..--> Ext .-2 c (Tor](C,V),k) --> Extc(C@BV,k) --> Ext~(V,k) --> Ext c . PROOF: The isomorphism (14) gives that the Ep'q of (13) (or (15)) are zero for q > 1 2 if gldim A = I. Therefore, in this case the spectral sequence (13) degenerates into a long exact sequence (16). The assertions about the left Ext$(k,k)-module structure are just reformulations of Ming's results, quoted above, in this special case.
109
Remark.- Theorem 3 can be applied to the Hopf algebra extension a Golod map R
> S. In this case A =
graded vector space { EXtR-l(s,k)
(I I ), coming from
T = the free associative graded algebra on the
)i>2 [ This graded vector space will be henceforth
be denoted by s-l~TR(S,k ) , i.e. the "suspension" of the elements of degree > 0 in ExtR(S,k).] , *
R = Ext~(k,k) and C = EXtR(k,k).Furthermore , for V = k, the isomorphism
(14) becomes: (17)
TorIB(C,k) --~ TorA(k,k) ~ s - 1 ~ ( S , k )
and here the left C = ExtR(k~k)-module
structure on Tor (C,k) corresponds to the
left EXtR(k,k)-modul e structure on s - 1 ~
(S,k), defined by Yoneda product
[18].
Therefore we have: COROLLARY.- Let R left Ext* E xtR(k,k) *
q) > S be a Golod map. Then we have a lon~ exact sequence of (k ,k)-modules :
> Ext*-2 (s-1~(S,k),k) Ext~(k,k)
> Ext* (k~k) Ext~(k,k)
)
(~8)
Ext* * Exts(k,k
where map
a 1 1
(k,k) - - >
t h e module s t r u c t u r e s
Ext *-I (s -l-----* EXtR(S,k),k ) - - > Ext~(k,k)
...
a r e d e f i n e d by Yoneda p r o d u c t s and by t h e a l ~ e b r n
Here is another application of the spectral sequence
(13) (or (15))and the Ming theory:
THEOREM 4.- Let (19)
k
> A
> B
~ C
> k
be an extension of cocommutative graded connected Hopf algebras such that gldim A =N N. From the Ming theory (i)-(iii) above, it now follows that for each t (0 < t < N), the
( Fl-tExtB (V'k) }i>O (put F s = F ° if s < O) are sub-Extc(k,k)-
modules of Ext~(V,k). Denote these submodules by F*-tExt~(V,k). F*-tExt~(V,k)/F*-(t-1)Ext~(V,k)
= E~ -t't
for 0 S be a
Golod map. Let V be a left finitely presented graded Ext~(k,k)-module. TTlen Ext* (V,k) is a noetherian left Ext* (k,k)-module, via the natural Ext~(k,k) Ext~(k,k) j** ring map Ext* (k,k) > Ext* (k,k) and the Yoneda product.
Ext~(k,k)
Ext~(k,~)
PROOF: We have an exact sequence of Hopf algebras:
(2o)
k
> T
> Ext~(k,k)
> ExtR(k,k)
> k
where T is a free algebra. Apply Theorem 3 to this sequence (20)! Using at some places the short notation B =
Ext~(k,k)and
C = Ext~(k,k) we obtain (a part of (16)) the
following exact sequence of Ext* (k,k)-modules: Ext~(k ,k) (21) ..--> Ext* (C~BV,k) --> Ext* (V,k) --> Ext *-I (Tor~(C,V),k) -->. Ext~ (k,k) Ext ~(k,k ) Ext~(k ,k ) Now V is a finitely presented left B-module and therefore we have an exact sequence (22)
0
> W
> F
----> V
- - >
0
of left B-modules, where F is a finitely generated and free B-module and where W is a finitely generated B-module. Tensor (22) with C! We obtain an exact sequence of left C-modules: (23)
0
> Tor~(C,V)
> CSBW
> CSBF
> C@BV
> 0
Here CSBF is a finitely generated free C-module. Therefore V ° = CSBV is also a finitely generated C-module. I claim that V I = Tor~(C,V) is also a finitely generated C-module. Since W is a finitely generated B-module, it follows as before that CSBW is a finitely generated C-module. But since C = Ext~(k,k) is noetherian (R is a local complete intersection) we have that C@BW is noetherian, and therefore its submodule Tor~(C,V) (use(23)!) is finitely generated.
Applying Corollary I to the
finitely generated EXtR(k,k)-modules V ° and VI, we obtain that EXt:xt~(k,k)(Vi,k) * (i=0,I) are both noetherian Ext* (k,k)-modules, and therefore the middle term Ext ~(k ,k ) of (21) is also so, and the Corollary 2 is proved. COROLLARY 3.- Using the notations and hypotheses of Corollary 2, we have that Ext~(k,k] is a (graded) coherent al@e~or_aa. (Left coherence and right coherence are equivalent since Exts(k,k) is a Hopf algebra.) Furthermore, for each finitely 6enerated S-module M, we have that the left Ext~(k,k)-module Ext S* (M ,k ) is coherent. PROOF: Put B = Exts(k,k). Recall that we proved in Corollary 2, that if V was a finitely presented left B-module [ i.e. if ,limk(TOr~(k,V)) < ~, 0~i~I, or , equivalently, if the dual vector spaces Ext~(V,k) had finite dimension for 02
[25]. The last part of Corollary 3 now follows
from Theorem I of [23], and therefore Corollary 3 is completely proved. Remark.- Taking M = S in Corollary 3, we obtain that k is a coherent presented)
Ext~(k,k)-module.
(thus finitely
Therefore Corollary 2 for V = k shows in particular
that Ext* (k,k) is a finitely generated Ext~(k,k)
(bigraded) algebra. This will be
applied in: COROLLARY 4.- Let S be a local ring that comes from a local complete intersection by a Golod map, and let S
~P > S"
be a second Golod map. Then Ext*
..... a noetherian
(k,k) i_~s
Ext~.(k,k)
(bigraded) algebra.
PROOF: Apply the middle part of the exact sequence
(18) of the Corollary of Theorem 3 .
to the Golod map S
> S ". We obtain an exact sequence of Ext
(k~k)-modules Ex't~(k,k)
(24)-,--> Ext*
Ext~(k,k)
(k,k)
~** > Ext*
(k,k) - - >
Ext~.(k,k)
Ext *-I
(s -I=-7,* EXts(S ,k), k )-- > .
Ext~(k,k)
Now according to Corollary 3 and the Remark following it, both Ext~(S'~k) and k are coherent Ext~(k,k)-modules.
Thus s - 1 ~ ( S ~ , k ) "
is also a coherent Ext~ (k ,k )-module,and
therefore Corollary 2 implies that Ext* (s-IE--x~(S~,k),k) is a finitely Ext~(k,k) generated Ext*
(k,k)-module.
This last ring is noetherian
(Remark following
Ext~(k,k) Corollary 3) and therefore Ext* (k,k) sits between two noetherian modules in Ext~(k,k) the exact sequence (24). Thus Ext* (k,k) is a noetherian Ext* (k,k) Ext~.(k~k) Ext~(k,k) module, and ~ fortiori Corollary 4 is proved. PROOF OF THEOREM 2: This is now immediate:
a) and c) follow from Corollary 2 and
Corollary 3 of Theorem 4 and b) Zollows from Corollary 4 of the same Theorem 4. § 3. RINGS WITH MONOMIAL RELATIONS. THEOREM 5.- Let k be a field, k[X],...,X n] the (commutative) polynomial rin 6 in n variables, let MI,...,M r be monomials in the Xi:s , and let R = k[XI,...,Xn]/(MI,...,M r) Then Ext* (k,k) is a (bigraded commutative) noetherian ring. Ext~(k,k) Remark I.-
The R of Theorem 5 is not local in general, but it has the same Ext-
algebra as the corresponding local ring k[[XI,...,Xn]]/(MI,...,M~) , and so the preceding theory can be applied to R and related rings.
113
Remark 2.- Since each variable X. defines a grading on R, it follows that R is 1 n-graded and that Ext* (k,k) is (n+2)-graded. In the course of the proof Ext~(k,k) of Theorem 5 we will obtain a more precise result about how the finite set of generators of Ext*
(k,k) can be chosen.
EXtR(k,k) PROOF OF THEOREM 5: Consider first the case where all the M.:s are squarefree. Fix a I t, I < t < n, and consider S t = R/(X t) = k[X I ~''" ,Xt,...,Xn]/(those
M.:s, where X t does not occur) j
(" means that the corresponding variable is omitted). For those Mi:s , where X t does occur, we write M i = XtM E (note that there is no X t in M E , since all Ms:S are squarefree).
Let --at be the ideal in St, generated by the images of these M E. Then
R = St[Xt]/a_tXtSt[Xt].
Of course we can suppose that all Mi:s have degree ~ 2, and
then ~t is generated by elements of degree ~ ]. Writing for simplicity S = St, ~ = ~t and X = Xt, we therefore have a Golod map: (z5)
six]
six]
> ~.x.s[x]
= R
(of course (25) is also a Golod map for more general S:s and ~:s). We now apply the Corollary of Theorem 3 to the Golod map (25)~ and we obtain from (18) the following long exact sequence of Ext*
(k,k)-modules:
Exts[x](k,k)
.... Ext
q0** (k,k) -->Ext* (k,k) -->Ext *-I (s-IE-~-* (R,k) ,k) EXts[x] (k,k) EXtR(k ,k) EXts[x] (k,k) S[X]
(26)
Ext *+I
(k,k)
>..-
~Ex~[X] (k ,k ) But EXts[x]* (k,k) = Exts(k,k) variable T of degree I
(27)
(T 2 = 0). rl~erefore
Ext* *
(k,k) = Ext*
EXts[x] (k,k) and ExtE(T)(k,k) bidegree
~kE(T), where E(T) is the exterior algebra on one
(k,k) ik
Exts(k,k)
*
(k,k)
ExtE(T)
= k[V], i.e. the commutative polynomial ring on one variable V of
(1,1). Thus (26) is an exact sequence of graded modules over
EXt~xts(k,k)(k,k)
~kk[V]. In particular
(26) inherits a grading from X and the
operations of V are compatible with this grading.We claim that V operates on the four modules surrounding Ext* (k,k) in (26)~ as it does on (27), i.e. Ext R (k ,k )
114
I) multiplication 2) each element of X-degree
U > I is a multiple
by V of an element
of the module
U-I.
Of course this
of (27)
by V is a monomorphism, of X-degree
is clear for Ext* (k,k) and Ext *+I (k,k), Ext~rx1(k,k) * (k,k) bL ] EXts[x]
in view
(we may even take U > I in (27)). But, since
EXts[x]~x._--~.S[xI,k)
~ s
Exts[x]tX-~-S[X],k)
this is also true for Ext *-I ,
~ s -1 Ext~(~,k)
8kHOmk[x](X'k[X],k)
-* (s - I -Exts[x](R,k),k) , and Ext*-2(s -I
,
...
Exts[x](k,k) Now it follows v = ~**(V),
easily,
using
I) and 2) and the exact
then every element
~ of X-degree
form ~ = v.~ ~ ~ where ~" has X-degree Returning
to the old notations
u-1. Furthermore
result
(n+2)-multihomogeneous
{ 6 Ext* *(k,k) ~xtR( k ,k )
products
of v.:s
I is at the verify
[ each v. has
( + 2 ) nd place
result
(n+2)-multidegree
] with elements
nonzero
algebra
is unique.(Use
the 5-1emma!)
the notation
is a linear combination
(1,1,0..,I,..,,0),
space,
of the bar resolution
of
where the last (al,...,a n)
spanned by these last elements
is an easy consequence
of the classical
- that if B is a non negatively
over k, then the graded vector
elements
~
whose last n ~mltide@rees
This last assertion
- proved by means
connected
if we put
for all t (I < t < n) we obtain that each
0 < a. < I (I < i < n). But the linear
is finite-dimensional.
(26), that
X t = X, S t = S and ~t = a, introducing
v t = v, and using the previous element
sequence
u > I in Ext* (k,k) is of the Ext~(k,k)
of degree < p. Thus Ext* EXtR(k,k)
graded,
spaces TorB(k,k) can contain no P (k,k) has a finite number of
generators,
if the M.:s are squarefree. Note that if one of the variables X.:s J J in all M.:s, then we do not need any extra elements above ~ith last n
absent
is
1
multidegrees
(al,...,a.j_1,1,aj+ I , . . . .,an)
We now pass to the general Xi:s
(we assume,
of course,
there are still canonical of multidegree
1
is at the
independently
due to FrSberg
(i+2) nd place
and Weyman
to a ring with lower m I . We can assume
iI that M I = X I M~ . . . . .
X I does not occur in Mk+1, =
-the last
(corresponding
to X i) (~ust use
Let m. = the maximal exponent of X. among the M.:s. l i j I, then we are in the preceding situation. Assume therefore m I > I.
reduce ourselves
R"
v. 6 Extl (k,k) l Ext~ (k ,k )
of Extl).
Using a procedure,
necessary!)
elements
(1,1~0,...,I,...,0)
the interpretation If all m i ~
case, when some of the M.:s might contain squares of the J that all deg(Mj) ~ 2). First of all it is clear that
, i I > 0 .....
... ' Mr • Now introduce ii-I ik-1
k[Xo,X 1 . . . . . Xn]/(XoX 1
Now Xo-X I is a non-zero
ik Mk = X I ~
divisor
M~. . . . .
XoX1
([10], p. 30), we will
(renumber the M:s if
~'
ik > 0 , but that
a new variable
~+1 . . . . .
X o and put
Mr)
in R" and R~/(Xo-X I ) ~ R . Thus we have a "large" map
115
R" ~'j > R in the sense of Levin [17]. It follows that the Hopf algebra map Ext~(k,k) - - >
EXtR.(k,k) is a monomorphism, and that (of. loc. cit. p. 212)
we have an isomorphism of left Ext~,(k,k)-modules: (28)
Ext~.(k,k) @
k ~
EXtR.(R,k)
mt~(~ ,k) NOW Ext~.(R,k) = 0, i > I and Ext~.(R,k) ~ k, 0 < i < I. Therefore we have an exact sequence of EXtR.(k,k)-modules (with trivial operations on k and s Ik): 0 -->
s-lk
> Ext~,(k,k) 'Z
> k
> 0
which gives rise to a long exact sequence of Ext* (k,k)-modules: Ext~.(k,k) .-- Ext * (k,k) w--->Ext* (s-Ik ,k) (Ext~(R,k),k)-->Ext* Ext~.(k,k) Ext~.(k,k) Ext~.(k,k)
(29)
Ext *+I
Ext~.(k,k)
(k,k)-->" • "
Inview of the formula (28) and the fact that Ext~.(k,k) is Ext~(k,k)-free, we obtain that the map w* can be identified with the ring map (30)
Ext* (k,k) EXtR.(k,k)
J-->
Suppose now, inductively, that Ext*
Ext~.(k,k)
(29 ~) that Ext*
*
EXtR.(k,k )
Ext*
Ext~(k,k)
(k,k)
(k,k) is noetherian. It then follows from
(Ext~(R',k),k) is a noetherian Ext*
Ext~.(k,k)
(k,k)-module, since
it is an extension of two such modules. Using the identification of w* with j** , we now obtain that Ext*
ExtR(k,k)
(k,k) is a noetherian Ext* (k,k)-module, and Ext~.(k,k)
therefore, ~ fortiori, it is a noetherian ring. Thus Theorem 5 is ~roved[ However, we wish to continue and obtain a more precise result about where the generators of Ext*
(k,k) are situated. Here is the result we are aiming at:
EXtR(k,k) THEOREM 5".- Let R = k[Xl,...,Xn]/(MI,...,M r)
where the M.:s are monomials in the
X.:s (of degree > 2) let v. be the element of multidegree (1,1,0,...,I ...,0) in i -~ i ....................... ' -Ext I (k,k) (corresponding to X. and defined in general above) mud let • 1 EXtR(k,k) m i be the maximal exponent of X i in the M :s. Then: a) Th___~em.
Therefore the (n+2)
-
variable Hilbert series of our (n+2)-~raded double Ext-al~ebra is: n p(ZI~Z2,YI~...~Yn)/ ~(I-Z~Z~Y.) where p(ZI~Z2~Y I ,Yn ) is a polynomial i=I I z i ' '" " " n+2 variables with non-negative
integral coefficients,
in
where furthermore
for any monomial ..... -IziIzl2vJ-2 -I I . . .y~n
with non-vanishin~ coefficient n j]<ml .... ,Jn<mn and max(il,i 2) < Z Ji " i=I
we have
PROOF OF THEOREM 5~: The theorem follows already from the proof of Theorem 5 if all m i < I. Let us assume m I > I etc. as above and consider R ~ Clearly R ~ is (n+1)-graded by grades grading) and the ring map R ~
> R = R'/(Xo-X~)
(bo,bl,...,b n) (we use Xo too, to define the
> R is n-multihomogeneous
(bo+bl,b2,...~b n) on R ~. Those elements
in Ext I Ext R.
if we use the n-grading (k,k) that are analogous to
(k ,k)
the v.:s for R will be denoted by v~. (0 < i < n). The long exact sequence J (n+2)-multihomogeneous, zero
scalar multiple
and it is easily seen that 6 is the multiplication
of v ° - v~
.
(29) is by a non-
If we c o u l d p r o v e that v" - v~ w e r e a n o n - z e r o o
divisor in Ext*
(k,k), it would follow from (29) that we have an (n+2)-graded
Ext R.(~,k) exact sequence of Ext* (k,k)-modules: EXtR~ (k ,k ) v °
(31) 0 - >
-
v~
Ext *-I (s-lk,k) - - ~ EXtR~ (k ,k )
and thus all assertions that m~ = I, m~ = m I
.
Ext Ext R. (k ,k)
(k,k) -->Ext* (k,k)-->O EXtR( k ,k )
in Theorem 5 p would follow by induction from (31) (recall I,
m E = mi, i > I ).
We therefore now endthe proof of Theorem 5 p by proving that v~ - v~ divisor. Assume the contrary,
i.e. that there is a non-zero c 6 Ext*
such that (32>
is a non-zero(k,k)
Ext~(k,k) (v~ - v ~ ) . c
We decompose c into homogeneous
= 0
components with respect to the degree
(denoted by
N
dego(
Z c~J )) defined by X ° : c = j=O
( CN # 0 ) "
'Fnus dego(Cj) = j ' dego(V~)=1 '
deg(v~) = 0 and it now follows by taking the deg ° -component of degree N+I of (32) that v~.c N = 0. But m~=1 and if N > O, it would follow from the inductive hypothesis o ~ (about multiplication by v~ ) that c N = O, which is impossible. Therefore N = 0, i.e. c = c o , and now (32) gives v~.c = 0 (take components of deg ° -degree 0 in (32)). But this is an equality of elements, whose deg ° -grading is zero. Now use the following trick of multihomogeneous
algebra:
Let R ~ o
be the subring of R', where
117
deg ° = O. We have (recall the notations
in the proof of Theorem 5 !):
R S = k[X I ..... Xn]/(Mk+ I ..... Mr) , where the Mk+ I ..... M r do not contain XI, so that R~o ~ A[XI]' where A = k[X2,...,Xn]/(Mk+ I~...,M r ). Clearly Ext* . (k,k) Ext R ~ (k ,k ) part of Ext
*
.
(k,k), where deg ° is zero. Since the e ~ a l l t y Ext~,(k,k)
o
= that
v~-c = 0 takes
place in this last part, and since R~ = A[XI]~ Ext* (k,k) = Ext* (k,k)[v~] Ext~(k,k) ExtX(k,k) o (v I is a polynomial variable), it follows that c = O, which is a contradiction and the Theorem 5 ~ is completely proved. Remark.- We had to work rather hard to get that 6 was a monomorphism or, equivalently, that the ring map (30) was an epimorphism.
There are reasons for that. Indeed, in
general, if (R',m ~) is a local commutative noetherian ring, x~C m_~ ~ ( ~)2 a non-zerodivisor, then R ~ j
J
> R'/(x ~) = R is still large [17], but it is not true that
zs an epimorphism or, equivalently that
(33)
Ext~ (k ,k ) Tor. (k,k) - - >
E x t ~ (k ,k ) Tot. (k,k)
is a monomorphism.
Here is a eounterexample,
due to Clas L6fwall and reproduced here
with his permission:
R" = k [ [ X , Y , Z ] ] / ( X Z -
y3)
,
Clearly X is a non-zerodivisor intersections,
R = R'/(X) = k[[Y,Z]]/(Y 3)
in R p. Furthermore, both R p and R are local complete
and their Ext-algebras
are generated by elements of degree I and
elements of degree I and 2 respectively
(34)
(thus x'= X).
Ext~(k,k) --
(cf,~.~.
[27]). Consider the inclusion:
> Ext~.(k,k)
and take an indecomposable generator T of degree 2 of EXtR(k,k). The image of T .
under (34) must be a decomposable of degree I. ~ u s ,
element, since EXtR~(k,k)
is generated by elements
already on the Tor I -level, the map (33) is not a monomorphism.
§ 4. FINAL REMARKS. OPEN PROBLEMS. It might be interesting to try to find other classes of local commutative noetherian rings R, for which Ext
(35 )
(EXtR(M,k),k) EXtR(k ,k )
is a noetherian Ext* (k ,k )-module for all finitely EXtR(k ,k ) generated modules M
PROBLEM I.- Could we "classify" those rings that satisfy (35) ? It is also possible to study the right EXtR(k,k)-modules *
EXtR(k,N)
(one of us did so
in [23] and Lescot also did so in [16]. This was applied to "Bass series" IN(z) = E dimk(EXt~(k,N))'Z i ~ i>O I
in [23] and [16]. In particular, Lescot has proved [16]
118
that the Bass series IR(z) is rational for any ring R with monomial relations as in § 3. He has proved this as a consequence of a very general theorem about rationality of P~(Z):s for multigraded M:s over such R:s, and indeed he has rationality of the multigraded version of PRM
(n+2 variables) too. There is probably a "double-Ext"
-
version of this, corresponding to the theory of § 3. Finally it should be remarked that the rings studied in § 3 contain the "StanleyReisner" rings (or "face" rings [28]) associated to a finite simplicial complex A . PROBLEM 2.- Give a combinatorial-geometrical interpretation of the coefficients in the rational function of n+2
variables in Theorem 5" for the case when R is the
Stanley-Reisner ring associated to a finite simplicial complex A. BIBLIOGRAPHY [I] ANICK, D.J.,A counterexample to a conjecture of Serre, Ann. Math., 115, 1982, 1-33. Correction: Ann. Math., 116, 1983, 661. [2] ANICK, D.J., Connections between Yoneda and Pontrjagin algebras, Lecture Notes in Mathematics, 1051, 1984, 331-350, Springer-Verlag, Berlin, Heidelberg, New York and Tokyo. [3] ATIY~, M.F. and MACDONALD, ~.~.,T ~ Introduction to Commutative Algebra, AddisonWesley, Reading, Mass., 1969. [4] AVRAMOV, L., Local algebra and rational homoto#y, Ast@risque, 113-114, 1984, 1543. [5] AVRAMOV, L., Differential graded models for ~°cal rings, RIMS Kokyuroku, 446,1981~ 80-88, Kyoto Research Institute for Mathematical Sciences, Kyoto, Japan. [6] BACKELIN, J., Les anneaux locaux ~ relations monomiales ont des s@ries de Poincar@-Betti rationnelles, Comptes rendus Acad. Sc. Paris, 295, S@rie I, 1982, 607-610. [7] B~GVAD,R. and HALPERIN, S., On a conjecture of Roos, These Proceedings. [8] CART#~ H. and EILENBERG, S., Homological Algebra, Princeton Univ. Press, Princeton, 1956. [9] COHEN, F.R., MOORE, J.C. and NEISENDORFER, J.A., Torsion in homotopy groups, Ann. Math., I09~ 1979, 121-168. [10] FR~BERG, R., A study of graded extremal rings and of monomial r i n ~ , Math. Scand., 51, 1982, 22-34. [11] GOVOROV, V.E., Dimension and multiplicity of graded algebras, Siberian Math. J., 14, 1973, 840-845. [12] GULLIKSEN, T.H., A change of ring theorem with applications to Poincar@ series -and intersection multi P licit , Math. Scand. 34, 1974, 167-183.
119
[13] GULLIKSEN, T.H. and LEVIN, G., Homolo@y of local rings, Queen's Papers in Pure Appl. Math., n ° 20, Queen's Univ., Kingston, Ontario, 1969. [14] JACOBSSON, C., Finitely presented ~raded Lie al@ebras and homomorphisms of local rin~s,
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[15] LEMAIRE, J.-M., Al~bres connexes et homolo@ie des espaces de lacets, Lecture Notes in Mathematics, 422, 1974, Springer-Verlag, Berlin, Heidelberg, New York. [16] LESCOT, Th~se, Caen 1985
and letter from J. LESCOT to J.-E. ROOS, June 14, ]985.
[17] LEVIN, G., Large homomorphisms of local rin~s, Math. Scand., 46, 1980, 209-215. [18] LEVIN, G., Finitely 6enerated Ext ~ e b r a s ,
Math. Scand., 49, 1981, 161-180.
[19] MACLANE, S., Homolo@y, Springer-Verlag, Berlin, Heidelberg, New York, 1963. [20] MILNOR, J. ~ud MOORE, J., On the structure of Hopf a l~ebras, Ann. Math., 81, 1965 211-264. [21] MING, R., Yoneda products in the Cartan-Eilenber @ change of rin~s spectral sequence with applications to BP,(BO(n)), Trans. Amer. Math. Soc., 219, 1976, 235-252. [22] MOORE, J.C. and SMITH, L., Hopf algebras and multiplicative fibrations I-II, Amer. J. Math., 90, 1968~ 752-780 and 1113-1150. [23] ROOS, J.-E., Sur l~alg~bre Ext de Yoneda d'um anneau local de Golod, Comptes rendus Acad. Sc. Paris, 286, s6rie A, 1978, 9-12. [24] ROOS, J.-E., Relations between the Poincar6-Betti series of loop spaces and local rin6s, Lecture Notes in Mathematics, 740, 1979, 285-322, Springer-Verlag, Berlin, Heidelberg, New York. [25] ROOS, J.-E., On th e use of ~rade d Lie algebras in the theory of local rin~s, London Math. Soc. Lecture Notes Series, 72, 1982, 204-230, Cambridge University Press, Cambridge. [26] S6minaire H. CARTAN, 11 e ann6e 1958/59, Invariant de Hopf et op6rations cohomolo@iques s6condaires, Paris, Seer. Math., 11 rue Pierre Curie, Paris 5,1959. (Has also been published by Benjamin, New York.) [27] SJODIN,G., A set of ~enerators for EXtR(k,k) , Math. Scand. 38, 1976, 1-12. [28] STANLEY, R.P., Combinatorics and Commutative Al6ebra,Progress in Mathematics, vol. 41, 1983, Birkh[user, Boston, Basel, Stuttgart. Department of Mathematics University of Stockholm Box 6701 S-113 85 STOCKHOLM (SWEDEN)
ON A C O N J E C T U R E OF ROOS
by
Rikard B~gvad and Stephen H a l p e r i n
i.
Introduction.
T h e o r e m A:
In this paper we prove the f o l l o w i n g two theorems:
Let R be a local c o m m u t a t i v e
is noetherian. T h e o r e m B:
ring whose Y o n e d a E x t - a l g e b r a
Then R is a c o m p l e t e intersection.
Let S be a 1-connected finite CW c o m p l e x and suppose the
P o n t r j a g i n algebra H,(~S;~)
is noetherian.
all but finitely m a n y degrees.
Then ~,(S)Q~ vanishes
(S is e l l i p t i c - cf.
T h e o r e m A was a q u e s t i o n of Roos
[14].
in
[8].)
T h e o r e m B is its trans-
lation to t o p o l o g y via the standard d i c t i o n a r y
([3],[4]),
and was
posed by Roos in [13]. The main tool in the proof is Sullivan's notion minimal models,
([12],[15])
d e f i n e d by him for the study of t o p o l o g i c a l spaces,
and adapted by Avramov
([3]) for the study of local rings.
The first key ingredient
is the n o t i o n of "category" of a m i n i m a l
model.
This was i n t r o d u c e d by F e l i x - H a l p e r i n in [7] for S u l l i v a n
models,
and shown to c o i n c i d e w i t h the c l a s s i c a l d e f i n i t i o n of
L u s t e r n i k - S c h n i r e l m a n n category. as well
of
(in sec.
Here we adapt it to A v r a m o v ' s models
2).
The second i n g r e d i e n t is the fact that a m i n i m a l m o d e l d e t e r m i n e s a g r a d e d Lie algebra whose u n i v e r s a l e n v e l o p i n g algebra is c l o s e l y related to the E x t - a l g e b r a or P o n t r y a g i n algebra the n e c e s s a r y facts in sec. In sec.
([3],[1]).
We recall
3.
4 we combine these ingredients to prove a single t h e o r e m
about models of w h i c h both T h e o r e m s A and B are corollaries.
In sec.
5 we deduce a t h e o r e m on graded Lie algebras. We thank L. Avramov and C. L~fwall for m a n y helpful discussions.
2.
The c a t e g o r y of a m i n i m a l model.
Let X = p ~ p
space over a field k
(possibly of c h a r a c t e r i s t i c
d e g x = p and
For n~0 we put X(n)=!pT>_nX p-
Ixl=Ipl.
be a g r a d e d vector >0).
If x~X
P
we say
By AX we shall mean the tensor p r o d u c t of the e x t e r i o r algebra on Xod d w i t h the s y m m e t r i c algebra on Xeven. where
APx=xA...AX
Then AX= • APx; w h e r e p!0
(p factors).
We shall use m i n i m a l model to m e a n a DGA of the form which
(AX,d)
in
121
(i)
X=X>0 or X=Xm
X
forward to
factor ~m as the composite of homomorphisms (2.1) (AX,d) i~(AX@AY,D) p.....(AX/A . >m X,d), where Y is a graded space and (i) (ii)
Y=Y>0 (resp. Y=Y0 (resp. X=XmX is a boundary.
then factors through
In"
lil>n.
Decomposing
The projection In as in (2.1) we
factor ~ as AX
AXeAY~AX/I Because H(~)
is an isomorphism
and ~ is surjective
an induction argu-
ment on the basis Yi of Y shows one can lift ~ through
0 to get
r : AX®AY~AX. For this paper we need an e l e m e n t a r y version of the mapping theorem
[7; T h e o r e m
of the p o s s i b i l i t y (AX,d)
5.1].
The proof of
that char k>0.
[7] needs m o d i f i c a t i o n
because
To state the result we note that if
is a minimal model and if we divide by the ideal generated by
elements xCX with
Ixl
1 is a KS basis
and dividing
sufficient
for all p.
vector
the exterior
model
~ cat(AZ,d).
space with basis u and degu=
algebra
on u if deg u is odd;
otherwise F(u) is the graded/, algebra with basis {7Pu}p!0 such that 71u=u, 7 P u .T q u = [P~q) TP+qu and degyPu=pdegu. Extend ( A X , d ) t o 7°u=l, a DGA
(AX®F(u),6)
(AZ,d)
factors
by setting'" ~(~Pu)=xl®TP-lu.
(AX,d)+
as (AX,d)
with
The projection
¢(7Pu)=O,
p>0.
~ (AX®r(u),~)-L
Because
(A(Xl)@F(U),6)
CAZ,d) is acyclic,
H(¢)
is an
isomorphism. Now suppose
cat(AX,d)=m.
Then for a suitable
factorlzation
(2.1)
of ~m we get morphisms (AX, d) -i--+(A X®AY, D) r-~ (AX, d) (AX/a>mX,d) with
H(p)
an i s o m o r p h i s m
®AxI(AXI@F(U),~)
and r i = i d .
,
P u t A=AX/A>mx a n d a p p l y
to get
(AX®F(u),6)
i' ;(AX®AY®r(u),D ') r' )(AX®F(U),6
J o'
(2.5)
A®r(u),6'), Clearly
Let s P = ~ o ( ¥ J u ) . A simple induction on p then 3: shows that each restriction (AX@AY®S p D') ~ (A®sP,6 ') of ~' gives an isomorphism
r'i'=id.
in homology.
Let IcA®F(u) Z, x[ and XlU if
Hence H(p')
IXll is even.
definition
of minimal
by I gives
a DGA with homology
degrees.
is an isomorphism.
be the ideal generated models
Thus the inclusion
by Z if IXll
In either
implies
is odd,
case condition
that 6'(I)cI.
~ in degree
Iek ~ A®F(U)
(iv)
Moreover,
and by in the division
zero and zero in the other induces
an isomorphism
in
homology. Similarly
if we set J=(p')-~(I) J@~ ~ AX®AY®F(u)
are homology
a morphism phism).
the composite
isomorphism.
we obtain
that
~ AX®F(U)
¢ : K~k ~ A X ® F ( u ) 9 ~ A Z
By induction
o : AZ + K~k such that ¢o=id
Thus
K~
isomorphisms.
In particular, tive homology
and K=(i')-1(J) and
(2.5) yields
is a surjec-
on a KS basis of AZ we obtain (and hence H(o)
the DGA diagram
is an isomor-
123
AZ' io ~J@k Cr' p,,
@ " the restriction of jective,
p'
Again H(p")
,AZ
is an isomorphism,
p" is sur-
and (~r')o(io)=id.
But by c o n s t r u c t i o n
I.I ..... I (m+l factors)=0.
tors through the p r o j e c t i o n AZ~AZ@AW÷AZ/A>mZ
AZ~AZ/A>mZ.
Thus p"i'a
If this is d e c o m p o s e d
as in (2.1) then the induced map AZ@AW+ISk
through
p" to a DGA m o r p h i s m AZ®AW+J~k.
desired
retraction
facas
lifts
Composing with ~r' yields
the
AZ@AW+AZ. O
Next observe that the proof of in our context 2.6
2.1] applies verbatim
to imply
Proposition.
dimX=~,
[8; T h e o r e m
If (AX,d)
is a m i n i m a l model with cat(AX,d)
+ (-i)IYl IXl<x,~>,x,yeX,~,S~L.
in L is then defined by <x;[~,8]>
By [3; Theorem 4.2],
= (-l)l$1.
(L,[,])
the homotopy Lie algebra of
is a graded Lie algebra: (AX,d).
(3.1) it is called
124
The grading of L induces a grading algebra,
UL.
In particular,
in the universal
enveloping
there is a resolution d
M.(i)
i 7 M.(i-l)
of the UL-module k in which each M,(i)
....
-+k
is a free graded UL-module,
and
d
is homogeneous of degree zero. This induces in turn a gradation in 1 each TorUL(~,~) which we write as Tor~L(~,k) : ~TorU L (~,~). 1 ' p l,p 3.2 Proposition. If L is the homotopy Lie algebra of a minimal model (AX,d),
then H~(AX,d 2) = Hom(Tor~L_p_q(~,~);~)
Proof: 280].
When L is ungraded
this is e s s e n t i a l l y
[6; Theorem
7.1, p.
The proof here is a modification. Denote by A
be the graded dual
the graded dual of AX ([16])
graded dual of left m u l t i p l i c a t i o n Ixl > lql.
: Aq P = Hom(APX)q;~).
Let
of d2, and for xsX let i(x):Ap+Ap_ 1 be the
Thus if xl~X,
by x.
Then i(x) vanishes
in A q if
~x~L are dual bases we can define D : UL®Ap~
UL@Ap_1 by D(zSa)
: (-I) IZlzesa - [(-I) 1z!IxXl ~ "z®i(xA)a. X We shall show that ---~UL®A D)UL®A D ...----+UL D E-E-+k p p-i is a resolution. The p r o p o s i t i o n is then immediate. A straightforward
computation
shows that D2=0.
(3.3)
Now write
(3.3)
in the form E : (ULQA,D)+k.
Denote by UicUL the linear span of words
of the form
s~i).
filtration
81.....8s
(~jsL,
Filter ULQA by the increasing
F,:
Fi(UL~Ap) = Ui_p@ApFinally,
notice that
A : x~-~x®l+l®x makes
the dual algebra structure
£(AI), on the graded space A I. identifies
the associated
AX into a coalgebra;
in A is the free divided powers
algebra,
Thus the P o i n c a r e - B i r k o f f - W i t t
graded algebra
(for the filtration,
theorem F,) of
UL®A as AL®F(AI). But by definition,
A~=L_q_I.
The filtration
respect to D and the associated differential, unique d e r i v a t i o n particular,
such that ~I(L)=0
H(AL®F(AI),6)=~
and 61(TPa)
and so (3.3)
F, is stable with
6, in A L Q ? ( A I) is the = a~TP-la,
aEA I.
In
is exact. O
4. 4.1
The proofs of Theorems Theorem.
Let
A and B.
We shall deduce both from
(AX,d) be a minimal model with homotopy Lie algebra
L such that (i)
cat(AX,d)
(ii)
UL is noetherian.
is finite,
Then X is finite dimensional.
and
125
Proof:
We suppose dimX infinite,
observe that both (AX(k),d),
(i) and
and deduce a contradiction.
First
(ii) also hold for the q u o t i e n t m o d e l s
because of P r o p o s i t i o n
(2.4) and the fact that the corres-
p o n d i n g Lie algebra is a s u b a l g e b r a of L.
In view of C o r o l l a r y 2.7 we
m a y thus suppose that H(AX,d)
has infinite dimension. >p (AX,d) by the ideals A-- X to produce a spectral
Now filter
sequence c o n v e r g e n t to H(AX,d).
In view of P r o p o s i t i o n
3.2 the columns
of the E 2 - t e r m can be i d e n t i f i e d w i t h the g r a d e d duals of the spaces Tor~,(~,k).
Since UL is noetherian,
each c o l u m n has finite total
dimension. But Lemma 2.3
(i) shows that
m+l columns c o n t r i b u t e to E ~. total dimension.
(since cat(AX,d)=m T
it
is
g> K((Z/~)3,1)
clear
129
such that f: Y
if
a triple
> K((~
there
is
some
the t r i p l e well
sequence
then f l i f t s
)7,1) space
a,b,c
defined
d,e,f~tIl(Y,~)
with
of
cohomology
elements
Massey
by a map
de=el=0.
degree
Hence,
as
I triple Massey
represented
product.
that T is a K ( ~ , I )
shows
represented
to T i f f
a non-zero
and n o n - z e r o
also
is
by g has
The l o n g
for
product, a
exact homotopy
some ~-group
~
of
order i 5 . Lemma
2.1
Any
connected
space
X with KIX -~
has
a non-zero
.. ~1 P6 is conclusion.
some f u r t h e r p r o p e r t i e s
(4.1). As X is of
(complex)
I is a l i n e b u n d l e linebundle
codimension
o n ~6. Its
~P(N)(I).
Let
I i n ~6 its d e f i n i n g
restriction
c16H2(X,Z/~)
to X is the t a u t o l o g i c a l
be the first
of Ir~. Then, as N i~ of ra~k 4, ~ ( i , ~ l ) free H ~ ( X , Z / l ) - m o d u l e step we n o w c o n s i d e r that < a ' , b ' , c ' ~ O
on g e n e r a t o r s a',b',c'
because
and a m o n o m o r p h i s m
~
in d e g r e e
oriented
i : H~(X,~) Apart
from the p r o j e c t i o n
need a well-known to p r e s e n t element
i
formula
a simple
class
I, ci, c21 and c~. As a f i r s t resp.
is an i s o m o r p h i s m
~c
and we find
in degree
I
2.
(real)
> Hm(~6,~Z)
Chern
is, t h r o u g h f ~, a
:= ~ a , ~ b
The next step is to p u s h a',b' are c a n o n i c a l l y
ideal
and c' into ~ 6
manifolds
of d e g r e e
formula f o r i~oi
As X and
~6
we h a v e a G y s i n - m a p
d i m ~ ~ 6 - d i m ~ ~ = 2.
x.i y = i (imx.y) . I will
p r o o f of it in the n e x t
we w i l l
take the o p p o r t u n i t y section.
The
131
element
i
(I)
is
the
class
As X is of c o d i m e n s i o n class
of
(4.2.1)
b"c"=0,
that
prepared
that
i
(1.1)
a",b",c"
and
(4.2.1)
we
-c13
Best
associ~e classiquement
(EE.M])
140
E 2 = EXtH~ (B) (H~(E),@)
=>
H (F)
appel~e suite spectrale d'Eilenberg-Moore. Or la multiplication de Yoneda fait de Ext
(~,@)
une alg~bre et de
Ext
H~(B)
(H~(E),~)
un
Ext
H (B) Th~or~me ~ . I . - S i l'appZ~ca£ion
l e s f o r m a l i s a t i o ~ de
B
e t de
Ext
(~,~)-module.
H~(B) p
est forma~able
(_~.S.]),
alors
E i n d u i s e n t d ~ ~omorphism~ (@~,Q) ~ H (f~B;Q)
H~(B) ~t
ext
(H~(E),~) ~ H (F;~). H~(B)
Le premier e s t un isomorphisme d'alg~bres, l e second un isomorph~me de modul~ S ~ ces a l g e b r a . En particulier, les suites spectrales d'Eilenberg-Moore d~g~n~rent (Iv]). A titre d'exemple, rappelons que toute application holomorphe entre vari~t~s k~lh~riennes est formalisable ([D.G.M.S~). Des th~or~mes 4.1 et 7.1, on tire imm~diatement un analogue d'un th~or~me de Levin ( [Le] ).
Coro£lai~e.- S i H-alg~bre gradu~e, ~ s£
H e s t une
EXtH(~,~)
~t
ExtH(H ,(~) est un
de Yoneda), a£ors
~-alg~bre gradu~e
l-connexe e t
H'
une
une alg~bre n o e t h ~ e n n e {po~ l e p r o ~ i t EXtH(~,~)-mod~e noethe~en.
De m~me, des th~or~mes 5.2 et 7.1, on tire une version gradu~e d'un r~sultat de Roos (ER]).
CoroZla~e 7 . 2 . - S o i t (H+) n # O
et
~ngendr~ par
(H+)n+l = O, 1
a/ors
H une
~-alg~bre gradu~e
ExtH(H / + n,~) t(~ )
est un
l-connexe. S i EXtH(~,~)-module libre
EXtH(H/(H+)n,¢ ) .
La representation d'holonomie fournit finalement une suite spectrale, appel~e suite spectrale d'holonomie qui ggn~ralise la suite spectrale de Milnor-
141
Moore d'un espace
EQ]"
Th~or~me 8 . 1 . - Pour chaque f i b r a t i o n , i l e x i s t e une s u i t e spect~ale du premier q u a ~ a n t v ~ r i f i a n t
E 2 = EXtH
Si
B ale
(~B)(~,H
(F))
-~-> H (E).
t y p e d'homotopie r ~ t i o n n ~ l l e d'une s u s p e ~ i o n ,
spect~ale d~g~n~reau t ~ m e
cette s~ite
~t on a un %~omorph%Sme d'espaces v e c t o r i e l s
E2
gradu~s ExtH
Le texte s'organise
÷
2.
Quelques points d'homotopie r a t i o n n e l l e .
3.
Calcul de l ' o p ~ r a t i o n d'holonomie r a t i o n n e l l e .
4.
Operation d'holonomLe noeth~rienne.
5.
Op~r~ion d ' h o l o n o ~ e f i b r e .
6.
Op~ra~ion d'holonom~e t r i v i a l e .
7.
Holonomie e t s u i t e
8.
S u i t e s p e c t r a l e d'holonomie.
PB
en
s p e c t r a l e d'Ei~enb~g-Moore.
de l ' o p ~ r a t i o n d'holonomie. F
÷
E
+
B
l'espace des chemins de
PB × E B
constant
:
D ~ f i ~ i t i o n de l ' o p ~ r a t i o n d'holonomie.
1 . 1 . - Soit
F
comme suit
~ H~(E).
I.
§ I - D~fi~on
Notons
(gB)(~,H~(F))
l'application b . o
topie pros par
F
envoyant
L'injection fournissant
d'holonomie de la fibration
une fibration. d'origine
B f
canonique
Supposons b
o
•
B
Dgsignons
polnte-~ en par
1
b • o
:
sur
(e b ,f) og cb dfisigne le chemin o o ~B x F ÷ PB × E se factorise ~ homoB
ainsi un morphisme
: ~B × F ÷ F
appel~
op~a~o~
142 1
\/
÷
PB×E B
$~B× F
Proposition
(~wJ).- L'op~ration d'holonomie u
H-~pace homotop~uement a s s o ~ i ~ i f Cette action
est s.h.m.
(~B,~)
est une operation du
sur l ' ~ p a c e
F.
([St]).
1 . 2 . - Example.
(I~
(2)
Si
f : E ~ B
est une application
fibration
homotopique
nomie
:
~
La fibration
canonique
l'injection
de base
d'holonomie
~B~
p
BS1
(Ef = B I × E) B ' par
b
la composition
L'op~ration
B
Ff ÷ Ef ÷ B
et
l'op~ration
la d'holo-
est d~finie
~B + PB ÷ B
du point
est simplement
(3)
associ~e
~B × Ff + Ff
continue
y
~tant dans
la fibration B,
o dans l'espace
= ~(l))
d'holonomie
des lacets.
dans la fibration
(p(~) = ~(0)
associ~e
l'op~ration
des lacets
libres
est la conjugaison
des
lacets -I ~(~,~') (4)
Le connectant
f~E
~P>
~
~B
de la suite de Baratt
>>
F
est par construction est compatible
D'apr~s S V T
Ganea +
~a
S × T
On v~rifie
= ~ o w' o w
J-~
ale
P>
B
la restrictiton
avec
~,
E
les operations
de
~
de
~ ~B
la fibre homotopique type d'homotopie
alors que l'op~ration
~B x {f }. o sur
F
x (~2S • ~T)
÷
F.
~(S) • ~(T).
est fournie
ration diagonale (~(S) x ~(T))
et sur
de l'inclusion
du joint
d'holonomie
~B
f~S • ~2T.
par l'op~-
143
§ 2 - Quelqu~ points d'homoto~e rationnet~e. Tousles
espaces consid~r~s sont suppos6s connexes par arcs et du type
d'homotopie faible d'un C.W.-complexe les espaces vectoriels
H~(S;@)
et
est la th6orie des modules minimaux. et proprigt6s 616mentaires.
de type fini. On notera H (S;@).
H~(S)
et
H~(S)
L'outil principal dans la suite
Nous rappelons ici quelques d6finitions
Pour plus de d6tails,
le lecteur est invit6 g se re-
porter ~ [Su, Ha, Ta].
2,1.- Le module miv~(mal de SulZivan. Tousles
espaces vectoriels et toutes les alg~bres sont suppos~es d6fi-
hies sur le corps
~.
Une alg~bre diffgrentk~£1e gradu@e commutative (a.d.g.c.) une alg~bre gradu6e commutative dA
de degr~
(xy = (_|)deg x.deg Yyx)
et de carr6 nul. Elle est dite libre si
+l
(A,d A)
est
munie d'une d~rivation A
est le produit tenso-
riel d'une alg~bre sym~trique sur un espace vectoriel
Y
par une alg~bre ext~rieure sur un espace vectoriel
concentr6 en degr~simpairs
on note
A = AX,
X
~tant la somme directe
vectoriel des mots de longueur
en
(x)~e A
de
X
Y @ Z.
Notons alors
A~X
dA(X) C A~2X
et s'il existe
:
l'espace
X.
est dire mi~im~e si
(A,d A) une base
i
Z
concentr~ en degr6s pairs
A = ]IX,
si
index6e par un ensemble bien ordonn@ tel que
dA(X ~) C AX< . Un morphisme
morph~me si Si
d'a.d.g.c.
~ : (A,d A) ÷ (B,d B)
: H (A,d A) + H (B,dB) (A,d A)
est un isomorphisme.
est une a.d.g.c, v~rifiant
H°(A,d A) = ~,
unique (~ isomorphisme pros) a.d.g.e, minimale (AX,d) (/iX,d)
+
(A,dA).
Le foncteur associe g chaque espace (ApL(S),ds)
(AX,d)
est appel6 un quaSi-iso-
munie d'un quasi-isomorphisme
s'appelle le module minimal de
PL-formes construit par Sullivan ([S~), S
il existe une
une a.d.g.c.
not6
ApL
Le modgle minimal de
(ApL(S),ds).
est par d~finition le module minimal de
(A,dA).
S.
144
2.2.- L'alg~bre de Lie d'un module minimal. Soit
L
une alg~bre de Lie gradu~e connexe (a.l.g) de type fini,
alors le complexe de Koszul (~Ta])
(C~(L),d)
est l'a.d.g.e.
(AX,d)
l-connexe
d~finie par (i)
X = Hom(sL,~)
(ii)
d : X
(iii)
~ A2X
= (-l) deg V<x,sEu,v]>.
C'est un modgle minimal. Par eontre, si
(I~,d)
est un module minimal, posons
= Hom(s(A+X/A~2x)P+I,@). Les ~l~ments u de L d~finissent des fonctions P su : A + X ÷ ~ par la formule = (-l) deg U<sb,u>.
L
L'espace
A2sL
s'interprgte alors comme
Hom(A~2X/A~3X;~)
= = (_])deg v ' d~finit ~le structure d'alg~bre de Lie gradu~e sur L D~composons v~rifiant
(AX,d).
d
o~
d.(X)~l AIX'
ont m~me alg~bre de Lie
d = d 2 + d 3 + ...
alors
(kX,d 2)
L
._(Ax,d 2) ~ C~(L).
et
," (AX,d)
et
(AX,d 2)
L du module minimal dtun
1-conne×e ~ti~omorphe ~ l'alg~bre de Lie d~homotopie r a t i o n n ~ l e
~ (~S) e ~
de
2.3.Soient et
d.l d[signe la d~rivation
est une a.d.g.e.
Th~or~me (ESu~, EA.A~).- L'alg~bre de Lie espace
~ e A+X,
L.
s'appelle alors £'alg~bre de Lie de sous la forme
et la formule
(A,dA)
s.
K.S.-extensions. (A,dA)
est augment~e par
chaque homomorphisme commutatif
et
:
(B,dB) gA
d'a.d.g.c,
sur
des a.d.g.c, telles que @.
Dans
H°(A) = H°(B) =
(EHa~), S. Halperin montre que
f : (A,d A) ~ (B,d B)
est associ~ un diagramme
145
f
+
(A,d A)
(B,dB)
(A OAX,d)
~
~
( ~ , d)
EAOid o~
: 1)
g
eat un quasi-isomorphisme
;
2)
i
eat l'injectiom
canonique
;
3)
Ii existe une base
(x)~e K
de
dx
~ < B =>
s'appelle un K.S.
Ix [ @ IxBI ,
eat alors unique ~ isomorphisme
la
pros
B
l-connexe
Th~or~me.-
(EG~)
Si
de f a ¢ o n ~ 6 1 p o t e n t e s u r m i n i m a l de
~ :
K.S.-extension
de
f.
eat dite minimale.
r~side dana le th~or~me
F ~
E
P-L B
et si
Elle
suivant d~ ~ Grivel
o~
e a t une f i b r a t i o n
f = ApL(p),
d'holonomie
Dana ce paragraphe, B
a/ors
(EH~).
~I(B)
(AX,d)
op~re
e a t un m o d u l e
fiB x F + F.
nous supposerons
Nous explicitons
H (~B) 8 H~(F)
6 7r (CLB) 8 ~
3. I . -
÷
rationnelle.
un C.W.-complexe
connexes par arcs. Nous calculons
H~(F)
toujours que
~ : F ~
E ~
l-connexe de type fini et
tout d'abord le modgle minimal
de
ensuite l'op~ration
d@finie dana l'introduction
Le m o d u l e m i n i m a l de en homotopie
B
E, F
et montrons
que pour
on a :T(~) 6 Der H~(F).
Traduisons
au § 1.1.
(Koszul-Sullivan)-mod~le
F.
eat une fibration avec
T :
tel que
et g S. Halperin dams le cas g~n~ral
H~(F),
§ 3 - Calcul de l'op~ra2ion
:
K
([H~).
L'utilit~ de la construction dana le cas
et un bon ordre sur
e A ~ [~ O,
-p.
@ Der (A), p~O P
munie du crochet
v
D,O~
=
@.6' - (-I) deg 8.deg
08'.@ et de la diff~rentielle
a.l.d.g. "Consid@rons la K.S.-extension module de la fibration pour la bigraduation
O~
de
Der(AY)
~, X =
(~X,d)
Notons
d
de
est une
k > (AY,d) A+X
homog~ne
d@finit des @l@ments
par la formule
De la relation
0 = I db
(b)~c A
La diff@rentielle
d]Ay = 1 0 d +
(~)
i + (AX 0 AY,d)
et choisissons une base @ (APx) q. P,q
~8 = 2 A , O ]
0 8~ + I
L
d 2 = O,
(_)Ib~lbR
~ b ~A
0 e~ .
on tire alors :
I
0 ~8 ~ + ~
l'alg~bre de Lie de
~
(t~,d)
(-l)
IbalbabB
et
cation lin@aire d@finie par ~(u) = ~ 8 ~ .
O [e$,O~.
:
L -+ Der(AY)
l'appli-
147
La premiere ~tape de la construction de
T
repose sur le lemme
suivant :
Lem~e. I)
~@ = 0
2)
L'applicatio~
;
~
: L ÷ H (Der(AY),%)
i n d u i t e par
@ ~t
un
homomorphisme d'alg~bres de Lie. D~monstration du le~me : I) La formule (~) montre que si comme
= O,
2) Rappelons
~Ta]
b
e AIx,
il est clair que
~@~ = O,
alors ~
:
o.
qua
= (-I) deg V<x,s~u,v~>.
Un simple calcul donne alors le r6sultat.
Composons maintenant H (Der(AY),~) ÷ Der(H ~ (AY,d)). dans
H~(AY,d)
~
avec le morphisme canonique
UL
Le morphlsme
: H(aB;~) 8 H~(F) + H~(F)
restre/~ tio~
da~
op6ration
L
de
:
: UL 8 H~(AY,d)
Th~or~me 3 . 2 . -
de
Nous obtenons ainsi une representation
par des d~rivations. Ceci se prolonge en une
l'alg~bre enveloppante
T
•
.... H~(AY,d).
~
coincide avec l e morphisme
d~fini da~ l'introduction.
En p a r t i c u l i e r ,
~(~B) e ~ d ~ f i n i t une a c t i o n de c e t t e alg~bre de Lie par d~rivaH~(F).
D~monstration du th~or~me 3.2 : Puisque l'alg~bre de Hopf
H~(~B)
est
T
T
^
primitivement engendr~e par
~ (~B) 8 ~ m L
et puisque les operations
sont naturelles, il suffit de faire la d~monstration lorsque
B
et
est une sphgre
a) C~. d'une s~h~re ~ p a i r e . Dans ce cas, la K.S.-extension de la fibration prend la forme
148
(Ab,O)
i
(Ab @ AY,d)
k > (AY,d),
Pour construire du quasi-isomorphisme
l'action
~
ne contenant
Soit maintenant
u
dans
avec
~
Puisque
cocycle
de
Db = b,
(AY,d)
tout d'abord
(AY,d)
une section
(Db = b)
par la formule
la composition
d
= I.
d = 1 @ d + b @ ~.
n!
pour
par
ensuite
÷
avec
(-i) n ~n 0 on(~).
n~O
On effectue
impair,
on construit
ql : (Ab ~ Ab ~ AY,D)
0(~) =
La formule
deg b
et
L
pas de
(alg~bre
on ~crit
b,
on peut
de Lie de
b = sb
identifier
(Ab,O))
o
~
l'~l~ment
et on a alors
¢. d~fini
= I
et
~ e H (F).
b) Cas d'une sphere paire. Notons
de la fibration de
(Ab/b 2 ,O)
:
d@finie
Un morphisme
~
par
ql
sur
induisant
o = id + by I + by 2 + cy 3 + y
on d~duit
v
÷
(AY,d)
un
(Ab~h 2 ~ A(b,~)
K.S.-mod~le
@ AY,D)
l'extension
DE = -bb.
est alors
~ AY,D) ÷ (A(b,~)
construit ~ AY,D')
en composant
la
avec une section
O
(AY,d).
Une telle section
y :
Soit
Db = b,
q2 : (Ab/b2 ~ A(b,~)
de la projection
et oh
(Ab~b 2 O AY,d)
d = I ~ d + b @ 0.
(Ab/b 2 ~ AY,d)
projection
÷
AY + (b ~ A+(b,c)
~
est n@cessairement oh les
Yi
de la forme
sont des endomorphismes
~ AY) @ (A~2(b,~)
lin~aires
~ AY). De l'~quation
:
@ + YI = ¥2 d - dY2"
de
od = Dd,
AY
149
En particulier, si cocycle
~
de
AY
u e L
et tout
a
satisfait ~
de
H (F),
alors pour tout
on a
=
(_|)deg u.deg ~+]
car
= -I. •
T.
1_j_+ (AXe AY,D) ÷ (AY,d)
DerAx(AX ~ AX) C Der(AX ~ AX)
des d~rivations s'annulant sur
une action par d~rivations
P
la sous-a.l.d.g, form~e
DerAX(AX @ AX) + DerAxsAy(AX 8 AX @ AY) de
H (DerAx(AX @ AX))
D'autre part, l'alg~bre de Lie B
de KoS.-mod~le
AX.
L'injection canonique
au module de Quillen de
(u @ e)>
(~H~),
sur
Der~,~(AX @ 5X)
induit
H~(AX ~ AX ~ AY) =H~(AY). ~tant quasi-lsomorphe
on a un isomorphisme d'alg~bres de Lie.
p~ : H~(DerAx(AX @ AX))
:~-+ L.
Ceci est donn~ explicitement par = (-I)deg @+lee(x), l'augmentation canonique de
~
e ~X.
Proposition 3 . 3 . - L'isomorp~sme ration
T :
~ (~B) e H~(F) ÷ H~(F)
D~mo~£TLcug£on
Les operations
~
et
:
~
d~signant
i)~ i d ~ n t i f i e l ' o p ~ r a t i o n
~
~ l'op~-
d ~ f i n i e dans £ ' i n t r o d u c t i o n .
II suffit de montrer que
~ = ~
(l'op~ration de 3.2).
~tant naturelles, il suffit de nouveau de faire la
150
d@monstration lorsque la base est une sphere.
a) C~dlune sphere impasse. Notons
(Ab,o) ÷ (Ab ~ AY,d) ÷ (AY,b)
d = I ~ d + b @ 0. = I.
Soit
L
l'alg~bre de Lie de
La d~rivation
un cycle v@rifiant
du
O~([d~)
Notons alors
un K.S.-mod~le de la fibration :
de
= u.
ql
et
DerAb(Ab @ Ab) Soit
o
(Ab,O)
Du
et
u
d@finie par
son image dans
dans
L
avec
du(b) = -I
est
DerAb@Ay(Ab~AbOAY).
la projection et section construites en 3.2.a.
a(¢) =
X ~,)nbnen(¢) • n~O
Alors
~(u)([~])=
~iDu~(~)]
=
= [q1( ~ ~(-I)n ~n@n+l(,))~
[O(~)] d@__ff ~(u,[~>]).
b) Cas d'une sphere paire. Notons fibration
(d = ] @ d + b @ O)
Soit
En par
u
particulier, du(b) = -I
Notons
Ab/b2 de
du
(Ab/b2,O) + (Ab~b 2
dans
avec
= -I. et
et +
~ AY
AY,d) + (AY,d)
un K.S.-mod~le de la
l'alg~bre de Lie de
= I.
La d@rivation
du(~) = - b e s t
ql
@ A(b,c) @ AY dans
L
et L
0
(Ab~b 2 ,0).
On reprend les notations de 3.2 b. du
de
DerAb(Ab/b 2 0 A(b,e))
un cycle v~rifiant
d~finie
p~([du]) = u.
la projection canonique et une section de celle-ci. D@signons par
DerAb/b2~Ay(Ab/b 2 ~ A(b,c) 0 AY).
Alors,
= [qlDu(¢ + b'Zl(¢) + bY2(¢) + cY3(•) + y(¢)]
=-[yl(¢)] = [0(¢)] d@f ~(u,[¢J).
,
Du
l'image
151
§ 4 - Operation d'holonomie noeth~6enne, Th~or~me 4 , 1 , - S o i t 1-connex~, S i
F + E ÷ B
dim ~(B) 0 ~ < ~
e s t un
H(F)
(2)
et
P(t)
dim H(E) < ~
a/0rs
H(~B)-module noeth~rien ;
La s ~ r i e de P o i n c ~ forme
une fibration e n ~ e espac~
de
/~(1-t 2i)
~t
H(F)
une f r a c t i o n rat~onnelle de l a
dim ~2i(~B) O ~
o~
P(t)
d~signe un polyn$me
coefficients entiers, Avant d'entreprendre d~finitions
extraites de EGu l , ~ .
Soit P(t)
la d~monstration de ce rgsultat, rappelons quelques
~(t) ~ £Et]
est dite
un polynSme v~rifiant
~-rationnelle
~(0) = ± I.
s'il existe un polynSme
R(t)
dans
Une s~rie formelle £Et]
tel que
P(t) = R(t)/~(t). Si
H
est un espace vectoriel gradu~ v~rifiant
on appelle s~rie de Hilbertde
H
IHI(t)
H
G =
$ G p~O p
est dit a) b)
Ps(t)
de
IHI
Un
G-module
si
est d~finie
Pour tout sous
S,
S.
un anneau connexe gradu~ inf~rieurement.
~-rationnel
La s~rie
~ dim Hi.t i i~O
d~signe la cohomologie d'un espace topologique
d~signe la s~rie de Poincar~ Soit
gradu~
H
i,
la s~rie formelle
IHl(t) =
Ainsi, lorsque
dim Hi< ~ pour chaque
G-module
; N
de
H,
la s~rie
INl(t)
est
q-ration-
nelle. Dans les l e w e s rieurement
(M =
Lemme I
I M ). p~O P
~u ~,-
1~ 2, 3 suivants,
les modules sont supposes gradu~s inf~-
Le lemme 2 est le dual de l'~nonc~ de Gulliksen.
sort
gradu~s, H e s t n o ~ h ~ r i e n e t
o ~.+ H' ~ H ~ H" + 0
une s u i t e e x a ~ e de
G-modules
~ - r a t i o n n e l s i ~ seulement s i H' #~i H" l e sont.
152
Lemme 2 gradu~s avec de
I
IGu 21.-
H' J-~ H'
Soit
de degr~ z ~ o t e l que
G de degr~ s t ~ c t e m e n t p o s i t i f .
~-rationnel, ~'(t)
H'
e s t un
J
I
H une s u i t e exacte de
G-modules
s o i t la mult~plic~gion par un ~l~ment
Alors, s i
G-mo~le n o e t h ~ e n
H
~
e s t un
G-module noeth~rien
~'-~onnel
avec
= (I - t deg g)n(t).
Lemme 3.- S o i t finie,
UL ~ t
alors
une
L
un
~-alg~bre de Lie gradu~e et connexe de dimension
UL-module noeth~rien e t
~L-rationnel avec
dim L2i ~L(t)
= ~ (I - t 2i) i
D~mo~tration du lemme 3 : Le r~sultat et soit
est ~vident
g e L
l'hypoth~se noetherien
pour
un ~l~ment
de r~currence et
Proc~dons
dim L = O,1.
~L/g-rationnel.
est un
Si
g
UL-modules
O + U(L).g ÷ U(L) + U(L/g)
Si
g
le r~sultat
UL-modules
UL
×g'~
UL
>
L
~B + F + E
E 2 = H~(E)
~ H (~B), et convergente Comme
H (F),
H (E)
est noetherien
UL.g ~ U(L/g)
avec le lemme
du lermae 2 appliqu~
spectrale vers le
est de dimension ~-rationnel.
Corollalre 4 . 2 . - S i
s
de L.
n > O
Par
UL-module) et la suite exacte
] fournit
le r~sultat.
~ la suite exacte
de
UL/g.
est une suite
et
de dimension
(et donc
D~monstration du th~or~me 4.1. : La suite spectrale fibration
sur la dimension
qui est donc dans le centre.
est impair,
provient
donc
U(L/g)-module
de
est pair,
Supposons
de degr~ maximal, U(L/g)
par r~currence
de
H~(~B)-modules
H (~B)-modu!e
finie,
de Serre de la avec
H (F).
le lemme 3 montre
que
E2
et donc
•
e s t un espace topologique v ~ r i f i a n t
a)
dim H~(S;~)
< ~ ;
b)
~(~S) ~ ~
c o ~ t i e n t une s o u s - ~ g ~ b r e de Lie l i b r e de c o d ~ e n s i o ~
f i n i e , alors la s ~ r i e de Poincar~ de
~s
est rat~nnelle.
153
De~o~Ybk~o~ ~
~>p(~S)
tel que
@ @
de Postnikov de
S.
c o r o ~ l ~ e 4.2. : Notons
soit libre. Notons alors La fibre homotopique
F
Sp
p le
de
le plus petit entier p~ @tage de la tour
~ : S ÷ S
est un bouquet P
de sphgres et d o n c : ~(~F)
D'apr~s Comme
~ ~ = ~(V)
R~a~que 4.5.-
(I - (P(F) - 1)) -I.
dim ~(B) @ ~ < ~
En effet, d'aprgs le r~sultat de Bogvad H (~B)
H (~B)
P(~S)
est rationnel.
dens le th~or~me 4.1. (EBo]), si
est un anneau noeth~rien
est de dimensi3n finie.
QB ~ PB + B,
P(~F) =
le th~or~me 4.1. montre que
L'hypothgse
C.W. eomplexe fini, alors ~ (B) 8 ~
P(F) = t. IV ] + I.
la form~le de Hilton-Steer,
P(~S) = P(~F).P(~So) ,
est n~cessaire.
avec
B
est un
si et seulement si
Ii en rgsulte que dens la fibration
ne oeut ~tre un
H (QB)-medule noeth~rien
que si
~(B)
@
est de dimension finie.
Question 4.4.- si avec
dim H~(E) < ~,
H (F)
Ex~ple~ 4.5.- Si spheres et si
H~(E)
F ÷ E ÷ B
est une fibration entre espaces
]-eonnexes
est-il un
H (~B)-module
?
F * E * B
finiment
engendr~
est une fibration de base un bouquet de
est de dimension finie, alors
H (F)
est un
H (~B)-module
finiment engendr~. En effet, dens ce cas un K.S.-mod~le
de la fibration est de la forme
(H~(B),O) ÷ (H~(B) @ AY,d) ~ (AY,d). La diff@rentielle
d
s'~crit donc de la forme
d = I 0 d + ~ b. @ @ i , i I
o~
bi
pareourt une base de
H+(B).
Si
plus ~lev~ que la dimension cohomologique d~ # O.
Ii existe donc
bi
avec
Di(~
~
est un cocycle de
de
E,
# O.
alors dens
(AY,d)
(H~(B) @
de degr~ AY,d),
154
Th~or~me 4 . 6 . 1-connexes, t e l l e que
Si
H~(F)
F J
E
est une fibrat~on de f i b r e e t base
P~ B
s o i t un
H ( ~ B ) - m o d ~ e fi~iment engendr~, alors l e s
conditions suivantes sont ~ q u i v a l e ~
:
(I) L'alg~bre de cohomologie e s t de nilpotence f i n i e . un c e r t a i n
((H+) n = 0
n).
(2) L'application
@n : ~n+1 (B) 0 ~ + H n ( F ) 0 @
(compos~e du connectant
de l a f i b r a t l o n avec l'homomorph~sme d'Hurewicz} ~ t n
sup~eur
~ un c ~ t a i n
D~m0~p~trat~0n : Dans
(EOp]),
I)
-----> 2).
2)
-~---> I).
Il
est un
Soit
xnc
q
et p o ~ t o u t
Hn(F)
n
A(~ X n) n
Notons
sur
un sous-espace dual ~
Corm~e ~
est un morphisme de
H (~B)-module ainsi que les id~aux
R = H+(F)/Im(6
gendrg.
Les
sous
: H (~B) ÷ H (F)).
H (~B)-modules
J
n
de
R R
In
d~finis par
est un formgs
tousles
g~ngrateurs
de
R.
J
n
est
des
done
R
~l~ments
un hombre
fini
r
(H+(F)) r+l C 11
de g~n~rateurs.
et
n
H (f~B)-modules,
In = In-]'H+(F)"
et
I
Im 6~
On a d o n e
(H+(F)) (r+l)n° = O.
•
orthogonaux
pour lequel
o
o
Finalement, le r~sultat de Oprea montre que sur
est un
H (~B)-module finiment en-
forment une suite croissante. II existe, d'autre part, un
rieure
Im @n"
A(~ X n) + H~(F) -----+ H~(~B) n II en r~sulte clairement que
Im 6~.
I I = ker 6 ~.
Notons
contient
nul pour t o u t
pa~.
Oprea d~montre que le compos~
isomorphisme de
pour
n
I n
~
J
n
o
= Oo o
est l'alg~bre ext~-
155
§ 5 - O p ~ r ~ o n d)holonomie l i m e . 5.1.- Soit
Notons
h : F ÷ Y
quement trivial, k
f : X ÷ Y
la fibre homotopique de il existe une application
en une application
~Cf
sur
une application continue de cofibre
~Cf x X
k :
et sur
~Cf x X ÷ F F.
g.
Le compose
k : X + F
H (~Cf) O H+(X)
+
Cf
Notons alors
(K~ 0 AZ,d)
F.
D'apr~s
une
K~A~
H+(K~ ~ AZ)
@tant le compos6 module de
k,
dans
est
X.
Y
eSt 1-connexe
i n d u i t un isomorphisme de
K.S.-extension
(EHa] ' § 20),
et
H (~cf)-module~
--+
:
~0!
Ay
~
A X.
([Ha]).
acyclique minimale. Dans ce cas,
K~ 0 (K@ O AZ)
Ay 0 AZ
÷ 0
est un module
d'espaces vectoriels dif-
la suite exacte
AyOA~
@rant nul,
Ay • AZ
Ay
f :
K~ = (Ker ~ 0 @)
0 ÷ K@ + Ay + q
f6rentiels induit par tensorisation
÷
g o f
est alors fourni par l'a.d.g.c.
La courte suite exacte
0
~Cf x X ÷ X
: Choisissons un modale surjectif de
Un module de
pour
k
Prolongeons
H+(F).
~@m0~£A~ti0n
AZ = H~(~Cf).
hk ~ f.
n o t a t i o ~ pr~c~de~tes, s i
conne×e par arcs, l ' a p p l i c a t ~ o n
@tant homotopi-
compatible avec les actions ~ gauche de
homotope g l'inclusion de la fibre homotopique de
x
avec
La seconde projection
Th~or~me 5 . 1 . - Avec l ~
gf
g : Y ÷ Cf.
--~-+ ~ O A ~
H+(~)
+
O.
est un isomorphisme.
> AX 0 AZ
- -
D'autre part,
q
0 AZ et
)
une a p p l i c a t i o n continue,
@ ~ I
un
le r@sultat s'en d@duit aussitSt. • r
Corollaire
: Soit
j : Y ÷ z =
Y ~
@
( v
e
n.+l l
i=l
alors 1)homologie r ~ d u i t e de l a f i b r e homotopique de H (~z)-module au module l i b r e
Ceci g~n@ralise ~ route cofibration le point de Halperin-Lemaire
(EHL]).
e s t i s o m o r p h e comme
j
r n. H (~Z) 0 ( 0 H+(S l ) ) . i=l b
du th6or~me suivant
156
Th~or~me.- S o i t al
b)
j
: x
~+ Y = x ~ ( v
est surjectif
~ (j)
de Lie l i b r e
~(v).
Pans ce cas,
V
une a p p l i c a t Z o n c o n t i n u e .
e ~)
si et seulement si
e s t un
Ker ~ ( j )
~t
une a l g ~ b r e
H (~Y)-module l i b r e .
5.2. - Exemples. ])
Les espaees
G
dfisignons par
G
l'espace
constante. Si
G
et
et
Gn+ 1
0
n
la fibre
n
de Ganea. Soit
X
un espace topologique,
•
et
r @ d u i t ~ un ~ o i n t
f : X ÷ G n n
f
: X ÷ 0
O
C n
sont d~finis, posons
h o m o t o p i q u e de l ' i n c l u s i o n
Gn ÷ Cn.
l'application
G
la cofibre de
f n
Le thgor~me 5.1. montre
alors : X
2)
Consid~rons l'injection
T(S3,S3,S 3) F
de
i
est
I-connexe,
S3 V $3 V S3
-~
d~signe le "fat-wedge" des trois spheres
T($3,$3,$3), S3.
o~
La fibre homotopique
admet comme 7-squelette dans une d6composition homologique rationnelle (S5 V
L'application envoie les spheres les spheres et la sphere S 97 de
H+(G n) ~ H~(~Cn_ 1) 0 H+(X).
si
S~, 7 S9
S5
f : F
S 5 V S5) V ( V s~). 1~i~9 ÷
S3 V S3 V S3
est d~finie comme suit :
sur les repr@sentants d'une base de
i ~ 8,
f
w5(S 3 V S3 V S3) O Q,
sur les repr~sentants d'une base de WT(S 3 V S3 V S3) O Q
sur le point de base.
provient par la longue suite exacte d'homotopie de l'~Igment non nul
~8(T($3,$3,$3)) @ Q. N~anmoins,
correspondant g
7 S9
H7(F) ~ H2(~T(S3,S3,S3)) 0 H5(F). est g la fois dans
L'~l~ment de
H2(~T(S3,S3 S3~.H5(F) et dans
H7(~T(sB,sB,s3)).Ho(F).
3)
Consid~rons la fibration de Hopf g~n@ralisge f~ • 2X
--+ EgX
.>
X.
H7(F)
157
Si alors
X
est coformel et de categoric de Lusternik-Schnirelmann
H+(f~X • ~X)
est un
H (f~X)-module libre.
5.5.- D~signons par
F
la fibre homotopique de la projection
(S3a V S3)
q9
e8
-~
[a, Ea,@] p
2,
p :
3 Sb •
n'est p as la cofibre d'une application. N~anmoins,
H+(F) ~ H (~B) O (u,v) = T(x) 8 (u,v)
avec
deg x = 2, deg u = 3, deg v = 8.
Une d~composition homologique rationnelle de
F = ( V
S 3+2n)
n>.O
L'action de
H (~S~)
an
sur
F
qJ
( V
[al, an]
H+(F)
peut ~tre d~crite comme suit :
e8+2n).
n~O
est d~finie comme suit :
x.S 3+2n = $3+2(n+l)
x.e
8+2n
= e
8+2(n+|)
,
x e H2(QS
~)
•
B
5 . 4 . - Probl~me. D~terminer des conditions plus g~n~rales que celles mentionn~es dans le th~or~me ! sous lesquelles
H+(F)
est un
H (~B)-module fibre.
158
§ 6 - Operation d ' h o l Q n o ~ e t ~ i v i a l e . Soit D~signons
~ : F
par
la seconde
J~
-~P
~ : ~B x F ÷ F
projection.
module minimal
de
$
Notons et
D~inition.(resp.
E
~ ~ ~2'
L
~
B
une fibration
l'op~ration alors
de base
d'holonomie
et par
(AX,d) ~ (AX ~ AY,d)
l'alg~bre
de Lie de
estt~viale
(resp.
B
| -connexe. ~2
P
: f~B x F -~ F
(AY,d)
un
K.S.-
si
~ ~ ~2
(AX,d).
@-triviale,
H-triviale)
H (~) = H (z2)).
est donc : L ÷ Der H~(F)
H-triviale
est nul.
II en r~sulte
H-triviale
dans une fibration
1-connexe,
car Hans ce cas
6. I . -
si, avec les notations
ou la fibre
Der F.
ment homotope Si k-~
~(~xid),
: Toute fibration
F
÷
par la classe d'homotopie
La fibration
seconde
~ : S n-| -~ ~S n
est l'adjoint
÷
Sn
de base une sphere est
de son morphisme
est rationnellement
~ l'application
d'o~ le lemme.
E
triviale
si
d'embrayage k
est rationnelle-
projection. de l'identit~
de
S n,
alors
•
Th~or~me.-
L ~ propositions s u i v a n t ~ s o n t ~ q u i v a l e ~
I)
~J e s t
~-triviale ~
21
Le morpheme d'alg~bres de L / e
@
:
:
L -> H (Der(AY))
e s t nul. 3)
~
admet un module pour l e q u e l
dYc
AY @ (A>~2x @ AY).
d ~ f i n i en 3.2.
159
l)
=>
: Sn + B,
2).
Si
la fibration
est donc rationnellement
2) = > bigraduation
3).
]IX =
@ p,q
II s'ensuit maintenant
u~ e L
donc nul et morphisme
y
est
image r~ciproque
Choisissons
sur une base
(APx) q
;
~
Ii suffit
alors
et
J~2
par
AX
homog~ne
d = l ~ d +
e AIX,$@ ~ = O.
alors
y e Y,
: y(z)
j : F + E
~tant homotopes,
Elle
pour la
~ b ~eA
y(y)
Ceci est
D~finissons
= y +
~ @~.
Si
~(u B) = D ~ "
alors un
~ b i (y) b eAX
et
- (z + ~ b i ( z ) ) ~A~2X ~ AY
(AX @ AY,D)
par
l'inclusion
FAx ~ A(y(Y)),D_].
de la fibre.
il suffit de voir que pour tout
et pour tout couple d'applications
f|'f2
: X ÷ F
si
Jfl @ Jf2'
fl ~ f2"
Pour chaque a.d.g.c. (AX ~ AX 8 AX,D) Soient et
pour
alors de remplacer
~-triviale.
i B e Der(AY).
On a a l o r s
D~signons
ju
pour
en posant
pour
I).
b
e 6 L,
est nulle.
de
peut donc s'~crira
deg iB_ dJ~
multiplicativement.
X
b
= ~ B '
en p r o l o n g e a n t
espace
d
@(~)
(voir ~, § 3) que pour tout
y : AY ~ AX @ AY
Les applications
~ une holonomie
et la d~rivation
satisfait
3) = >
alors pour chaque
triviale
@B = iB ~ _ (-I)
z e AY.
@-triviale,
o~
Dx = dx,
alors
on a
la d~rivation
de degr~
s(x) = x,
s(x) = O,
D~signons ~(x)
= p(~) = O.
~'°
Par contre
par
Dx = ~
notons et
une homotopie
~IX@y = gp
-|
d~finie
s(1) = O,
p
et
(AX,d)
I
l'a.d.g.c.
D~ = O.
g, h : (AY,d) ÷ (AZ,D)
~ : (AX 8 AY,d) I + (AZ,D)
que, par d~finition,
(AX,d),
deux homomorphismes
([H~)
entre
gp
¢.eSd+ds IX@Y = hp,
et ok
s(y) = y,
AY I ÷ AX @ AY I
que
hp. s
Rappelons
d~signe
par : s(y) = O,
s(9) = O.
: (]IX ~ AY) I + AX @(AY) I la projection
Nous montrons
d'a.d.g.c.
~
se factorise
est un morphisme
g travers
d'adgc
p
d~finie
par
en un morphisme
et nous montrons
que
160
le composa
AY I + hx @ iY I ÷ AZ
a) D~signons par s(A2X)
et
X.
l'id~al de
(AX @ AY,d) I
Un petit calcul montre que
b) La relation
c) Pour tout d'une K.S. base Comme
J
est alors l'homotopie recherch~e entre
O(x) = 0
x e X,
(x)~e A
(sd)(x)
e I,
de
J
est stable par
x a X
montre que
~(1) = O. (f~,d).
d'aprgs (a)
par hypoth~se de r~currenee
pour
engendr~ par s
g
et
h.
A2X,
et par
d.
~[X = O.
Ceci se d~montre par r~currence le long
Supposons
~(iB) = O
pour
(sd)P(x) e I
pour tout
p ~ I.
~((sd)P(x )) = O,
V p ~ I.
~ < ~. On a donc,
Ii r~sulte alors de la
fo=ule [HaJ hp(x ) = gp(x ) + ~(x ) +
que
~(iC~) = O.
d) Puisque
X c l'id~al engendr~ par
La d~composition en somme directe d'~crire
d = d| + d2,
En particulier, Puisque dans
~. @2
Posons 0
=¢oe
J
alors
d'a.d.g,c,
y
de
d2(Y) e AY,
est stable par
X,
~(J) = O. @ (Ay)I
permet
Y :
dl(Y) = dl(Y) = d2(Y) = O,
d2(~) = ~.
et
s,
l'image de
@ 2 = sd2+d2s.
eI
On a alors
sd I + dls
est contenue
Im(en-@2 ) C J
et donc
.
~' : (AY) I ÷ AZ
~'d 2 = D~'
entre
et
Im d i C J .
@ = sd+ds
e) Soit (AY) I,
X
(/IX @ iY) I = E(AxI) + ~ (AY)~
avec pour tout
dl(Y) ¢ A~2X 0 AY,
¢oe
~ - L ~((sd)P(x )) p~l p!
g
et
d~finie en restreignant
et il r~sulte de h.
(d)
que
~'
~
g la sous-alg~bre
est une homotopie
m
6.2.- Classifiants. A chaque espace 1-connexe ~F
:
F
÷
EF ÷ BF
F
est associ~ une fibration universelle
(BF = B aut F, EGo_]). Les fibrations ~ fibre
F
et ~ base
161
|-connexe sont alors classifi~es par les classes d'homotopie d'applications continues de
B
dans le rev~tement universel D~signons par
de
Der(AY)
(AY,d)
d~finie oar
Der
-
BF
de
un module de = ~Der|
et
BF.
F
et par
Der
o
Der(AY)
= Der p
pour
la sous-a.l.d.g.
p > O
;
p
H (Der) = H+(Der).
BF = I _ ~ B~,
B~
~ (~BF) = @ ~
On a
une fibration de fibre ~
:
~(~B
B ~+ BB p
parcourant les sous-complexes finis de
F
(~B)
sur
B
) 0 ~ ÷ H D~(AY)).
.
L'injection
~ B~
B
BF.
d~finit par pull back
et donc (§ 3.1) un morphisme ea compatibilit~ des
~
avec les inclusions
d~finit par passage ~ la limite un morphisme d'alg~bres de Lie
: ~ (~BF)
H~(Der(~Y)).
O ~ ÷
Th~or~me 6 . 2 . - Avec l e s notations p r e c e d e n t s , I) P o ~ t o ~ e
fibration
~
de f i b r e
on a
e£ de base
F
B
l e diagramme
s ~ v a n t commute ~ (~BF)
o~
~
~
~
-
H (Der(AY))
'+
d~signe l e morphisme c o n s t r ~
f i a n t e de l a f i b r a t i o n 2)
0
en 3.2
~
1 'application clas-
~.
e s t un isomorphisme d'alg~bres de Lie.
D~mo~t~ation : I) provient de la construction de 2) Soit
~
Sn ÷ ~BF
:
L'application adjointe de base
Sn+! .
(Lemme 6.1) et
Comme ~ = O.
avec
et de la naturalit~ de
~.
p(~) = O.
~' : sn+l ÷ BF
~(~) = ~ ~ ( ~ ' = ) p
~
O,
est donc injective.
~
d~finit une fibration
est rationnellement triviale
162
La surjectivit@ de ~e = 0 base
et
[O] # O,
S n+l
avec
e
~
se voit comme suit : si
induit une fibration
[@] ¢ I m
~.
[0]
n'est jamais
que
F
(si
~n(F) O ~ # 0
Elle peut cependant ~tre
6.3.-
Si
dim w ( F )
< ~,
~>n(F) ~ ~ = O,
on a
H-triviale
elle
: supposons, par exemple, F
@tant F.
T.N.C.Z.
L'holonomie
H-triviale.
Fibratio~
de b ~ e
Proposition 3.- Si d'holonomie est
Q-t~iviale
un 2-c~ne.
~ : F J-~ E e t o~
B
P+ B
e s t un
e s t une f i b r a t ~ o n o~ l ' o p ~ r a t i o n
2-c~ne, a l o r s i l
existe
une l o n g u e
e x a c t e en cohomologie : Hn+l(F)
&n+l
D~mo~t~ation (A,d A) * (A
>
B
[H+(B) @ H~(F)]n
@tant un 2-cSne,
q~ (AY,d)
d(A I) C A2,
est envoy@ dans
supposer
:
AY,d)
0
AI.A 2 = A2.A 2 = O, A~2X
triviale de
l'op@ration d'holonomie
il en est de m&me de la fibration universelle de fibre
est darts ce cas
type
et
satisfait
Im ~ . .
soit une sphere paire, alors toute fibration de fibre
(ETh]),
suite
F -~ E~ + B~
H~(D---$T(AY)) + H+(Der AY)).
~-triviale
H (Der(AY)) # O). n
non rationnellement
appartient donc aussi ~
bans la fibration universelle est donc l'isomorphisme
~
~ ¢ Der (Y) n
A 2.
d(A 2) = O.
La
~-trivialit@
d(Y) C AY $ (A 2 ~ AY).
O
Comme
avec
+
~
Hn(E)
j~
Hn(F)
admet (EFT])
n ....
un module du
A = @ @ A 1 $ A2, AI.A I C A2, bans le module minimal
hX ÷ A,
de l'op@ration montre qu'on peut
Consid@rons alors la courte suite exacte
---+ Ker q
~ (A @ AY,d)
Ker q = (AI$ A 2) ~ AY
et
---+ (AY,d)
>
O.
d IKer q = d A O I = l @ d,
on obtient
la longue suite exacte +
oN
A
EH+(B)
0
est induit par
H~(F)] n
'
Hn(E) J---+ Hn(F) ~
d2 : Y ÷ A 20
Y. •
[H+(B)
0
H*(F)] n+1
. . . .
163
R~arque H~(B)
et de
:
H~(F).
Si
on a
est de dimension finie, il e n e s t
En effet, d~composons
H+(B) = HI(B) $ H2(B) HI(B) O H~(F)
H~(E)
sous la forme
~ partir de la d~composition
s'injecte alors dans
dim H~(F) < ~.
H+(B)
H~(E).
Ii en r~sulte que
de m~me de
A + = A 1 @ A 2.
Com~le dim H~(E) < ~
et
HI(B) # O,
dim H~(B) < ~.
Ceci soul~ve la question suivante
:
Ques£J.on : si
est une fibration avec holonomie
triviale et
F
dim H~(E) < ~,
J > E
-P-+ B
a-t-on
§ 7 - Operation d'holonomie ~
dim H~(F) < ~ ?
o p e r a t i o n de Yoneda.
7.1.- Pour toute fibration Eilenberg et Moore ~-M]
F -~
E --p-+ B
entre espaces l-connexes,
ont construit une suite spectrale v~rifiant
E 2 = ExtH~(B)(H
Rappelons que l'application
p
(E),~) =>
H (F;~).
est dit f o r m a ~ a b l e
(EL~)
s'il existe
un diagramme commutatif
H (B;~)
~
H (E;~)
"
(~E,dE)
P
(~B,dB)
avec
~I _
et
-@2
Si
p
d'Eilenberg-Moore
des quasi-isomorphismes
est formalisable,
alors,
"collapse" au terme
E2
et
p
(IVY),
un module minimal de
la suite spectrale
et l'on a
p.
164
Ext
(H (E),@) ~ H (F;~). H~(B)
Sous cette m~me hypothgse de formalisabilit~,
Ext H~(B)
Th~or~me 7 . 1 . -
Si
H (~B) @ H.(F) ~--~-+ H (F)
p
B
est formel et l'on a
(@,@) = H (~B;@) ~ .
e s t format%sable, l'op~ration d'holonomie
colncide, via les isomorphi~mes precedents, avec l'op~-
ration de Yoneda Ext
(~,@) 0 Ext H~(B)
~0~£/ta£210N H~(B)-module et si
P'
(H~(E),~) ~ Ext H~(B)
:
Si
P
est une r~solution projective de
est une r~solution projective de
alors la cooperation de Yoneda
(H~(E),~). H~(B)
A
sur les
Tor
~
H~(E)
comme
comme
H~(B)-module
est d~finie ([Le])
par la commu-
tativit~ du diagramme suivant :
A
Tor H~ (B) (H~(E),~)
~~l@g-1
H~(P ~@ ~) H (B)
o~
g
%
d~signe l'augmentation
Notons et
, H (P ~@ P') H (B)
÷
TorH~(B)(@,@) @ TorH~(B)(H~(E),~)
+
H~((p ~@ ~) H (B)
@ @
(P' ~@ @) H (B)
P' ÷ ~.
(TLX,d) -~ (AX @ AY,d) + (AY,d)
(AX,d) -~ (~X @ AX,D) ÷ (AX,O)
un K.S. module de la fibration
p
est un K.S.-mod~le de la fibration des chemins
PB÷B. Comme (H~(B) @ AY,d)
B et
est formel, il existe un quasi-isomorphisme (H~(B) @ AX,D)
(AX,d) ~
sont alors des r~solutions projectives
(H~(B),O). P
et
P'
165
de
H~(E)
et
~
comme
H~(B)-modules.
traduction de la d~finition
7. f.- C0ro//a/re
Soit +
a/0rs
H une
(3.1) de l'op~ration de l'holonomie, i
(version gradu~e d'un th~or~me de Roos (ER]).
~ - a l g ~ b r e gradui~e connexe. S i
ExtH(~(H+)m,Q) e s t an ~)[m0yL~t%~on
sh.sh' = 0,
sh.h' = O,
La projection un quasi-isomorphlsme.
0
montre alors que
Le diagramme ci-dessus devient alors la
:
Ext H(@,@) -module ~ b r e
Notons dh = O,
(H S s(H + )nl,d)
(H+) m # 0
(H+) m+! = O,
et
engend~ p~
ExtlH(H/(H+)m,@~).
ii a.d.g.c, d6finie par :
dsh = h.
q : (H @ s(H+) m) ÷ H/(H+)m_
de noyau
(H+) TM % s(H+) m
est
La eourte suite exacte :
÷
(H,O)
(H,O)
÷
(H $ s(H+)m,d)
i> (H/(H+)m,O)
÷
(s(H+)m,o)
+
0
est un module de la eofibre d'une appli-
cation. Ce module ~tant formel, l'homologie r6duite de sa fibre homotopique est + EXtH(H/(H+)m,~) , qui est donc (§ 5) un EXtH(~,~)-module fibre engendr~ par s(H + ) m .~ Ext HI ( H /(H+)m,~ ) .
7.3.gradu~ de
H,
•
C o r o l l a i r e . - S o i t H une @-a/g~bre gradu~e connexe, I un i d e a l + a/ors Extl(H,@) e s t un E x t ^ ( ~ , @ ) - m o d u l e l i b r e engendr~ par H/[ , I
D~monst~ation :
Ceci r~sulte du § 5 et de la courte suite exacte
0
÷
I
+
H
÷
H/I
÷
0.
166
§
8
-
S u i t e s p e c t r a l e d'holonomie d ' u n e ~ i b r a ~ i o n . Soit
~ : F
J~ E
un K.S.-modgle minimal et
L
P' B
une fibration (AX,d) i ~
l'alggbre de Lie de
En filtrant l'a.d.g.c.
(AX @ AY,d)
(AX N AY,d) ~
(AX,d).
par la longueur des mots en
on g~ngre une suite spectrale du premier quadrant v~rifiant
E~ 'q =
et convergeant vers
P,q El
s'~crit :
H~(flX ~ flY,d).
dl= dL + d~
avec
dL
d~finie par la representation
d@(~) = (xi)Ie I
X
iel
La diff~rentielle
@ u..~ 1
dl:
la diff~rentielle de : h ÷ Der HX(AX,d)
x.
,
(flY,d)
~
~
EAPx +
X, H~ (Ay,d~ p+q
E~ +l'q
(flX,dL) S C~(L)
et
d~
avec,
H~(AY,d).---
1
d~signe une base de
X
et
(u. g L)
la base duale :
i
<xi'suj> = ~ij " ([GHV~, § 5.25)
Ii r~sulte alors clairement de
E~ 'q
~ ~ q ( L ; H ~ ( F ) ~ p+q
que
Ext~q(@,H~(F)).
Nous pouvons donc ~noncer :
Proposition 8. I . - I1 e x i s t e une su~%e spectAale du premier quadrant
E p'q = E x t H ~ B ) ( C , H ~(F))
=>
H ~(E)
que n o ~ appelons s u i t e s p e c t r a l e d'holonomie de l a fibrat~on
~.
R~m~que • La formule de dualita de Cartan-Eilenberg construit une suite spectrale duale :
H(~B) E2 = Tor P,q P,q
Corollaire 7.- S i
(~,i (F))
F~
E~
~>
B
i (E).
e s t une f i b r a t i o n ,
et si
B ale
£ype
d'homotopie rat~onnelle d'un bouquet de spheres, alors l a s u i t e s p e c t r a l e d'holo-
167
nomie d~g~n~re au terme
E2 : H(~B) H(E)
Coro££~6~e
f i b r a t i o n avec r i e n , alors
2
H(E)
de finitude sur I
H~(E)
B
:
H~(F)
SoY~i
F
e s t un
÷
E
÷
B
une
H (~B)-mod~e no~h~-
Notons
(fiX,d) un module minimal de
montre qu'il existe un quasi-isomorphisme AX
(AXil~ @ AY,D)
v~rifie
E~ = O p,~
D'autre part, les espaces
(R~ciproque du th~or~me 4.1).-
e s t de dimension f i n i e .
est un ideal de
pour module
(H,(F),~).
B un C.W.-complexe f i n i . S i
D@mons£ra£ion
o~
= Tor
contenant = d~f
(~/l,d)
pour
H~(F)
~ : (AX,d) ÷ (AX/I,d)
pour un certain
@ (AX O AY,d) (AX,d)
L'hypoth~se
r.
E
admet alors
et le gradu~ associ~
p > r.
~tant un
ToriH~(~B)(H~(F),@)
est donc de m~me de chaque
A~rx
B.
H (~B)-module noeth~rien,
sont de dimension finie pour chaque
i.
II en
Ei, ~.
BI
BL I OGRAPHI
ANDREWS P.and ARKOWITZ M.
E
- Sullivan's minimal models and higher
order ~nitehead products. Can. J. of Math. 30, n ° 5 (1978), 961-982. AVRAMOV L. and HALPERIN S. - Through the looking glass : A dictionary between rational homotopy theory and local algebra (These proceedings). B~GVA D
R.
-
Graded Lie algebras in local algebra and rational homotopy. Thesis Stockholm (1983).
DGMS]
DELIGNE P.,GRIFFITHS
P.,
MORGAN
J.
and SULLIVAN
D.
-
Real homotopy theory
of K~hler manifolds. Invent. Math. 29 (1975), 245-274.
168
EILENBERG
S. a n d
MOORE
J.C.
Homology and fibrations
-
I. Coalgebras,
cotensor
product and its derived functors. Comment. Math. Helv. 40 (1966), FELIX Y. and
HALPERIN
S.
-
199-236.
Rational L.S. category and its applications.
Trans. A.M.S.
273 (1983),
|-37.
FELIX Y., HALPERIN S. et THOMAS J.C. - Sur certaines
alg~bres
de Lie de
d~rivations. Ann. Inst. Fourier, FELIX Y. et THOMAS
J.C.
GANEA T.
|43-150.
Sur la structure des espaces de cat~gorie
-
A para~tre
a-O
32, (1982),
2.
Ill. J. of Math.
- A generalization
of the homology and homotopy
suspension. Comment. Math. Helvet. 39 (1965), 295-322. GANEA T.
- On monomorphisms Topology,
EGo]
GOTTLIEB D.
in homotopy
Vol. 6, (1967),
theory.
149-152.
- On fiber spaces and the evaluation map. Ann. of Math. 87 (1968), 42-55.
FG.H.V
GREUB
W.,
HALPERIN
S. a n d
VANSTONE
R.
cohomology
-
Academic Press, GRIVEL
P.P.
-
Connections,
1976.
Formes diff~rentielles Ann. Inst. Fourier,
Su|]
GULLIKSEN T.
et suites spectrales.
29 (1979),
17-37.
- A change of ring theorem with applications Poincar~ Math.
~u-2~
curvature and
III.
GULLIKSEN T.
to
series and intersection multiplicity.
Scand. 34 (1974),
167-183.
- On the Hilbert series of the homology of differentiel graded algebras. Math.
HALPERIN
S.
-
Scand. 46 (1980),
15-22.
Lectures on minimal models. M~moire de la S.M.F. n ° 9/10 (1983).
HALPERIN
S.
et LEMAIRE
J.M.
-
Suites inertes dans les alg~bres de Lie.
Preprint
(1983), Nice.
(To appear in Math. Scand.)
169
LEMAIRE J.M. et SIGRIST
F .
-
Sur les invariants d'homotopie rationnelle
li@s ~ la L.S. cat@gorie. Comment. Math. Helv. 56 (1981), 103-122. [Le]
LEVIN G.
Finitely generated Ext-algebras.
-
Math. Scand. 49 (1981), 161-180. [Me]
MEIER W.
Some topological properties of K~hler manifolds
-
and homogeneous spaces. Math. Z. 183, (1983), 473-481.
[0p]
OPREA J
- Infinite implications in rational homotopy theory. To appear in Proceedings of A.M.S.
[Q]
QUILLEN D.
- Rational homotopy theory. Ann. of Math. 90 (1969), 205-295.
R]
ROOS J.E.
Homology of loop spaces and local rings.
-
Proc. of the 18 th
scand, congress Math.
Aarhus (1980).(Progress in Mathematics, n ° 11, Birhguser, 198].) [St]
STASHEFF J.
Parallel transport and classification of fibrations.
-
Lect. Notes in math. N ° 428, (1974). ES~
SULLIVAN D.
- Infinitesimal computations in topology. Publ. I.H.E.S. 47 (1977), 269-331.
TANR D.
Homotopie rationnelle : ModUles de Chen, Quillen,
-
Sullivan. Lect. Notes in Math. n o 1025 (1983), Springer Verlag.
IT@
3.c.
Rational homotopy of Serre fibrations.
-
Ann. Inst. Fourier 31 (1978), 71-90.
[v]
VIGU M
-
R~alisation de morphismes donn@s en cohomologie et suite spectrale d'Eilenberg~ioore. Trans. A.M.S. 265 (1981), 447-484.
[W_]
WHITEHEAD G.
- Elements of homotopy theory. Graduate texts in math.
Yves
F E L I X
(I 978), Springer Verlag. Jean-Claude
T H 0 MA
S
UNIVERSITE CATHOLIQUE DE LOUVAIN
UNIVERSITE DE LILLE I
1348 - LOUVAIN-LA-NEUVE
59655 - VILLENEUVE D'ASCQ CEDEX
(Belgique)
(France)
Flat families of local , artinian algebras with an infinite number of Poincar@ series
by
Ralf FrSberg, Tor Gulliksen and Clas LSfwall.
Introduction. For a local ring (R,m,k) let PR(Z) denote the Poincar@ series i~0dimkTor~k,k)z i. The origin of the present work is a question how Poincar6 series may vary in a flat family of local artinian k-algebras. In particular we were interested in knowing if such a family might have an infinite number of Poincar6 series. We will show that this is indeed the case by exhibiting a one-parameter family {Rl}16 Q of local artinian Q-algebras of length 85 such that the corresponding Poincar6 series form an infinite set. We also get as a bonus an example of an augmented 2-algebra A, free of rank 85 as 2-module, such that A/p are local rings and PA/p(Z) are different for all primes p, and also TorA(2,~) has p-torsion for all primes p.
It turns out that it is possible to construct families of local artinian q-algebras whose Poincar6 series vary quite vividly and depend on various algebraic and/or arithmetic properties of the parameters. For instance we show that there exists a family {RI}16 C and a power series f(z) such that PRI(Z)= f(z) if and only if I is transcendent over Q, and that there exists a family {$I}16Q2 such that the calculation of PS (z) for all I is equivalent to solving Fermat's equation n n 11 + I 2 = I for all n and 11,126~. It was natural for us to start looking at local k-algebras (R,m,k) with m 3 = 0. Let C be the class of such k-algebras and let B
be the
class of algebras of type k/(gl,...,gs) , where k p
is the free associative (non-commutative) algebra and the gi s are linear combinations of the elements T~,m I S i S n, and TiT j + TiT i ,
171
I ~ i < j ~ n. It follows from results of LSfwall [LS] that if we can construct a family {B l} in B with infinitely many Hilbert series Bl(z) = i>0Xdir~(B~).zIA i , we get a family {A I} in C with infinitely many Poincar@ series. Each element BEB is the universal enveloping algebra U(G) of a graded Lie algebra G. Anick and L6fwall-Roos, see [L6-Ro], have a construction, which to any graded (non-commutative) algebra N gives a graded Lie algebra G, such that U(G)(z) is determined by N(z). If N is generated by elements of degree one and has relations of degree two 0nly, then U(G)EB. Thus we have a construction available, which to any family of non-commutative algebras with generators of degree one and relations of degree two and with infinitely many Hilbert series gives a family of confutative local rings with m 3 = 0 and with infinitely many Poincar@ series. For this reason we were lead to the study of noncommutative graded algebras. Exhibiting a family in B with infinitely many Hilbert series also makesit possible for us to construct a family of topological spaces {XI} , in fact mapping cones of maps between wedges of spheres 6~S 3--~-~ yDS2, with 1 I infinitely many series Z ( d i m ~ w . ( X ~ ) z l, w. denoting homotopy groups. i~O ~ i ~ l
I. Poincar@ and Hilbert series of families of sraded k-alsebras. A graded algebra will in this paper mean an algebra which has a presentation k/l. Here k is the free associative (non-commutative) algebra in the variables TI,...,T n of degree one and I is a homogeneous two-sided ideal in k , k a field. Of special interest to us will be the case when I is generated by elements of degree two. We call such algebras 2-related. For a graded module M = ~ M. over a graded algebra we define the i~0 l Hilbert series of M to be i M(z) = i~odimkMi'z . The set of elements of positive degree in a graded module M will be + denoted M .
172 For an augmented k-algebra A (or a local ring (A,m,k)) we define the Poincar~ series of A to be
PA(Z) = Z di~Tor§(k,k)'z i i~0
~
l
By a famiily of k-algebras {A~}, ~ = (XI,...,~m)C k m, we will mean a set of k-algebras together with a finitely presented k[X]-algebra A, X = (XI,...,Xm) , such that AX =A/(X-X)
for all ~ E k m. The family {A~} is
called a flat family if A is k[X]-flat. We call the family ~raded if A has a presentation A=k[X]/(fl, .... fr ), Y = (YI,...,Yn), where the fi's are homogeneous in Y. Thus all A~ are graded k-algebras in a graded family {A~}. Finally we call the family commutative if A is commutative.
First we examine how Hilbert series A~(z) may vary in a commutative graded family {A~} of algebras. Claim. In a commutative graded family there are only finitely many Hilbert series. In fact a much more general statement is true as the following proposition shows. We note that in a graded family {A~} there is a uniform bound for the number of generators and the degree of the relations, namely if A can be presented as a free algebra in n variables over k[X] with relations of degree Sd in Y, then each A~ has a presentation k modulo forms of degree Sd.
Proposition
I. Let n and d be fixed integers. There are only finitely
many possibilities for A(z) when A belongs to the class of graded algebras of the form k[Xl,...,Xn]/(f I .... ,fr ) where k is a field and the fi's are forms of degree Sd. Proof. Let B=k[XI,...,Xn]
and let A be a graded factor ring of B. The
syzygy theorem of Hilbert states that A has a minimal graded resolution b bI 0-~ i~_~B[-ni,r ] --~...--~ i~=iB[-ni, i]~-~ B ~-~A.-~ 0 for some r S n, where the brackets stand for a shift in degree,
(i)
173
(B[-k]) d = B_k+d. To construct a step in this resolution is equivalent to solve some linear system of equations with coefficients which are forms in B. It is shown in [He] (also c.f. [Se] and [La]) that there is a bound M, only depending on n and the degrees of the coefficients in the system, such that all solutions can be generated by solutions of degree the resolution is of length
~ M. Since
S n, this gives a bound N = N(n,d) for all n.
Since the degrees of the syzygies are uniformly bounded it follows that the number b. of syzygies are uniformly bouude4. In each fixed degree the resol
lution (I) is an exact sequence of vector spaces and thus their alternating sum of dimensions is zero. Taking generating functions we get the formula b~Izni,1 + ~2zni,2 _ ... + (-I )r ~rzn i ,r)/(1 _z)n i=I i=I i=I and hence we see that there are only finitely many possibilities for A(z).
A(z)=(1-
We are interested in the following property of a commutative graded family {AI}:
(P) The set {PAl(Z)} of Poincar6 series is finite.
We will show that there exist flat families of local graded artinian k-algebras not satisfying (P). There is another, seemingly weaker, property for a commutative graded family {At}:
AI AI (P') There is a number N such that, if dimkTor i 1(k,k) = di~Tor i 2(k,k) for all i ~ N ,
then PA
(z) = P A 11
(z). 12
In fact (P) is equivalent to (P') for a family {A~}. Of course (P) implies (P'). But if {A~} is a graded family, then there are only finitely many AI possibilities for dimkTor i (k,k) for fixed i. This follows by the same reasoning as in the proof of proposition I: Let A be a graded k-algebra in
174
n variables and with relations of degree ~d. Constructing a step in a graded free A-resolution of k is equivalent to solving a system of linear equations over A. This can be lifted to B. By induction over i it follows from the theorem of Herrmann mentioned above that there are only finitely many possibilities for
N A zi ' Z dim. Tor. X(k,k) Hence (P') implies (P). i=O ~ i
In the study of non-commutative
families of graded algebras we will
be interested in the following two properties for a family {BI}: (H) The set {Bl(z )} of Hilbert series is finite. (H') There is a number N such that, if dimk(B~1 )i =dimk(Bl2)i then
B),I
(z )
= BI2
for all i ~ N ,
(z ).
The properties
(H) and (H') are equivalent for a family {B)). Of
course (H) implies (H'). If n is a bound for dimk(B~) I for all I, then dimk(Bl) i ~ n z, thus there are only finitely many possibilities for i=0
dimk(B~)izi
hence (H') implies (H).
In next section we will show that (H) is not satisfied for all families of non-commutative algebras.
2. Hilbert series of non-commutative graded families. In this section we give two methods of constructing graded algebras with badly varying Hilbert series.
Construction
I. Let A be a graded (non-confutative)
algebra and let
A L and AR be two graded vector subspaces of A +. Let T be the coproduct of A with k/(a2). If W = {I}UW + is a graded k-basis for A, then +
+
+
{WaW aW ...aW aW} is a graded k-basis for T. Let I~-T be the two-sided ideal generated by aA L and ARa and let ~ = T/I. We note that if A is 2-related and AL,ARCAI,
then ~ will be 2-related. Let W I (and W3, respectively) be a graded
has, for a complement to AA R (and ALA , respectively)
in A and let W 2 be a
graded basis for a complement to ALA + A A R in A +. Then a k-basis for A is WU{WIaW2aW2...W2aW 3) and hence A(z) =n~0W1(z)W3(z)zn+1(W2(z))n
+
A(z) =
175
(A(z)-AAR(Z))(A(z ) -ALA(Z))Z[I ~ ( A ( z ) - I - ( A L A + A ~ ) ( z ) ~ -I + A(z)
Example I. Let A=k/(bc-cb-lc2),
16k. As a k-vector space A
(2)
is generated i
by c i ,c i-lb, ...,cbi-l,bi hence A(z) S (l-z) -2. In fact we have equality since k/(bc-cb) :k[b,c] is made to a cyclic left A-module by b*cib j =cib j+1 +ilci+Ib j and c.cib j =ci+Ib j and hence A(z) ~ (l-z) -2.
Let AL: (c-b)k and AR : b . k ,
then
ALA(Z) : AAR(Z)=
z(l-z) -2
since, as is easily seen, (c-b)ci,(c-b)ci-lb,...,(c-b)b i (and cib,ci-lb.b .... , bm.b, respectively) are linearly independent. Finally we have (ALA+~&R)i+ I = Aib + (c-b)ci-k = Aib + (1-iX)c i+1.k. Thus if
~=0
or I-I~{1,2,...}
we have
(ALA+AAR)(Z) = (l-z) -2 - Io If chark = 0 and I-I : q6 {1,2,...} we have (ALA+AA R) (z) : (l-z) - 2 - I - zq+1. If chark = p and I -I = q 6 {1,2,...,p-I} we have
(A~A+AAR)(Z) = ( I-z)-2
I - zq+1(1-sP) -I .
Hence if I= 0 or I-I ~ {1,2,...}, then A. =k/(a2~ be-cb-lc2,ac-ab,ba) it
has Hilbert series (by formula (2)) At(z) = (1-z)-2(1+z). If chark = 0 and I-I = q 6 { 1 , 2 .... }, then
~1(z) = (1-z)-2(1+z-z
q+~ • ){1-zq+2) -I .
Finally, if chark = p and I-I = q 6{1,2,...,p-I}, then Ax(Z) = ( I-z)-2( 1+z-zP-zP+1-zq+~)(I-zP-zq+2) -I .
If we replace k by • and put ~ = 1 we get a ~-algebra B = ~/(a2,bc-cb-c2,ac-ab,ba)
(3)
176 such that the Hilbert series of B/pB are different for all primes p (and also different from the series of B ® g ~ ) .
This phenomenon can not occur
in the commutative case according to proposition I.
Example 2. Let A=@/(bc-~cb),
~E@
and let A L = A R = (c-b).~. This gives
A~= ~/(a2,bc-~cb,ba-ca,ab-ac). It is possible to compute ALA(Z) , AAR(Z) and (ALA+AAR)(Z) as in example I to get
~(z) = (1+z-z2)(1-z2)-1(1-z) -2
if In # ] for all n > 0 and Al(z) = (1+z-z2-zn+2)(1-z2-zn+2)-1(1-z) -2 if ~ is an n'th primitive root of unity.
Construction 2. (This is an alternative to construction I, it yields algebras with smaller Hilbert series but needs one more generator.) + Let as before A be a graded algebra and A L and A R be subspaces of A . Let T* be the coproduct of k/(L 2) and A and k/(R 2) modulo the twosided ideal (*L, R*)
where * stands for anything of positive degree. As a
graded vector space this algebra equals
k/(L2)~k Let I # be t h e t w o - s i d e d
A ~ k k/(R2). ideal
generated
by LAL and ARR. Then
I" = LALA + AARR + L(ALA+AAR)R and if ~ = T*/I" then ~(z) =A(z) + z ( 2 A ( z ) - A L A ( Z ) - A A R ( Z ) ) + z2(A(z)-(ALA+AAR)(Z)) •
(4)
We note that if A is 2-related and AL,ARCAI then A will be 2-related.
To be able to give simple descriptions of the spaces ALA, AAR and ALA+AA R we will put restrictions on the algebra A. If B is an algebra and M a B-bi-module, the trivial extension B U M
is B ~ M
as vector space and
has multiplication (b,m)(b',m') = (bb',bm'+mb'). From now on we put
A = B ~ ( V ~kB), where B = k<S> and S is a finite set of elements of degree one and V is a
177 k-vector space of finite dimension. We make V @ k B a B-bi-module in the following way. For each s6S there is given a linear transformation Js: V--~V. To each monomial B = SlS2...s n in B we consider the composite map JB= Js1°Js2 °'''~Jsn" For B = I we let JB denote the identity map. This defines V as a non-6raded left B-module by B.v = JB(v) and k-linear extension. For each s6S we define s(v@ b) = s v ~ s b which extends as above to an operation of B to the left on V ® k B. The operation of B to the right on V ® k B is the obvious (v® b ) b " = v e ( b b ' ) .
these two operations are compatible. To define V ~ k B module we let the degree of v ~ b
as a ~ B - b i -
be I + d e g b , that is we consider V as
concentrated in degree one. With this definition V ~ k B indeed becomes graded, since deg(b1(v @ b2) ) = deg(blVgblb2) = deg b I +deg b 2 + I = deg b I + deg(v~ b 2) and deg((v~ bl)b 2) = d e g ( v ~ b l b 2) = deg b I + deg b 2 + I = d e g ( v g b I ) + deg b 2 •
Proposition 2. Let E be a basis for V. The algebra B~/((ee ~,se-J s(e)s; e,e'6E, s6S}) and hence B ~ < ( V ~ k B) is 2-related. Proof. First consider the surjection p: k<SUE>
>BW(V®kB)
defined by p(s) = (s,O) for s6S and p(e) = (O,e®1)
for e6E. Then
p(ee ~) = (O,el~ 1 ) ( 0 , e ~ ' ~ 1 ) = ( 0 , 0 ) and p(se-Js(e)s) =p(s)p(e) -p(Js(e))p(s) = = (s,O)(O,e~ I) - (O,Js(e)~1)(s,O) =
178 = ( 0 , s ( e @ 1 ) ) - (0,(Js(e)@ 1)s)= (0,Js(e)@ s)- (0,Js(e)@ s ) = 0 . Thus p factors through H and we get
H(z) ~ B~(V~kB)(z). Using the relations ee" = 0 only, we see that words of type Wlelw2e 2...wrerwr+1, ~here w.1 is a word in S and e.EEI for all i, generate H. Using also the relations se = Js(e)s we see that it suffices to take words w I and ew2, where Wl,W 2 are words in S and eEE, to generate H. Hence
H(z) S B(z) + zdimkV.B(z) = B ~ ( V ® k B ) ( z ) . We have shown t h a t i(z)
: B~(V@kB)(z)
which g i v e s H ~ B ~ ( V @ k B ) s i n c e B~(V(~.B)K i s a f a c t o r o f H. To be a b l e t o a p p l y t h e c o n s t r u c t i o n +
two subspaces A L and ~
1 ( o r 2) we need t o s p e c i f y
of A . We will choose these as subspaces of
VcAI and call them V L and V R respectively.
Let {~} be the k-basis of
monomials for B and let B~ = B'k. We have ALA = V L ~ B = ~ (VL~Bs) and AA R = im(B~VR--~ V ~ B ) = ~ J s ( V R) @kBs whence
A~A+~ R :~( (%(V R )+V~)':~kB~). Taking k-dimensions we get (~)
ALA(Z) = z.dimkVL.B(z) AAR(z)
= ~di~JB(VR).zdeg B +
(ALA+AAR)(Z) = ~dimk(Js(VR)+VL).zdeg B + I If J $ is an isomorphism for all sES
(7)
then (6) may be simplified to
~(z) = z.dimkVR.B(z) The formulas (5), (6) (or (6")) and (7) may be inserted in either epnstruction
(6)
I or 2, formulas (3) or (4), to get formulas for A(z) or
(6")
179
~(z) depending only (if Js is iso for all s) on dimkV L, dimkV R, the number of elements in S and the crucial term • ~d~mk(J~(V R)+VL)z1+deg We have ~(z) = (1+zl2B(z)(1+zdi%V)-
z2(di%VL.B(z) + ~di~Je(VR)Z aeg e) _
_ z3(~di~(JB(VR)+VL)zdeg B
Suppose now B = C<s>, the free algebra over ~ on one generator of degree one, and suppose V = ~2. Then there are two canonical forms for Js' namely
I ~2
and
. This motivates the first two examples
below. The properties of the rings in examples 3 and ~ are similar to those in examples I and 2. The set of algebras obtained in examples I and 2 have one generator less, they can not A=B~(V@k
however
be'obtained as A where
B) for any B and V.
Remark. We note that the above reasoning is also valid in the more general case when B is a graded monoid algebra, that is B = k<S> modulo a set of monomials and differences of monomials of equal degree. If the monoid algebra is 2-related, then A will be 2-related. In particular we could have 2 2 used B=k[Sl,S 2 .... ,sn] or B =k[sl,s2,...,sn]/(s~,s2,...,sr).
This of
course imposes restrictions on the maps J . s
Example 3. Let B=~<s>, V = @ 2, VR:(I,0).~, ~ ~n Then OTnf1~ s ~ =~n~n-1
0
r~t~ ~nn 1 I=
21
VL=(I,1)'@
and Js =(~
,hence dim~(J~(VR)+V L) = I if X = n > 0
~)" or
X = 0 and n > I~ and dim@(J~(VR)+V L) = 2 otherwise. Thus we have a family {Ax) with "generic" value of Ax(z) if X ~
and with A~(z) different for
all X@JN. (When ~ = 0, Js is not iso so the series AAR(z) has to be computed by means of (6).)
180
1 I,
~I I" Hence dim¢(Jns(VR)+VL) = 2 if
#I and
dim~(~s(VR)+V L) = I if In= I. This gives a family {Ax} with "generic" value of A}(z) if and only if ~ is not a root of unity.
Example 5
Let B=~<s>, V=¢3, VR •
and let J
= s
(I 3 1 3 ~ 3~'~ ="
I"
2'
VL
3 ~ ~'
(I,0,-I)~+ (0,I,-I)~ =
(~ 0 0 i 01 X 0 I Then Jns(VR)= (xn+3 ~n+3 >n+3~ 0 02 13 ! " ~ I ' 2 '- 3 ~.~ and
dimQ(jn(VR)+V L) = 2 if the determinant In+3
I
0
0 _~+3 _ I
I -I
xn1+3
1
n+3 + ~n+3 n+3 . n = 3 otherwise. = 11 2 - ~3 = 0 and dlm~(J s (V)+V_) R L
Moreover dimQ(J~(VR))= I if (~I,X2,~3) # (0,0,0). For 13= I this gives exceptional values of At(z) if and only if Xn+3+~n+3 I -2 = I for some n. We have one value of At(z) for I= (~i,~2)= (1,0) or (0,1), another value for X= (-1,0) or (0,-I). The statement that At(z) is independent of X for all other values of i is equivalent to Fermat's last theorem.
Example 6. Let {~0=I,~I,...,~N } be the set of monomials in {~I .... ,~m } of degree Sd, let S= {s0,...,SN,S~,...,s~} , let B=@<S>, V=@2, V L = V R = =(1,0)-¢ and let Js z= Ii
and J s.,= i
B b e a m o n o m i a l i n S, t h e n JB = l ( a )
-~i
I
for i=0,1,...,N. Let
where l(a)
is a linear
combi-
nation of ~O,...,~N with coefficients in g. Any such linear combination can be achieved by appropriate choice of B. Thus we have exceptional values for A~(z) if and only if i(~) = O, that is if and only if (~1,...,~m) satisfies a polynomial equation of degree Sd. If we restrict to m = I we get a family {A~} with exceptional values of A~(z) for algebraic numbers ~ of degree Sd, and if we restrict further and also let d = I, we get exceptional values for IC~.
181
Example 7. Let S= {So,S~,Sl,S2,S2,S3,S3,...,Sm,Sm} , let B=@<S>, V=C2, V L = V R = (0,I).~ and let • j
So= 0
1
I -I}
s~ =
I} ~ Jsi ={Ol~ ~} for i = 1,2 ..... m and Jsf=l 01
for i= 2,3,...,m. If B is a monomlal zn S, then JS = 0
" where
p(A) is a polynomial in {AI,A2,...,km,A -I 2 ,A-I 3 " " ' k m -. } and we have an exceptional value of AA(z) if p(A)= O. Claim. We can get any polynomial p(A) Cg[AI,...,X m] in this way. Proof. First we see that we can get any q(~)E~[~,...,A m] in this way. Let ~ be any monomial in A2,...,Am, then
,0
0lj~ Is obtained as an appropriate product of the Js. "s, it follows
Since (0
that there is a B such that J~ =' 0 Also if
j
i BI = 0
lJ"
and J@2 =110 Y) then j@182= 0I X l ~ " Hence any matrix of
the type (I0 q ( ~ ,
q(A) 61[~ 2 ..... Am] is obtainable as a J B. Suppose now
d • ,Xm) D(A) :Do(A2 ..... Am) +AIpI(A 2 ..... Am) + ... ÷ X1Pd(A2,..
then p
~
0
1
and the claim is proved.
In the presentation of A1 which follows from proposition 2 the only relations containing
,...
are the relations s e I = A
for i =
m
2,3,...,m. If we replace these relations with ~is~el =else, i =2,3,...,m, we get a family { ~ } with exceptional values of A~(z) for all ~ = (~1,...,~m) which satify some polynomial equation over Z at least if AI~2...A m # O. But if AIA2...A m= 0 we get, as is easily checked, a value of dimk(A~) 3 which does not agree with the generic
182
value. Hence the family {A~} has exceptional values of A~(z) if and only if ~= (~1,...,Xm) satisfies some polynomial equation over g. In particular when m = I we get a one-parameter family {A~}~@ with exceptional values of Ax(z) if and only if ~ is an algebraic number over ~.
3. Families of graded Hopf algebras with infinitely many Hilbert series. We now recall the construction in [L6-Ro] mentioned in the introduction. Let N=k/(gl,...,gt)
be a 2-related algebra. Let
ml,...,mn2_t be a set of monomials in {TI,...,T n} of degree two whose images in N constitutes a k-basis for N 2. To the algebra N we define a graded Lie algebra G in the following way. The Lie algebra G is generated by a set {T I ,... ,Tn,L I ,.. .,Ln,Y,R I ,...,Rn,Zml ,. "',Zmn2_ t} of variables of degree one and has the following relations (I)
T~=O,l I S i S n ,
(2)
[LI,T j ] = [Ti,Rj], 1 < i , j < n
(3)
~ c
i,j 10
and [Ti,Tj] =0 , I S i < j
[Li,Tj]= 0 if and only if
Z c..T.T.E i,j 1j i J (gl '" "''gt )
(4)
[Y,Ti] =0, I . The third homotopy group of a wedge of 2-spheres
m 2 v S., is known i=I i
to be a free abelian group on m(m+1)/2 generators (see e.g. [Gr-Mo, 2 2 p. 240]). The generators are the Whitehead products [si,sj], iS j, where s 2 denotes the canonical element in ~2($2). Thus elements in
m~ z3(~S ) may be identified with elements of ~<TI, .... Tm> which are linear combinations of TiT j +TjTi, i~ j. Let fl,...,fr be elements of
g which are such l i n e a r combinations. Then f t , . . . , f r
gives
rise to a continous map r 3 m 2 v S:---@ v S.. i=I ± i=I i let X denote the mapping cone of f. It is implicitely proved by Lemaire f:
that there is a rational correspondence between the Hilbert series of
190
~/(f 1,...,fr ) and the homology algebra of the loop space X of X, which we denote by H~(~X;~)
(see [Le, p. 64] and [Le ~, p. 117]
and for an explicit formula [Ro, p. 450, formula (9)]). It is a wellknown fact that H~(~X;~) is the enveloping algebra of the graded Lie algebra ~ (wi+1(X)@~) i=I a i = dimQ(wi+1 (X)~Q)
defined by the Whitehead product. Hence if
then
H~(~X;~) = U (1+z2i-1)a2i-1(1-z2i) -a2i. i=I We now choose the Hopf algebras U(G)~ from the end of section 3. From these we get a family {X~} of topological spaces, namely X~ is the mapping cone of a map f~: oo
69 3 15~2 vS~-~¥~ , with the property that the series I 7
i=idzm~ ( ~" ~ i + 1 ( X ~ ) ~ ) z i takes infinitely many values when ~ varies in ~.
Remark. During the work on this paper we have been informed that D. Anick independently has constructed a family of graded noncommutative algebras over ~ with infinitely many Hilbert series. He also has a family which generically is of ~lobal dimension 2. This can be achived also by our methods. More precisely, consider the following slight variation of our example I: Let D~ = k/(ab-ac,bc-cb-~c2,bd). 2 if and only if X = 0
or ~-I ~ (1,2,...).
Then D~ has global dimension
191
References.
[He], Herrmann G., Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), 736-788.
[La], Lazard D., Alg~bre lin$aire sur K[X I .... ,Xn] , et @limination, Bull. Soc. Math. France 105 (1977) no. 2, 165-190.
[Le], Lemaire J.M., Alg~bres connexes et homologies des Espaces des Lacets, Lect. Notes in Math. 422 (1979), Springer.
[Le'],Lemaire J.M., A finite complex whose rational homotopy is not finitely generated, Lect. Notes in Math. 196 (1971), 114-120, Springer.
[L6], LSfwall C., Une alg~bre nilpotente dont la s@rie de Poinear@-Betti est non rationelle, C.R. Acad. Sc. Paris 288 (1979), s@r. A, 327-330.
[L6-Ro], LSfwall C. et Roos J.-E., Cohomologie des algSbres de Lie gradu@es et s@ries de Poincar@-Betti non rationelles, C.R. Acad. Sc. Paris 290 (1980), s$r. A, 733-736.
[Mo], Mount K.R., Fitting~s invariants and a theorem of Grauert, Math. Ann. 203 (1973), 289-294.
[Ro], Roos J.-E., Homology of loop spaces and of local rings, Proc. 18th Scand. Congr. of Math., Progress in Math. 11 (198~)~ 441-468.
[Se], Seidenberg A., Construction in algebra, Trans. Am. Math. Soe. 197 (1974), 273-313. Ralf FrSberg and Clas LSfwall Department of Mathematics University of Stockholm Box 6701, S-113 85 STOCKHOLM (SWEDEN)
Tor Gulliksen Department of Mathematics University of 0slo Blindern - OSLO 3 (NORWAY)
A N O T E ON I N T E R S E C T I O N
MULTIPLICITIES
by Tor H. G u l l i k s e n
Let
(R,_m)
denote
be
finitely
generated
R
is r e g u l a r we d e f i n e
a local
[3 ].
If
DEFINITION. surjective
R
The p u r p o s e R-modules ding"
is not r e g u l a r
Let
~
of this note M and N,
REMARK.
If
R
be computed the P o i n c a r ~
n
is the
=
R
of
as
follows:
We c h o o s e
R.
is a r e g u l a r
xR(M,N)
a
local
only depends
of m i n i m a l
intersection,
regular
then
on t h e
"embed-
xR(M,N)
in the f o l l o w i n g way.
can
Consider
[] ] t h a t
= lim P ~ ' N ( t ) ( ] - t 2 ) n - d i m R t÷-I
over
dimension
of
also
R, i.e.
the d i m e n s i o n
of
R/m.
One may ask if it is p o s s i b l e
TorR(M,N)
If
~ l e n g t h ( T o r ~ ( M , N ) ) ti i;0 4.2 in
embedding
m / m 2 as a v e c t o r s p a c e
of
N
series
from c o r o l l a r y
QUESTION.
and
length.
from the t h e o r e m below.
TorR(M,N)
xR(M,N) where
M
Then we put
follow
of
xR(M,N)
completion where
is to s h o w t h a t
p~'N(t)
It follows
R -+ ~
is a local c o m p l e t e
in terms
let
finite
multiplicity:
and n o t on the c h o i c e
R ÷ ~. T h i s w i l l
has
we will d e f i n e
d e n o t e the m - a d i c
dimension.
ring and
M~N
= Ei(-l)ilength(Tor~(M,N))
ring h o m o m o r p h i s m
ring of m i n i m a l
such that
the i n t e r s e c t i o n
xR(M,N) as in
(noetherian)
R-modules
for rings
R
to e x p r e s s
xR(M,N)
which
are not l o c a l c o m p l e t e
local
(noetherian)
in t e r m s inter-
sections.
THEOREM. and
N
Assume
Let
C
be an a r b i t r a r y
be C - m o d u l e s that
A ÷ C
of finite t y p e and
B ÷ C
are
ring, has
and let
such that
M®N
finite
surjective
ring h o m o m o r p h i s m s ,
M
length. A
193
and
B
being
dimension
regular
of
A
and
local that
rings
of
B
of m i n i m a l equal
the
dimension,
embedding
that
is the
dimension
of
C.
T h e n we h a v e xA(M,N)
PROOF.
We m a y
Hence
so is the
A
XcB
have
ass~ae
that
A,
fiber-product
is a h o m o m o r p h i c a commutative
= xB(M,N).
B
and
A
XcB.
diagram
--~
a: = K e r ( R of
c m
R.
and
By means of
÷ A)
b c m
of t h e s e
m/m 2 .
and
Since
B and
give
rise
maps
local
local
structure ring
rings. theorem
R, t h u s
we
ring homomorphisms
B
, C
b: = K e r ( R
A,
complete
By C o h e n ' s
of s u r j e c t i v e
A
ideal
are
image o f a r e g u l a r
R
Put
C
R
are
÷ B). all
Let
m
regular,
b e the m a x i m a l the
inclusions
to injections a/ma
÷ m/m 2
b/mb
÷ la/m 2
we will
consider
a/n~
and
b/n~
as s u b s p a c e s
Put s = dim R - dim A
Since
dim A = dim B
both
elements.
Let
a] ,...,a r
respectively
5] ..... '~r
representing mal
sets
a
and
(r(s)
~ow
are
minimally
be a basis
b] , . . . , b r
5] ' "'''~r"
b
for
be elements
extend
these
generated
a/ma in
two
by
N b/mb.
~
s
Let
respectively
sequences
to mini-
of g e n e r a t o r s a],...,ar,...,a s bT,...,br,...,b s
for
a
and
b
respectively.
For e a c h
i
(04i<s)
the e l e m e n t s
a | , . . . , a i, b i + I, .... b s represent
linearly
of a regular they
generate
system
independent
of p a r a m e t e r s
each
for
in
m / m 2. H e n c e
R. L e t
~i
they are
denote
the
part
ideal
and put A
Then
elements
Ai
is a r e g u l a r
A s = A.
In the
following
suffices
to prove
1
:= R / a .
--i
local
let
ring.
|(i(s.
Ai_ ] X
Observe To prove
Ai (M,N)
= X
CM,N)
that
A0 = B
the t h e o r ~ n
and it c l a r l y
194
Here
we w i l l
pose.
To
be the
use
a technique
simplify
ring
~ich
the n o t a t i o n
R/[
where
£
was
we p u t
is the
used P
ideal
in
:= A
[2] for i-] and
generated
a similar Q
:= A
pur-
i Let
L
by
a l, .... ai_ I , a i , b i , b i + I, .... b s. Observe
that
L
need
not be
regualar.
a. 0--~ P ~
where ly.
ai
From
and the
bi
P-~ L-~
0
0 - ~ Q bi~ Q - + L - ~
O
denote
sequences
We h a v e
multiplication
above
we o b t a i n
by
exact
ai
standard
sequences:
and
bi
spectral
TorL(~i,TorP(N,L))p q
=> T O r p ~ q ( M , N )
Tor~(M,Tor~(N,L))
=> T o r p ~ q ( M , N )
respectivesequences
and
where equals
TorP(N,L) q z e r o for
• ..TorL(M,N)
Tor[C. from w h i c h
it
and
TorQ(N,L) q q % 0,1. Hence ÷ T o r i P] (M,N)
÷ Tori% follows
equals we
N
obtain
for exact
q = 0,1 sequences
÷ ToriL| (t,i,N) ÷ ToriLl (M,N)
MN) ÷
or2+ ,
÷
and
orih
÷ ...
*
that xP(M,N)
= xQ(M,N).
REFE RE NCE S
[}]
T.H. G u l l i k s e n , A c h a n g e of ring P o i n c a r ~ s e r i e s and i n t e r s e c t i o n (1974) }67-183.
theorem with applications to m u l t i p l i c i t y , Math. Scand. 34
[2 ]
M.-P. M a l l i a v i n - Brameret, Une r e m a r q u e sur les a n n e a u x r ~ g u l i e r s , S e m i n a i r e D u b r e i l - P i s o t ( A l g ~ b r e et T h ~ o r i e Nombres), 1970/]971 no. 13.
[3]
J.P. Serre, A l g ~ b r e L o c a l e M u l t i p l i c i t ~ s , Mathematics, ]], S p r i n g e r - V e r l a g , Berlin,
Lecture 1965.
Notes
locaux des
in
Department of Mathematics University of Oslo Blindern - Oslo 3 (nORWAY)
195
REDUCING
THE
T O THE
POINCARE CASE
SERIES
OF L O C A L
OF Q U A D R A T I C
RINGS
RELATIONS
by Tor
Let
(R,m,k)
be a c o m m u t a t i v e ,
a finitely
generated
the
power
formal
a given
attempts cases.
M
have One
R-module.
tion
of the
over
two
been made
Poincar~
different
R(M)
Later
G.
series
series
of
modules
over
In the where
the
artinian
DEFINITION. quadratic
where
we
relations
say if
R0
is a field
generated
by
ideal
that
it has
closed
the m × m - m a t r i c e s with
of
M
M
be
we m e a n
respect
P RM
and m a n y
to h o p e f u l l y 2] w h e r e
to the case
we h a v e
reduction
cf.
R
showing
simpler
computaM = k, b u t
the
formula
by
M.
how
the P o i n c a r ~
of P o i n c a r ~
series
of
[2]. the
artinian
quadratic
a local the
of
in terms
reduce
ring
case
to the
case
relations.
is d e f i n e d
by
special
form
Xm, Y I . . . . .
or a c o m p l e t e
is g e n e r a t e d
the q u a d r a t i c
case
reduced
extension
by certain
i,j where
was
rings,
shall
is d e f i n e d
We w i l l
maximal
let
k (|_tp~)-] = PR"
ring
R0[[X I.....
whose
and
series
in [ ], t h e o r e m
expressed
local
following
ring
ring
to c o m p u t e
the g e n e r a l
presicely
a beautiful can be
difficult
M
More
trivial
made
of a m o d u l e
very
appeared
rings.
is the
Levin
By the P o i n c a r ~
to r e d u c e
reduction
k PR(N) where
local
[ dimkTor~(M,k)t q p) 0
it is u s u a l l y
such
noetherian
series P RM =
For
H. G u l l i k s e n
Ym] ] / I
regular
by a p r i m e
local
ring
number,
and
of d i m e n s i o n I
is an
I ideal
forms ~..X.Y. 13 i 3 '
(~ij)
~ij
£ R0
run t h r o u g h
to t r a n s p o s i t i o n .
a set of m a t r i c e s
which
is
196
THEOREM.
Let
generated R*
÷ R
tions
R
be an a r t i n i a n
R-module.
where
Then
R*
local
there
is a local
ring
defined
PROOF.
By C o h e n ' s
properties
÷ R
and
let
by
~,i be
a finitely
surjective
special
homomorphism
quadratic
rela-
= p R pM
R*
R0
and
a local,
and pM
with
ring
exists
induces
R.
Let
in
theorem
above,
an i s o m o r p h i s m
v] , .... v m
an b-module, entries
structure
described
and
R* R R
and
is an a l g e b r a
such
between
that
the
the
residue
b e a set of g e n e r a t o r s
consider
the
following
over
a ring
structure class
for
m,
R0
map
fields
of
R0
considered
set of m x m - m a t r i c e s
as
(aij)
with
RO: A =
{(aij) I [ = i j v i v 9 1,]
= 0}
N o w put R* = R0 [ IX] ..... X m , Y ] ..... Y m ] ]/I where
I
where
(aij)
in
R*
is the
and
structure minimal lifted in
ideal
6 A. let
map
f:R*
resolution
to an R * - f r e e
m * , the m a x i m a l For
as a free q differential
with
entries
the
following entry
a I , . . . , am
way.
whose
in
R0
depending
we and
in
er~tries of t h e p r o d u c t where
aij
a way that
m,
replace , c =
the o t h e r
hand
of
since
F
[aijXiYj
X i (resp. extending v i.
show
lift
Let
that has
Yi ) the
F
F
be
a
can b e
coefficients
c
that
c =
~aiv i 1
entry if
c q
by
c
c*
is even.
are quadratic
a basis with
to m a t r i c e s We
is a m i n i m a l
on
each
associated D
entry
i ai i
select
D*q_] D*q q= 0. F
such
we
matrix
to
that
we
where Then
f([aijXiYj)
is
the D*
resolution,
can
fix e l e m e n t s
c * = ~ i a i X-i clearly
expressions
the
of t h e t y p e
[~ijvivj
zero s i n c e
of
do this qin
We h a v e
f ( [ a i j X i Y j) = On
to
first
so for e a c h
Dq_] Dq
£ R O.
F
D q now h a v e
We
Observe
Yi
differential
q be the
Let
is
forms
of
R* .
component
Dq
image
homomorphism and
F*
of
of
D* q is odd,
Xi
complex ideal
in q n_~ in such * q-]"
if
[~ijXiYj
unique
the
sending
the
We shall
÷ F
To o b t a i n q
be
the q u a d r a t i c
be
M.
R-module. F
9 i)
of
each homogeneous
F
each
and
by all
(resp.
Xi ÷ R
R0 ÷ R
R-free
generated
Let
it is an e n t r y
of
197
Dq_]Dq =
0.
This
shows
that
D*q-] D*q = 0. established. Now
let
standard lic,
Y
that
The
existence
be a minimal
spectral
sequence
so it is a m i n i m a l Tor
R*
(M,k)
(~ij)
fi A,
of the
R*-free
argument
R*-free
so
[~ijXiYj
lifted
complex
resolution
the
total
resolution
= 0, w h i c h
of
R.
complex
of
M.
F*
means
is now
Then
by a
F*® Y
is a c y c -
Hence
= F * ® Y ® k = ( F * ® k ) ® ( Y ® k) R* R* R* k R* W
= ( F ® k ) ® ( Y ® k) R k R* This
proves
the
formula
in the
= TorR(M,k)®TorR(R,k). k
theorem.
RF~iARKS. ].
Any
R-free
all 2.
p
There from
complex
C
sufficiently exists
which small,
a minimal
is b o u n d e d c a n be
R*-algebra
fact
that
the h o m o m o r p h i s m
of L e v i n
[3],
i.e.
the
TorR(k,k) is s u r j e c t i v e . Rochandel
Then
[4],
the
R* ÷ R is a m i n i m a l 3.
The
integer
length than
of
m m.
or e q u a l
is a p r i n c i p a l R0-module estimate
to
m2
ideal
"best
the
also
of
(This
and
possible"
L. A v r a m o v
ideal
elements. ring
R.
is
This
large
A with
and
H.
of the a n g m e n t e d
See
in the d e f i n i t i o n
of all m × m - m a t r i c e s is
of
closure
case
of
÷ R
follows
in the
R* I
[3, T h e o r e m
2.5].
c a n be c h o s e n can be
comes
entries
Rahbar-
algebra
f r o m the
in R 0. )
from the
to be the
generated fact
is an R 0 - s u b m o d u l e
is c l e a r
by less that
of the That
following
* R
m = n.
Then
this
/(t] ..... t n ) 2 we h a v e
k[[X| ..... X m , Y ] ..... Ym] ] =
/( .... X i Y j .... )
i~i~m,
free
example:
kit] ..... tn] R = Choose
sense
÷ TorR(k,k)
resolution.
In that
R*
map
by a r e s u l t
acyclic
i.e. C = 0 for , P to an R -free complex.
resolution
the
induced
below,
lifted
] ~j~m.
R0
198
REFE RE NCE S [] ] [2 ] [3 ] [4]
T.H. Gulliksen, Massey Operations and the Poincar~ Series of C e r t a i n Local Rings. J. Algebra 22 (]972), 223-232. G. Levin, Poincar~ Series of Modules over Local Rings. Proc. Amer. Math. Soc. 72 (]978), 6-]0. G. Levin, Large homomorphisms of local rings. Math. Scand. 46 (]980), 209-215. H. Rahbar-Rochandel, Th6se, Caen ]979, appendice p.52, prop. 2]3.
(Note by the editor: For complementary results, c f. the mathematical introduction to these proceedings.)
Department of Mathematics University of Oslo Blindern - Oslo 3 (NORWAY)
THE RADICAL OF
~,(~S) ~ ~ , II
by Stephen Halperin
I.
INTRODUCTION
Let
S
be a simply connected
dimensional
in each degree.
with the Samelson product) Quille
topological
space whose rational homology
The rational homotopy Lie algebra is then also finite dimensional
[Q] showed that if no further restrictions
possible graded connected Lie algebras(finite By contrast,
severe restrictions
rational Lusternik-Schnirelmann
are imposed on category of
(equipped
in each degree.
are placed on
dimensional
is finite
~,(~S) ® ~
S
then all
in each degree)
~,(~S) @ ~
can occur.
if we assume that the
S , cat (S) , is finite.
(This is defined
O
in general zation
in [F-H] and coincides with the classical
S~
when
S
is a
L-S
category of the locali-
CW complex-[T].)
This article continues the study of the Lie structure of hypothesis
cat (S). finite) begun in IF-H-T] and [F-H-T-T].
~,(~S) @ ~
(with the
The techniques derive from
O
the earlier two papers,
and the fact that I have carried out this stage of the inves-
tigation alone is due only to the several thousand miles of ocean between Europe and Canada which made impractical
a collaborative
In [F-H-T] we showed that if graded ideals in series of
R
~,(~S) @ ~
(solv length)
Cato(S) _< m
If
if either
then the sum
is itself solvable, is
R , of all the solvable
and that the length of the derived
~ 4 + 21og2(m) . The ideal
homotopy Lie algebra. We also established Theorem.
effort.
R
is the radical of the
the
eat (S) < m then a graded idea] I c ~,(~S) ~ @ o dim ~,(~S) @ ~ is finite or if for all k
is solvable
if and only
2k-I I dim 12i < m . i=k Here it will be shown that the radical is in fact nilpotent
(and thus that every
solvable of
R
ideal is nilpotent) and the length of the lower central series (the nil length) 8+41og2m is < [ 2 ~ m ( m + 1 ) ] , if cat (S) < M . Explicitly we have --
Theorem
I. Suppose
O
S
simply connected,
each degree, and that
eato(S) ! m . Let
lowing conditions
are equivalent:
--
with finite dimensional I c ~,(~S) @ @
rational homology
in
be a graded ideal. The fol-
200
2k- I Z dim I. < [2m(m+1)]2 . i=k i Every finitely generated subalgebra of I
(i)
For all
(ii)
k ,
(iii) I
is solvable.
(iv)
is nilpotent.
I
is finite dimensional.
Under these conditions, moreover 8+41og2m nil length I < [ 2 ~ m ( m + I)] 2d < [2/~m(m+1)] where
d
is the length of the derived series of
Corollary. Let C > 0
I
be a graded ideal in
I .
~r,(f~S) ® ~ . Then either there is a constant
such that
P Z dim I i < C log 2 p , p > 2 , i=I -or there is a constant K > 0 such that (I)
P Z dim I. > K p ,
(2)
i=I
p > I .
l--
The ideal
I
is solvable if and only if it satisfies the first set of inequalities.
Proof. If
I
is solvable the first set of inequalities follow from Theorem I(i). If
I
is not solvable, then Theorem 1(ii) yields a finitely generated subalgebra,
infinite dimension, and the second set of inequalities results, Suppose now Denote by
RS
and
Theorem 2. If ideal in
S
RT
and
RT
(~I)(RT)
¢#(R S) .
ker ~# c R S
satisfies (~) and since
Im ~#
ker ~ # c R S
is an ideal in
J(~)
~1(j(~))
is an
also satisfies (I) it follows
is an ideal, for every homogeneous
is an ideal
~ ' ( J ( ~ ) ) . Thus
Im ~#
then
satisfies (I). Hence it is a solvable ideal; i.e.
[~'~#(Rs)] T ~#(RS)
RT
m
~#: ~,(~S) ~ ~ ~ ~,(~T) ® ~ .
satisfy the hypotheses of Theorem I, and if
while
Conversely, since
so does
is continuous, inducing
the radicals. Then we have
T
~,(~T) ® ~
Proof. Since that
~: S ~ T
I , of
in
c RS
Im ~# . This ideal satisfies (I). Hence and
~,(~T) ® Q . Since
(~I)(RT) c R S . ~ E ~,(~T) ~ 9:
J(e) c ~#(Rs) . This shows that
~#(R S)
satisfies (I) it is contained in
D
Corollary.
Let
F
~ E ~-~ B
be a Serre fibration for which
F
and
E
satisfy the
hypotheses of Theorem I. Then
(i~) -I (%> = R F Proof. Clearly, dim ker 7#
Im i# = ker ~
is finite, so
is an ideal, as is
ker ~ # c R F .
m
ker ~
. Moreover ([F-H])
201
The study of rational homotopy iterated integrals.
theory has, as one of its sources,
Either this way or via the standard methods,
compute the real homotopy Lie algebra forms on a smooth manifold,
Proposition
I. Let
S
satisfy the hypotheses
If
able ideals in
~,(~S) ® Ik.
Proof. Write
I~ ~ ~
{v }
proof of Theorem
of
and let
Ik. For
N
I
~
I and let
R c ~,(~S) ®
R @ 1~ is the sum of the solv-
be the sum of the solvable ideals in
x E I
I still works with
there is a constant
of Theorem
is an extension field then
L = ~,(~S) ® ~
q-basis
from the algebra of differential
S . This does not affect the radical:
be the radical.
Fix a
~,(~S) ® ~
Chen's work on
it is possible to
write
x = ~ x
instead of
such that for any
~
® v
x
L ® ]k.
C L . Now the
as coefficients;
in particular
k,
# {i E [k, 2k-1]IIi # O} < N . This will then apply to the ideals Theorem
1(ii),
Remark.
Part of Theorem
Lie algebras)by I
I
c R . In particular,
I
generated by the
each
I may be phrased
x
I
and
x C R ~ ~ .
I
of
locally nilpotent ~=~ I
~,(~S)
referred to
A X = exterior algebra The ground field id
~
m
nilpotent.
I. At points we shall the reader is
convention:
if
symmetric algebra
X
is a graded space then
(X even) .
throughout.
The main ingredient
S
(X °dd) ~
and by
and a review of the facts we require. Here I
shall simply recall that we use Sullivan's
Theorem 3. Let
I
the theory of minimal models and rational category:
[FHT; §3] for the notation
L
@ ~ :
solvable ~
The rest of the paper is devoted to the proof of Theorem need explicitly
in
(in the language of infinite dimensional
saying that for a graded ideal
locally finite ~
C R
x
in the proof of Theorem
I is
be as in Theorem I, and suppose
I c~,(~S)
@ ~
is a graded solv-
able ideal. Then for each k 2k I dim 12i+I ! [2m(m+ I)] 2 - 2m 3 . i=k In what follows, Theorem 3 will be first reduced about minimal models.
Two key lemmata are then proved
mata of [FHT; §5]), and are subsequently Theorem
If, moreover, cause ([F-H],[H])
of Theorem
~,(~S) @ ~
statement
(they are analogues of the lem-
used in § 4 to complete the proof. Finally,
I is deduced from Theorem 3 in § 5. The space
to satisfy the hypotheses
(in §2) to a technical
S
will be suppose d throughout
I and 2.
is finite dimensional,
then Theorem 3 is trivial be-
202
dim lod d ~ dim ~ o d ~ S )
® @ ! dim ~even(~S) @ ~ ! m
Thus in our proof of Theorem 3 we suppose that
~,(~S) @ ~
. is infinite dimensional.
2. THE REDUCTION STEP
In order to prove Theorem 3, it is clearly sufficient to establish Z dim I2i+i _< 2 m 2 ( m + 1 ) 2 - m 3 i=£ Fix
~ ~ ~'
degree of
[~,B]
with
3~ ~ 2%'. If
if
3% ~ 2%' .
~, B C ~ i=%I2i+I
(2.1)
are homogeneous then the
is even and lies in an interval of the form
[2r,4r-2] . It follows
from [F-H-T-T] that all such brackets span a space of dimension at most obvious i~Iductive procedure now gives a sequence homogeneous elements in
~i=%I2i+i
[Bi,B j] = 0, and
BI,B2,...,B N
m -I . An
of linearly independent
such that
i # j ,
(2.2)
m N ~ ~i=£dim I2i+i . We are thus reduced to proving that N < 2 m ( m + I) 2 - m 2. Moreover,
it follows from (2.2) that the sub Lie algebra, E , generated by the !
~i's consists of the span ~f the
B i s (in odd degrees) together with an evenly
graded space of dimension at most
m- I
Finally, for any homogeneous
spanned by elements of the form
~ C ~,(~S) @ ~ , and for any sequence
[Bi,Bi ] . i¥ ( 1 ~ i y ~ N )
we observe that [Bit[Sit_1[...[~i1,~]...] I n d e e d , by ( 2 . 2 ) we may s u p p o s e
= 0 ,
deg Bi
t ~ 4m. ~ ...
(2.4)
J deg ~i
t Yv = [ g i v [ ' ' ' [ B i 1 ' ~ ] ' ' ' ]
(2m < v < 4m)
a r e e v e n , and w h i c h l i e the
Yv
is zero,
Now l e t
and so
(AT,d)
i n an i n t e r v a I ¥v = 0 ,
have distinct of the form
v ~ 4m ,
product.
[r,2r-2]
which proves
b e t h e m i n i m a l model o f
is equipped with a scalar
" The s e q u e n c e o f e l e m e n t s 1 degrees at least m of which
Moreover,
S . if
~x
.
By [F-H-T-T] one o f
(2.4).
Then each p a i r E A2T
Tp+1, ~ (aS) ~ Q P is the quadratic part of
dx , x E T , then (2.5)
= ±<x;[~,S]> . Suppose the
~.
have been numbered so that
i
a = deg 131 < _ . . . .
a quotient
= Hom (xP+I;~)
Moreover,
(because
-satisfying
of
(AT,d) . The dual
sub Lie algebra,
(and hence
E)
L , of
by construction.
we have
pja. has a basis
(iii) For degree reasons
YI' .... YN
d
maps
satisfying
X [2a+1'3a+1]
dy i = 0
into
and
= ~ij "
A2x[a+I ,2a] ; moreover
this map
is injective. deg
Since in
X
that basis
E
to
[~N, BN] j 4~' + 2 < 6% + 2 j 3 a
is contained
X [2a+1'3a+1]
in
X J~a.
has as basis
to a basis
~1,...,~t
In particular
we have
it follows
In view of (iii),
elements
for
- I
wl,...,w t
that the dual
space
and (2.5) we may conclude
(t ! m - I )
which form a dual
Eeven .
L = E @ L>3 a ; i.e.
E
is complemented
by an ideal.
Now
put Y = X [a+1'2a] Thus
U
W = X [2a+1'3a+1]
is dual to the ideal
L>3 a , W
U = X k3a+2 is dual to
Eeven , Y
is dual to
Eod d , and
AX = AY ~ AW @ AU . It follows quadratic
part
from d2
(i),
(ii) and
(iii) above,
that the differential
d , and its
satisfy
d(Y) = d2(Y)
(2.6)
= 0 ,
(2.7)
d = d2: W ~ A2y . Let
32
and let
be the quadratic 8i, 8~ : U ~ U J
part of the differential
be dual to
d2u = ~i *Yi @ 8i(u) From
2 d2 = 0
we deduce
adB i , a d ~ . . J
in the quotient
+ lj ±w.j ® e!(u)j + I ® ~2 u .
(for the extension
of
8.1
to a derivation
however
u E U ;
(2.8) in
AU)
(2.9)
We shall use these formulae First,
(AU,d) ,
Then by (2.5) we get for
d2ei = ± 8 i ~ 2 .
3 - in § 4.
model
to complete
we establish
the proof of (2.3) - and hence of Theorem
our key lemmata.
204
3. THE KEY LEMMATA
We retain
the notation
of § 2. For
p ~ 0
put
S P = {~ = (o I , .... ON) IO i 6 ~ , o i _> 0 , I o i = p} . If
o 6 S
put
Iol = p , o! = H (o.!) i
P O
I
Y (Recall . . , o N)_
g Yi
=
Yi
is the ~asis
is written
Similarly A
If
(i)
we s e t
then
Y
and A
dual to the
o
= {4}
O
~ 6 Aq
There
satisfy d~'(~;o)
Bi.)
is written and f o r
~i ~ = (ml...gi...~q)
= 4 • If
3. I. Lemma. which
of
i+e
we write
If
1 < i < N
then
(ol,..,q i + I,.
~.(i~ + o ) .
1 < q < N
q = {~ = (~I , ...,~q) Imi 6 ~ ,
e 6 Aq
~i(~i)
and
I
mX, d) , Because
cat
(AX,d)
to a Sullivan model
< m
p:
there are elements
O
vT 6 AX(T 6 Sm+1) Then
dp v
o 6 S
~(i;o) p~(i;o) Thus
define
m
= vi+ ° - v!I + O
= 0, ~(j;i +o)
in particular
is aeyelie ker p
dvT = yT
= 0 , and so there are cocycles
pv$ = pv T . For
Then
such that
~(~;~)
= y
in
AX ® AZ
and
~(i;o)
and
d~(i;o)
such that
6 AX ® AZ
by
"
- ~(i;j +o)
= 0
yj~(i;o) -yi~(j;o)
it is a coboundary.
v'T
We now define
is a cocycle elements
= y in
= yi y
= yi~(4;o)
ker p . Since
~(~;o) ,
I~I = 2,
.
ker p
lol = m
in
as follows:
(a)
If
(b)
If
ok = 0 , k > j
d~(i,j;o)
is the biggest
is any element
in
ker p
for which
.
integer for which
ok # 0
set
= ~(i,k;j + $ k o) - ~ ( j , k ; i + ~ k o) .
It is then straightforward The same construction ~(~;o)
~(i,j;o)
= yi~(j;o)-yj~(i;o)
k > j
~(i,j;o)
then
6 ker p,
to check that the (applied
m 6 Aq , o 6 Sm
~(i,j;o)
inductively
satisfying
over
(i) and
satisfy q)
gives
(ii). Since
equations for
q > 2
Cato(AX,d)
(i) and
(ii).
elements i m
there
205
is a retraction
~: (AX @ AZ,d) ~ (AX,d) . Put
P(~;~) = ~ ( ~ ; o )
For the next lemma we need a little more notation. module.
Fix an isomorphic
suspension of algebra and Bigrade
Y AY'
copy,
with basis
Y', of
Y
with basis
sYi: deg sy i = deg Y i - 1
is a polynomial
N = AsY ® AY' @ M
.
Suppose
M
Y'i
and let
. Thus
AsY
is a free sY
AY-
be the
is an exterior
algebra. by putting
N q'p = AqsY @ APY ' @ M . The elements C S
f 6 N q'p
can be identified with the collections
f(~;a) C M ,
w 6 Aq,
, via P
I ~ sy~ A... A s y ~ ® ~ . (y')O @ f(~;o) ; ~,~ 1 q o. here (y,)~ = ~ (y[) 1 t NOW d e f i n e o p e r a t o r s ~1 a n d g2 i n N , h o m o g e n e o u s o f b i d e g r e e s f =
(I,-1) ~(y[)
as follows. = sy i
Let
and put
~
be the derivation
62 = ~ ® i d .
in
AsY ® AY'
g i v e n by
(3.2)
(1,0) ~(sy i)
and
= 0 ,
Then set
N
61(~ ~ ~ ~ m) = A short (AsY ~ A Y ' , ~ )
calculation is
~ sYi^~ i=1
® ~ ~ y:'m.~ 612 = 622 = ~1~2 + ~2G1 = 0 .
shows that
the classical
contractible
Moreover,
m o d e l a n d so (3.3)
H(N,~ 2) = 1 ® I ® M . On the other hand, because H(N,61) where
M
is a free
AY-module,
we have
= sYl A... A s y N @ I ® F ,
(3.4)
M = AY ® F . (This is essentially Lemma 5.6 of [FHT]). In particular Hq'P(N,~I) Finally,
= 0
we interpret
if
61
a simple calculation gives for
q < N . and
62
(3.5) in terms of the decomposition
f E N q'p
(~If) (~;~) =q~1(_/)1-1y~..f(~i~;~) and
i=l
q+l (~2f)(~;c~) = ~ (-l)l-lf(~i~;c0 i + o ) i=l 3.6. Lemma. Suppose f 6 N q'p (2 < q < N) q+l . (-l)a-lyi~ -f(aim;~) = 0 and i=l l for all
co C A q + l , a 6
Sp
l
m, ~ . There is then an element
co C A q + l , o 6 Sp_l . satisfies q+l . Z (-l)l-lf(;i~;~i+~) i=l
g 6 N q-1'p
such that
= 0
(3.2). Indeed,
206
q
q
i-I
X (-I) i=I for all
The lemma is stated
lied. Using g
the formulae
such that
in "component
X (-|)l-lg(~i~;~0 i +0) i=I
g 6 N q-1'p,
61g = f
because
Suppose
and
= 0 ,
-62f
= 0
and
hypothesis
(3.5)
q-I
> I
and we are required
to
we have by (3.5)
it is automatic is proved for
that
has bidegree
f = ~I g ,
~2g = 0 .
f 6 N q'p',
f = 61g1 ' gl 6 N q-1'p. ~2g I
that
p' < p , and that
Then
(q,p-I)
~i(~2gi)
=
our induction
such that
62g 2 = 0 .
(3.3)
g = gl + 6 1 g 3 . Then
Put
that
= 0 . Since
and
we apply
p = o
p = 0
that the lemma
g2 6 N q-1'p-I
~ig2 = 62g I Since
p . If
implies
~2(52gI)
yields
that it is how it will be app-
~I f = ~2 f = 0
62g = 0 .
on
q < N . Since
by induction
f 6 N q'p. As above
form" because
above we see that
We do this by induction
=
and
e ,~ .
Proof.
find
= f(m;o)
y~ -g(3im;c) z
to find
g3
with
61g = 61g I = f
~2g3 = g2 "
and
62g = ~2gi - 8 1 6 2 g 3 = 6 2 g I - 6 1 g 2 =0. D
4.
PROOF OF THEOREM
Recall
2.
the notation
at the end of § 2.
In particular,
e.: U ~ U 1
adS.. Extend the ~. to derivations in AU . Denote by F (AnU) i 1 p of the elements of the form 0. o ... o 0. ~ , ~ 6 Anu . Set 11 zp Fr(Anu) and note that,
= Anu,
that each
~
is dual to e~ :F
J Further,
r _< O ,
span
(4.1)
(4.2)
= 0 .
Recall J
the linear
in view of (2,4)
F4mn(AnU)
Since
is the dual of
a.
J ads. 3
is a linear combination its extension
(AnU) ~ Fp+2(AnU)
to
.
Anu
of vectors
of the form
[~i,si].
satisfies (4.3)
P
by definition 8 i :Fp(AnU)
~ Fp+I(AnU)
(4.4)
~ F 0 (An+Iu)
(4.5)
while by (2.9) 72: Fp(Anu) Next,
define graded
spaces
Ak, r c AX
by
207
= A kk+1X
@
AiY 0 AJw ® Fr_i_2n(AnU)
Z
i+j+n=k By (4.2) we have (4.6)
A~kx = Ak, 0 = Ak, I = ... ~ Ak, (4m+2) k = h ~k+1 X. From equations
(4.3),
(4.4), and (4.5) we deduce
d: Ak, r ~ Ak+1,r+ 2 . On the other hand, if
(4.7)
F~(AnU)
is a graded complement for
F (AnU)
and if we
put Bk, r = AM(E;Q) If HM(E;Q)~0, While the pair
: 0
and
H>N(F;Q)
and HN(F;~)~0 then
= 0.
(M,N) is a d i m e n s i o n pair for R.
(M,N) may v a r y with E, the d i f f e r e n c e M-N d e p e n d s o n l y
on R (given F'~E'+R c o n s i d e r E'×E). R The correct t r a n s l a t i o n of T h e o r e m A reads Theorem B:
Let R be a simply c o n n e c t e d CW complex of finite type w h i c h
is semi-finite w i t h d i m e n s i o n pair
(M,N).
If n>_max(M+l), 2N+3) then the
h o m o t o p y fibre of the inclusion Rn+R of the n skeleton is r a t i o n a l l y a wedge of spheres. Corollary:
If a 1 - c o n n e c t e d CW complex,
r a t i o n a l c o h o m o l o g y ring,
R, has f i n i t e l y g e n e r a t e d
then for all nhn ° (some n o ) the h o m o t o p y
fibre of Rn+R is r a t i o n a l l y a wedge of spheres. Proof:
We show R is semi-finite.
of H*(R;~).
They define
Let ~l,...,~r be the e v e n g e n e r a t o r s
~:R÷K=ZK(~;Iail ).
The fibre,
E, fibres over R
1
w i t h fibre ~K(9;l~il-l)
w h i c h has finite d i m e n s i o n a l cohomology.
The
1
E i l e n b e r g - M o o r e spectral sequence c o n v e r g e s to H*(E;Q) (H*(R;Q);Q).
Since H*(K;~)
f i n i t e l y g e n e r a t e d H*(K;9)
is a p o l y n o m i a l algebra, module,
from Tor H*(K;9)
and H*(R;~)
is a
this is finite dimensional.
This c o r o l l a r y may be regarded as the strict analogue of T h e o r e m A.
We are, however,
not limited to the spaces in C o r o l l a r y i, and
indeed we have T h e o r e m C:
Let R be a simply c o n n e c t e d CW c o m p l e x of finite type such
that for some mo, ~i(R)@~=0, Corollary.
i>m o.
Then R is semi-finite.
Under the h y p o t h e s e s of T h e o r e m C there is an n o such that
for nLn o the h o m o t o p y fibre of Rn+R is r a t i o n a l l y a wedge of spheres. Remark:
The heart of the proof of T h e o r e m B is T h e o r e m 2.1 in the next
section, w h i c h is its t r a n s l a t i o n into the h o m o t o p y t h e o r e y of cgda's. As is usual in this kind of exchange,
the basic idea of
[L] is still
the p r i n c i p a l element of the proof, but the d e t a i l e d t e c h n i q u e s n e e d e d are quite different. Remark:
r Suppose R = i~iK(Q;nl ) (the case c o n s i d e r e d by Ruchti).
Then
R satisfies the h y p o t h e s e s of both c o r o l l a r i e s and so the q u a l i t a t i v e V e y - R u c h t i result follows from either. N : ~(2ni-l)
In this special case we can take
and M=0 and so our r e q u i r e m e n t
is n~z(2ni-l)+3;
in fact in
t h i s l c a s e V e y - R u c h t i get a better bound for n. The authors w i s h to thank L. A v r a m o v and W. Singer for several helpful discussions.
213
2.
D i f f e r e n t i a l algebra and t h e m a i n theorem.
to [B-G],
[Ha],
[Su],
c o n n e c t i o n w i t h rational h o m o t o p y theory. rior algebra
The reader is r e f e r r e d
[Ta] for the theory of m i n i m a l models and its
(X °dd) ® symmetric algebra
Here we recall that AX=exte-
(X even) denotes the free com-
m u t a t i v e g r a d e d a l g e b r a over a graded vector space. ~, here and t h r o u g h o u t is assumed of c h a r a c t e r i s t i c
(Our g r o u n d field, zero.)
All g r a d e d
spaces are s u p p o s e d c o n c e n t r a t e d in d e g r e e s ~0, and ® denotes tensor p r o d u c t w i t h respect to ~
(as opposed,
e.g.,
to ®A ).
A c o m m u t a t i v e g r a d e d d i f f e r e n t i a l algebra ted if A°=k,
simply c o n n e c t e d if also AI=0.
q u i s m if H(¢)
is an isomorphism.
(A,d A)
is connec¢, is a
The e q u i v a l e n c e class of a cgda
(under the e q u i v a l e n c e r e l a t i o n g e n e r a t e d by quisms) h o m o t o p y type.
(cgda)
A cgda morphism,
is c a l l e d its
A cgda is said to be a w e d g e of spheres if it has the
h o m o t o p y type of a c o n n e c t e d cgda H w i t h d i f f e r e n t i a l zero and satisfying H+.H+=0. If A is a cgda,
a KS e x t e n s i o n of A is a cgda m o r p h i s m A~A®AX
where X admits a well o r d e r e d basis x
such that dx sA®A(XN(Ax)
Assume IcA is a d i f f e r e n t i a l
= 0
ideal such that A ~ n + i c I c A hn
for some n>_max(M+l,2N+3). Then the S u l l i v a n fibre of A+A/I is a wedge of spheres. 2.2 A.
Remark:
Let A ® A X A ~
Then @A(A@AXA)
be a S u l l i v a n m o d e l for the a u g m e n t a t i o n of
is a functor from g r a d e d d i f f e r e n t i a l A - m o d u l e s to
g r a d e d d i f f e r e n t i a l vector spaces,
and it sends c o h o m o l o g y i s o m o r p h i s m s
of m o d u l e s to c o h o m o l o g y isomorphisms. M®AXA,
As a g r a d e d space, M ® A ( A ® A X A) =
and we use this n o t a t i o n for simplicity.
Note that a special case occurs w h e n
¢:A+M is a cgda morphism,
so
214
that M is a differential
In this case M ® A X A has the homotopy
A-algebra.
type of the Sullivan
fibre of ~.
type of the Sullivan
fibre of A+A/I.
2.3
Lemma:
In particular
A / I ® A X A has the homotopy
Let r=n-N-l,
Then the inclusion >r ISA(A@AX A) + A-- ®A(A®AXA)
is zero in cohomology. Proof:
Let d be the d i f f e r e n t i a l
in t~ and put
F = (Ax)M.
and by 2.2, H(W®A(A@AXA))=0
because
Now consider
FJ=0
for ker d in B n-l.
Moreover, as well.
W is a differenNote that
(j>N) and r=n-N-l.
the commutative
diagram
(I.B)@A(A@A~A)
=
W~A(~AX A ) >r ~ -(A-- -B)®A(A®AX A) in which the horizontal arrows (2.5) and remark 2.2.
(2.5)
.(I@AX)®A(A®A~A) [J
= >(AAr~AX)®A(A®AXA ) induce c o h o m o l o g y isomorphisms
It follows
that the inclusion,
by (2.4),
j, is zero on
cohomology. On the other hand, because A@AX A is acyclic, ordered basis,
induction on the well
x , of X gives an i s o m o r p h i s m
(A®AX)®A(A~AXA)
= A®AXA~AX (A@AXA, d)@(A X,d)
as d i f f e r e n t i a l
A ® A X A - algebras
I (respectively
by A ~r)
([Ha]).
identifies
Multiplying
on both sides by
j with the map
incl.®id:(I®AXA,d)®(AX,d)C-+(Alr@AXA,d)®(AX,d). It follows that H ( i n c l ) : H ( I ® A X A) + H ( A ! r ® A X A) is zero. Proof of 2.1:
The commutative
diagram
= >r @AX- A A--
~ AQA~ A
proj >r ~AX--A i A/A--
215
and the acyclicity
of A~AXA,
imply that
H(proj):H+(A/I®AXA ) ~ H+(A/A!r®AXA ) is zero.
Thus there are cocycles
>r
in A-- /I®AX A which represent
a basis
of H+(A/I®AXA). Since n~2N+3, cocycles
2r=2n-2-2N!n+l,
is zero.
This
and the product of any two of these
shows that H + ( A / I ® A X A ) has zero multiplication,
and also shows that the cocycles we chose define a quism H ( A / I ~ A X A) =~A/I®AX A. O
3.
The second main theorem.
Theorem C. 3.1
A
KS complex
Theorem.
Let
Here we establish
(AX,d)
be a 1-connected KS complex
There is then a KS extension AX+AX@AY (i)
Y is finite dimensional
(ii)
in w h i c h dimX nEB' -- --N+I is analoguous
I~I
ker
such
that
BN+ I
is a m o n o m i a l
we t h e r e f o r e
have
basis a
~fln,m -u~m ~EBN+ I
by c o n s t r u c t i o n
E BN+ I
which
XCN
by the
n {~--}~EBN+ I
(15)
(16)
divided
for lifting
a monomial
unique
Since
RN+ I
commutative.
B~+]. for
is
SN+ I
to (9).
= 0
this
implies
for e v e r y
225
Write,
again,
_N+] =
(f]
(mN+2+(f~+] fN+] -'''' r )~(mN+3+m( " N + ]_ _r]
~N+2' =
. fN+] ), , ~N+]
''
Pick
ker
'
r
/m%r]
a monomial
fN+])
,..,
basis
IN+2 = .m N + 2./ ( m.N + 3 +.m N + 2
for
n _u--
{~--}~6B +2 , where
form
is of the
B~+ 2 = B~+]U
BN+ 2
relation
~+2
in
m Uk* ~ -
form
For
n u-- =
(]7) of the
same
@ IN+ 2.
r
n
of the
,..,fN+]))r
for
every
we may some ~
assume m
that
6 BN+ ]
with
0 m(~]+] ''" ,fN+])) r
lal
and
~ N+2
for
n 6 BN+ ' 2 '
some
k.
we have
Put a unique
, m ~' fN+] ~-, 8 n,m _u-- + [. n, j ~£BN+ 2 3 -J
form as (]0).
Let -fN+2 fN+] 3 = CN+](fJ ) = J
(is) Again, ~+2
by d e f i n i t i o n
of
o,
the o b s t r u c t i o n
.
(]9)
for
lifting
X#N+2
to
fN+2
3
= j
Definition
(].3).
the Massey these
(20)
Clearly k ) 0
n ~ c u-. ~ ( B ~ + 2 3,a --
is o(X#N+ ] , ,
With
+
The
products
notations,
~j+2 =
this
#N+] <x;
we have
the
cj ,n--ua)
j
is called
n> = Zjcj,~
Y j*®
([
yj
_
a defining 6 A 2, for
system n
for
£ BN+ 2.
following
m + [ <x. l>u ! + [ yj<x * ;m>u-[ yj<x* ;m>u--n ~B~ ~B~+~ a~B~+2
process
we obtain
map
Y~ " fN+]+ ~ 3 ~6~+2
m a y be c o n t i n u e d
a diagram
indefinitely.
For
every
226
T2
~
T1
(21)
~N~k
RN+k+]
+
+~'
H
~N%k
N+k+]
SN+k ¢~N+k SN+k-]
a monomial with
basis
J~J
( N+k
(22)
inducing
the
{u~ -- J~6BN+ k
for
such
SN+k'
there is a unique n m U~ u_-~~n,m -~ E B N + k -- _
relation
that
in
for e v e r y
n
SN+ k
,
identity
(23)
~
~n,m<X*;n>
= 0 ,
m E BN+ k.
n E B ~ + k -- _ And
there
IN+k+ ] such
is a c o r r e s p o n d i n g
of
that
ker in
~' N+k+]
RN+k+ ] n u_-- =
(24)
where
we have
=
put
basis
{~}m£B~+k+
] for
the c o m p o n e n t
(_N+k ,fN+k. N+k N+k f] .... r )/~(f ..... fr ) • IN+k+ ]
we have
for every
~
with
, n 8' 0n,m -u- + [ n , j
~_
I~I
( N+k+]
M~+k 3
BN+k+ 1 = BN+k U BN+k+ ] .
Moreover,
(2s)
fN+k+1
= ~N+k(fj)
= fN+k +
~
3
nEB'
3 The
obstruction
(26)
for l i f t i n g
X~N+k
to
~j~
--
RN+k+ ]
N+k+]
n u--.
--
is
) = ~ y~ . fN+k+] j
o(X0N+k,~+k+l
3 = ~ y~ . ~j+ k J
Definition
(].4).
the Massey In particular,
The
map
products we
find
~N+k
+ [ ,
is called
a defining
<x ;n> = Zj Wj,n yj for every
* We,_n~_) n ([j YJ"
£EBN+k+ ]
k,
E A2
for
system ~
for
E BN+k+ ] •
227
fN+k+] 3
(27)
Notice
that by (4.2.4)
(28)
H
of [La] ] we have
lim
=
k+1 ~ ~ yj<x*;n>u n I=o n6B~+ 1
=
SN+k
therefore (29)
H = k[[u~
. . . . .
Ud]]/(~ ~
~d )
. . . . .
where
(30)
fj = lira f~!+k. k÷= 3
Formally we may therefore write c0
* > n ~j = ~ ~ ~]v'<x- ,_±-n._u-" i=o _6BN+ n ' 1
(3|)
§2
Massey products
for
ExtA.(E,E)
In this paragraph we shall shall
let
X,
in §1, be some
let
A
be any
A-module
concerned with the deformation
k-algebra
and we
E. We shall thus be
functor of
E
as an
A-module
DefE: ! ÷ Sets defined as follows, { S ~kA DefE(S ) = ~ A As is well known,
÷
End(Es)i %
ES ~
÷
End(E)
E S ~S k = E
the corresponding
is
cohomology
S-flat?
/ iso.
is
A i = Ext~(E,E) The deformation [La| ], parallels global
theory
for modules,
the corresponding
theory and a relative
of [La] ] holds.
as hinted at on page ]50 of theory
theory,
There
and the main theorem
There are no surprises,
leave the details to the reader.
for algebras.
is a
(4.2.4)
and we shall therefore
228
Pick
any
free
resolution
the a s s o c i a t e d HomA(L.,L.).
single
L. of
complex
By d e f i n i t i o n HomAP(L.,L)
Let
d i : L i + Li-]
be
dP: is d e f i n e d
as an
A-module,
and
consider
of the d o u b l e
HomA(L.,L.)
complex
we h a v e =
H H o m ( L m , L m _ p) m>o
the d i f f e r e n t i a l
Hom~(L.,L.)
of
L.,
then
÷ HomP+I(L.,L.)
by
dP({=~}i~o) Clearly
E
HomA(L.,L.)
= di0=~_ I _ (-I)P J 1 od z - p
is a g r a d e d
differential
associative
A-
0
algebra,
Lemma
multiplication
(2.1).
There
being
is a n a t u r a l
i EXtA(E,E)
Consider
the c o m p o s i t i o n
any s u r j e c t i v e
of
Hom
(L.,L.).
isomorphism
i . = H (HOmA(L.,L.)),
morphism
i ) 0.
~: R ÷ S
in
1 , such
that
m °ker ~ = 0. --R Assume
there
{L.,di}, This
exists
i.e.
means
a lifting
of the
that
free
there
{L.o k S, di(S) }
resolution
exists
L.
of
a commutative
0
where
÷-- Ho(L.
+-- L o ® k S
÷--
for e v e r y
shall
see that
resolution E = Ho(L. )
of to
of the
form
÷
L] ~ k S
d2(S) ~
L2®k S
d3(S) ÷
d] ÷
L]
d2 ÷
L2
d3 ÷
i, the c o m p o s i t i o n di+ ] (S)
We
LO
complex
E.
diagram
d] (S) 0 "~-- H o ( L . ® k S )
of the
any
o di(S) such
Ho(L.~kS) S.
=
lifting
= ES
, and
0. is,
in fact,
that
ES
an
A ~k S - f r e e
is a l i f t i n g
of
229 Both contentions may assume
are obviously
they hold
corresponding
for
true
S
statements
for
di(s) }
conversally to
S
A ~
of
S-free
obstruction
For every
are
is a c o r r e s p o n d i n g
S. By a s s u m p t i o n
we
the
{L.~kS,
of
E
lifting
we have
di(S) ) and
ES
of
is,
{L.,di} itself,
an
E S. di(S) }, and let us compute
{L.~kS,
an o b s t r u c t i o n
i, pick a lifting
L i ~ k S + Li-] ®k S, to
to prove
Given a lifting
E S = Ho(L.~kS)
{L.~kS,
for lifting
clearly,
there
lifting
of
lifting
problem.
to
a lifting
resolution
Pick one such
is then,
{L.,di}
that any such
determines
so by induction
R, we are through.
it is easy to see that
{L.~kS,
S = k,
S. If we then are able
But first we have an existence to
for
di(S) }
R. This o b s t r u c t i o n
for lifting
di(R): '
R. This
to
the
E
to
S
R.
Li® k R + Li_ ] k®k R
is obviously
possible,
of
di(S):
since all
Li
A- free.
Since
di(S)0di_] (S) --: 0
maximal Li_2~k R
ideal
mR
of
is induced
and
since
I = ker
R, the c o m p o s i t i o n
~
is killed
by the
d~(R) 0d~_] (R) : Li~kR
+
by a unique map
Oi: L.l ÷ Li-2 ~k I The
family
{Oi}i) 0
defines
an element
O E Hom2(L.,L.) One checks defining
that
d20 = O, so that
It is easily d~(R)'s
Moreover,
O is a 2 - c o c y c l e
of
HomA(L.,L.),
an element O(Es,~)
the
~k I
if
seen that lifting
6 Ext~(E,E) o(E S, ~) the
o(E S, ~) = O,
£ Homl(L.,L.)
~k I.
is independent
of the choice
di(S) 'sthere exists
~k I such that d ~ = - 0
.
an e l e m e n t Put
of
230
d, (R) = d'.(R)+~. , l 1 1 then one
finds di(R)
and Now
0 di_ ] (R) = 0
{L.mkR , di(R) } let
is a lifting
{L.mkR;di(R) }
R, then
there
be any
is an exact
of
{L.~kS , di(S) }
lifting
sequence
of
a long
exact
Hn(L.@kR)
÷
H n (L" ®kS)
÷ H n _ ] ( L . ® k I)
÷
....
÷
H] (L. ~kS)
÷
÷
H
from w h i c h
o
(L.~
I)
it follows
÷
E ~k I
o
(L.®
R)
+
÷ Ho(L.®k s )
0
that
Hn(L.~kR) 0
di(S)} ~ 0
sequence
Hn(L.®kI)
H
to
of c o m p l e x e s
÷
÷
R.
{L.®kS , di(S) I
0 ÷ {L.-kl, di~1 I} ~ {L.®kR, di(R)} ÷ {L.%S, inducing
to
*
= 0
Ho(L.~
for R)
n ) ], ~
ES
+
is a l i f t i n g
of
and 0
is exact. Therefore
Ho(L.~R)
Moreover,
given
= E
two
liftings
The
family
element
{ni}i) O
÷ Li-] is a
liftings induce
R.
1 = ],2 E~
and
of E~
of
maps
~k I.
]-cocycle
of
Hom'(L.,L.)
defining
~ 6 Ext~(E,E).
In this w a y we o b t a i n {liftings making space
to two
di(R)]-di(R) 2
ni: Li
to
{L.~kR , di(R) l} ,
{L.~kS , di(S) }, c o r r e s p o n d i n g E S , the d i f f e r e n c e s
ER
the
a surjective
of E s to R}
set of
liftings
( t o r s o r ) over
We h a v e
established
the
map
x Ext~(E.E)
÷ {liftings
of
R
Es
Ext~(E,E). following,
to
of E S to R}
a principal
homogenous
an
231
Propositi0n
(2.2).
Let
E S £ DefE(S)
{L.~kS,
di(s) }
of
defined
obstruction O(Es,~)
L.
to
correspond
S. T h e n
E Ext~(E,E)
there
to the
lifting
is a u n i q u e l y
~k I o
given
in terms
defined to
above,
such
R
if
that
O(Es,~)
is a p r i n c i p a l
T h u s we h a v e entirely
O(Es,~)
= 0
in t e r m s
of the c o m p l e x
products,
apply
<x
turn out to be
the d i f f e r e n t i a l Pick of
a basis Ext
for
xi
and
(E,E)*
n>
for
in §] mn/m~
let
an
£ = the
graded
Denote
for
HomA(L.,L.) iff
~ I
m a y be
ES
lifted
calculus its
by
for
ES
to
Ext~(E,E). Def E
given
liftings.
of §]
In fact, "ordinary"
and the
compute <x ,n>
the of
§]
Ext~(E,E)*
Ext I
and
X i £ Hom~(L.,L.) Yj
Massey
products
of
HomA(L.,L.).
{x ..... x~} of
over
of
N+k"
k-algebra of
liftings
(torsor)
and
n 6 B'
j = ] ..... r,
(n] ..... nd) k-algebras
with S
that we h a v e
and a b a s i s
and
{y] ..... y r }
{y ..... y r }
the
Ext 2. be a c o c y c l e
6 Hom~(L.,L.)
representing
be a cocycle
d I~I = Z i = ] n i = N
and
R
. Fix
in
Rn
the
the b a s i s following
and c o n s i d e r {~]---~p}
as of
slightly
identities v I = Uil,
ui = 0 insisted
set of
yj.
. Recall
(])
the
space
some g e n e r a l i z e d
i = ] .... ,d,
confusing
on.
= 0
L.
--
dual b a s e s
representing Pick
of
the c o n s t r u c t i o n s
{x] ..... Xd}
corresponding Let
then
homogeneous
"
will
O
at h a n d a nice o b s t r u c t i o n
this we shall
Massey
2-cocycle
R.
Moreover,
Using
of the
upon because
1 = ] ..... p.
if
i ~
it m a k e s
{i I ..... ip} the n o t a t i o n s
more
streamlined
later
232
We shall pick a monomial
basis
for the
k-vectorspace
S
n
of the
form in] md {u] -oou d i 04mi 2. (3),
~j,n
=
yj<x
w
;n>
where,
by a s s u m p t i o n
<xi],xi2, .... x i ;~> is (uniquely) d e f i n e d . P u~ and c o n s i d e r the d i a g r a m Put ~j = ~i~i= N a j . n _ %N- ]
~(],0)
E.
formal
x~
{di)i>0'
family
~2 ' is a d e f i n i n g
a set of g e n e r a t o r s
moduli
that
=
represents
may be w r i t t e n
fj
~(0,0)
step by step
as
where
,
k [ [ x ] ..... x d]]
for
which
a purely
the
)
~(0,])
Thus
products
~]'~2
induces
I~I < N,
defining
and
systems
corresponds
for
all
therefore
Massey
§]
products
to a f a m i l y
(7).
=
The map
w <x ; n >
236
(2)
{a m } m E ~ N _ ]
of d.
1
]-cochains
of
HomA(L.,L.)
, and
~ is a cocycle e. --I N__. Moreover, for every
(3)
[
~
such
that
representing m
for every
i > 0,
~i,o
x . , e = (0,.~_~_~,0,.. 1 --I i
O)
6 BN_ ]
0 ~
= 0
~iEBN _ ] Let
di(SN-]):
defined
Li~SN-]
+ Li-] ®k SN-]
be the
A ® SN_]-linear
map
by m
d i ( S N _ ] )I L i ® ] = ]
(3)
Then
~.
~BN-]
implies
universal
[
that
deformation
u-- •
~ ' ~ --
{L.o k SN_] ; di(SN_]) I of
L.
to
S 2 defined
is a lifting
of the
by the map
#]: H ÷ k[u] ..... Ud]/m2. Recall
that
$]
if one wishes,
corresponds to
S2
to the d e f o r m a t i o n
defined
by the
d [ X~® ~, 6 E x t ~ ( E , E ) ~ i=l 1 l By c o n s t r u c t i o n E~N_]
L.,
or of
E
element
m/m 2 = DefE(S2).
{L.~ k SN_ ] ; di(SN_] ) }
induces
the d e f o r m a t i o n
E DefE(SN_ I ).
Sticking
to the
a6B~
=
the
2-cycle
notations
{m£~dllml=~ }
Y(n)
of §] , and
the M a s s e y
=
~
(8)
and
(9)
Zn (B~ ~n,m Y(n)
translates
=
into
noticing
product
that
<x*;n>
for every
is r e p r e s e n t e d
by
o
ml+m2=n ml
§1
of
the
is a coboundary.
m2
following.
For
every
_m E B N
,
237
Now,
pick
for every
m £ B --
a
]-cochain
N
m
such
(HomA(L.,L.)
that (4)
d am = --
and consider
[
~n,m Y(~)
n6B' -N
-- --
the family
(5) {~m}m
Let,
for every
6 BN
i ) O, di(SN):
Li~ S N + Li_1 ~ S N
di(SN) I Li~ ] = Z~£BN~i,m~_ _ _u~m.
Then
be defined
(4) translates
by:
into
di(S N) 0di_ ] (S N) = 0. Consequently
{L.~ k SN; di(SN) }
{L.SkSN_] ; di(SN_]) } E%N~ ~ DefE(S N)
of
to
SN
is a lifting
, and induces
E~N_] . E N' again,
of
therefore
corresponds
a lifting
to a map
:
4~N
H + S N which we now fix. According
to (].2)
products
<x ;n>
induces,
a family
defining
system
By definition,
is a defining
for
~
see
(5), we shall
(].2),
refer
~
these
for the Massey is induced
to any such
Massey
by,
family
and
as a
J <x ;n>, ~ 6 BN+ ]•
products
see §] (]]),
this obstruction
system
6 B~+] . Since
for the Massey
of the obstruction, By (2.2)
~}I
products
are given
in terms
o(E~N , ~N+]). '
is defined
by the
2-cocycle
O =
{Oi}
where
0 i = d i ( ~ + 1)0di_1 (~+~), dl(RN+]): dl ( RN+ ]
Li® RN+ ] ÷ Li_]® RN+ ]
any
lifting
of
di(SN).
such that
di(~+,)J then
being
streight
forward
Li® ] =
calculation,
~_ = i , m ~(B N using
~
§]
(]0),
shows
that
Pick
238
Oi =
,7
(
Y,
Y
n6B'N+ ] ImJ ~N+] ml+n_12=m
r J=]
+
Remember
[ (
~] Iml
that
Comparing
~N+I
di(S N) 0di_ ] /S N) = 0. (].2)
(2.6).
and §] (]]), we have proved the following
Given a defining
Massey products by the
3 iY+m--2=m~3-~' j " "i,m~ ai-] 'm2 ) f~'~
m
this with
Proposition
8'm,n'~i,mOai_],m2 ) u n ]
system
<x ;n>, ~ £ ~ + ]
{a }mE~N+]
, <x ;n>
for the
is represented
2-cocycle ~[
Y(n) =
y
#'
a
lmJ ,
(18),
to ~
a
defining
system,
i
E BN+ 2 .
(19), and we may copy the
above.
We end up with the following, Proposition
(2%7).
Given a defining
Massey products the
*
,
system
{am}m6~N+k_] W
<x ;n>, n 6 BN+ k , <x In>,
2-cocycle Y(n}
=
~
~
Iml u ~
refering
system
to §]
for the M a s s e y
(28),
(29),
(30),
products
<x ;n>, n
£ ~+k+]
sum up the c o n t e n t
of t h i s
follows
(2.8).
Given
is d e t e r m i n e d
an
A-module
E,
the
by the blassey p r o d u c t s
H = k[[x I .....
Xd]]l(f I .....
formal of
moduli
ExtA(E,E).
H
of
E
In fact
fr )
where . = ~ [ yj<x*;n> f3 1=2 £ ( B ~
Cqrollary field
(2.9). k
Any
complete
is d e t e r m i n e d
by
xS.
local
k-algebra
Ext~(k,k),
A
with
i = ],2
and
residue its
Massey-products.
Proof.
Obviously
A
is the
formal
moduli
of
k
as an
A-module. Q.E.D.
"
240
BIBLIOGRAPHY [Car]
Cartan H. Seminaire ]960-6]. Paris, ]962.
Institut Henri Poincar6,
[La] ]
Laudal,
O.A., Formal Moduli of Algebraic Structures, Lecture Notes in Mathematics No 754, Springer-Verlag ]979.
[La2 ]
Laudal,
O.A., Groups and Monoids and their Algebras. Preprint Series, Inst. of Math., University of Oslo, No 12 (]982).
[M]
Massey,
W.S., Some Higher Order Cohomology Operations, Symposium International de Topologia Algebraica, p.p. ]45-]54, La Universidad Nacional Autonoma de Mexico and UNESCO, Mexico City ]958.
[May]
May, J.P., Matric Massey Products, (]969), p.p. 533-568.
Journal
of Algebra
]2
Palamodov, V.P., Cohomology of analytic algebras (russian), Trudi Moskovskogo Matematitseskogo ob~Qsi¢~ (Vol 44) ]982, pp. 3-6]. [S&S ]
Schlessinger, M. & Stasheff, J., Deformation theory and rational homotopy type & The Tangent Lie Algebra a Commutative Algebra, Manuscripts, ]982.
of
Department of Mathematics University of Oslo Blindern,OSLO 3 (NORWAY)
A METHOD FOR CONSTRUCTING BAD NOETHERIAN LOCAL RINGS
Christer Lech
We wish to introduce some new ideas for obtaining results involving socalled bad Noetherian local rings. Taken together, these ideas form an alternative to the well-known method by Rotthaus,
first applied in [7]-
We shall display them by indicating a proof of Theorem I below.
THEOREM I. A complete Noetherian local ring S is the completion of a Noetherian local domain if and only if the following conditions hold. (i)
The prime ring of S is a domain that acts on S without torsion;
(ii)
Unless equal to (0), the maximal ideal of S does not belong to (0) as an associated prime ideal.
Before going further, let us state, without proof, a more comprehensive theorem, which,
in essence, comprises not only Theorem I but also the
main results in the articles [3] by Brodmann and Rotthaus and [5] by Larfeldt and the author. The same basic principles can be used for the proof -- not, however,
in the same simple guise.
THEOREM 2. Let S be a complete Noetherian local ring and ~ a n
ideal of S.
Then, in order that ~ be generated by a prime ideal in some Noetherian local ring having S as completion,
it is necessary and sufficient that
the following conditions hold. (i)
For W = {x~ S; ~ : x = ~ } ,
the ideal ~S W of S W is generated by a prime
ideal in a flatly embedded subring of SW; (ii)
Unless equal to ~,
the maximal ideal of S does not belong to
as an associated prime ideal.
242
Theorem ] rounds off a line of research beginning with Akizuki [1], Sect. 3, and having Brodmann-Rotthaus [2], Prop. (15), as its most recent exponent. The author has been strongly impressed and spurred by the last-mentioned result. He has also had the benefit of stimulating discussions with Christel Rotthaus and Ralf FrSberg.
For the proof of Theorem ] we shall need certain notions concerning subrings R of a local ring (S,m). In reality these notions will merely concern the corresponding local subrings R R ~ ~ . We shall say that the injection R - - ~ S rs#O
is flat if S is flat as a R-module,
for r ~ R -{O1,
torsionfree if
s ~ S - {01, unramified i f ~ S ( R ~ ) ~
rational etc if the field extension ~ ( S ) / k ( R R ~ )
residually
is rational etc. Here
k( ) indicates the residue field of a local ring. Similarly A will indicate the completion of a local ring and -- the algebraic closure of a field. Following Well in [8], we shall say that a field extension K/k is re~lar
if K and k are linearly disjoint over k.
In the sequel, let S be the complete Noetherian local ring of the theorem and ~ its maximal ideal. Let further ~ be the minimal finite subset of Spec(S) whose union consists of all zerodivisors in S.
The theorem has the form of an equivalence between two statements about S. Let us introduce the following third statement as a connecting link:
There exists a ring injection R - - > S
which is
(I) flat, (2) unramified, (5) residually regular, (4) torsionfree.
This statement is implied in an obvious way by the statement that S is the completion of a Noetherian local domain. Moreover,
it implies,
243
through (4) and (2), the validity of the conditions (i) -(ii) theorem. When discussing the reverse implications,
of the
we may strengthen
condition (ii) into
(iii) ~
does not belong to (0) as an associated prime ideal,
as the possibility ~ = (0) causes no difficulties. theorem,
it suffices to prove the two implications
(i)-(iii) (*)
Thus, to prove the
-~
~>
(*);
'S is the completion
of a Noetherian local domain'.
The second implication is of minor interest from our present methodological point of view: k(S) is countable,
it can be treated by rather general means, and when
it can be got round entirely by strengthening (~) so
that rationality takes the place of regularity.
In the proof that follows
some of the details will be omitted.
The desired Noetherian local domain is obtained by a ring construction embodied in the concept of straightness defined below.
DEFINITIONS.
A morphism A ~
called telescopic
B of commutative rings with l-elements
is
if there exists a well-ordered generating set {xiliE I 1
for B over A such that, for each j ( I, A [ I x i l i ~ j } ]
is free as a module
over A [ { x i l i < j}]. A morphism A --~B of local rings is called straight if it is unramified and can be presented as a composition of a telescopic morphism and a subsequent localization.
Since the given injection R - - > $
can be assumed to be local and since
every straight local morphism is obviously flat, we get the desired result by combining the following two propositions.
244
PROPOSITION. Let k m ~ K
be a separabel field extension, and let R m ~ R 1 ,
R - - ~ R 2 be two unramified flat local morphisms, both inducing k ~ K residual field extension. Then there is an isomorphism
~R
as
2 respecting
the ~ - a l g e b r a structure and the given identity of the residue fields.
PROPOSITION. For any field extension K/k and any local ring R with residue field k there exists a straight local morphism R ~ R k ~K
I with
as residual morphism. For any such morphism R - - ~ R ~ the statements
indicated below hold true. R Noetherian
~
R I Noetherian;
K/k regular
~
~R I prime (V~ (Spec(R)).
The first proposition can be obtained as a consequence of [6], Thm. 82, which ensures the existence of a q - a l g e b r a h o m o m o r p h i s m ~ I --~R~ respecting K. It is not hard to see that any such homomcrphism is indeed an isomorphism. The second proposition is in substance contained in [4] apart from the very last assertion, which can be made evident by the device of extending local domains into valuation rings. (The regularity hypothesis might be unnecessarily strong, but separability, at least, is needed.)
Let us now turn to the more fundamental first implication. Each of the properties (I) -(4) expresses a condition on R. Assuming (i) and (iii) to hold, we shall show how to construct a subring of S that satisfies them all. The first three can be summarized as follows: All S-linear maps S n - - > S and S n - - ~
given by matrices over R and ~(RR~?~) resp.
must have kernels generated by elements in R n. Thus R must exhibit a sort of completeness in its relation to S, namely by providing
~ultimate'
solutions for all linear equations of certain types. Clearly S, as a subring of itself, meets these demands. However,
there is a simple way
245
for obtaining a countable
subring with the same property.
It consists in
forming the union of an increasing sequence of countable subrings so chosen that the linear equations arising from one ring have appropriate solutions in the next. The existence
of such sequences is clear from the
fact that each submodule of sn is finitely generated ( n = 1 , 2 , 3 , . . . ) . The described procedure for complying with the demands of (1) -(3)
forms
the basis of our construction.
Concerning the property (4), which means that R ~ $ begin by making a few observations. of S, by (i). Secondly,
First,
simple adjunctions
: (0) ($ E~),
let us
(4) holds for the prime ring of the form R - - > R [ s ]
will
preserve the validity of (4) (in the natural implicative sense) if the element s either represents a transcendental
over R in each S/~ ( ~ ( ~ )
satisfies a relation of the form a s = b with a,b (R,
a/O;
or
let us refer
to these two types of adjunctions as transcendental and fractional resp. Finally,
the property of a ring extension to preserve the validity of (4)
is ~transitive' with respect to arbitrary well-ordered
towers, not only
finite ones.
Put together,
these facts allow us to conclude that (4) holds for any
subring of S that can be obtained from the prime ring by a possibly transfinite
succession of transcendental and fractional adjunctions.
It suffices to show that our basic procedure can be performed within that framework. Thus it is enough to prove the following assertion: For any countable subring R of S enjoying property (4) and any linear equation over R or k ( R R ~ )
as considered above,
it is possible to
incorporate a S-linearly complete system of solutions by means of simple adjunctions of the two permitted types.
What resources in transcendentals assumptions
do we have? It follows from the
(notably (iii)) that S contains a complete discrete valua-
tion ring which maps injectively into each of the rings S/~ ( ~ ) .
246
Hence S contains indeterminates
an uncountable
in each S/@.
ically independent rings
Suppose
we can maintain
a countable
subset
of sn determined
of a generating
permitted
adjunctions.
arbitrary
(minimal)
Remembering vectors
of R,
by the equation
existence
say with a n ~ 0 . alx I + ... + a
generating
if
set.
Let E be the submodule
x = 0. We must n n
system in accordance
of transcend~ntals,
O,-al) , (O,an,O .....
show the
from R by
with Nakayama' s lemma.
and observing
-a2) , ....
(0 . . . . .
adjunctions.
that the n-1
O,an,-an_l)
system whose all n-1
the remaining n:th components
first comp-
In view of the
can then be adjoined
adjunctions.
for equations
of modification
are larger.
can be used as a modifying vector. exclusively
(uncountable)
system for E which can be reached
A similar argument applies possibilities
by omitting,
subring of S for which (4) holds and
onents can be adjoined by transcendental
by fractional
for given countable
the independence
belong to E, we can find a generating
defining equation,
but,
Such a system can be obtained by modifying an
the abundance
(an,O .....
sense,
of the original
then that R is a countable
that al,...,a n are elements
independent
It is true that these elements are algebra-
only in the absolute
of constants,
necessary,
set of elements representing
transcendental.
over k ( R R ~ ) .
But here the
In fact, any element
This leads to adjunctions
of ~
that are
n
247
REFERENCES
[I] Akizuki, Y., Einige Bemerkungen ~ber primate Integrit~tsbereiche mit Teilerkettensatz.
Proc. Phys.-Math. Soc. Japan 17, 327 -536 (1935).
[2] Brodmann , ~., Rotthaus , C., Local domains with bad sets of formal prime divisors. J. of Algebra 75, 386 -394 (1982). [3] Brodmann, M., Rotthaus, C., A peculiar unmixed domain. Proc. Amer. Math. Soc. 87, 5 9 6 - 6 0 0
(1983).
[4] Grothendieck, A., El@ments de g@om@trie alg@brique, Chap. O, ~ 10. 3 . Inst. Hautes Etudes Sci. Publ. Math. N°SS (1961). [5] Larfeldt, T., Lech, C., Analytic ramifications and flat couples of local rings. Acta Math. 146, 201 -208 (1981). [6] Mat sumura , H., Commutative algebra, 2nd ed. Benjamin/Cummings, Reading, Mass. 1980. [7] Rotthaus, C., Nicht ausgezeichnete, Math. Z. 152, 107-125
universell japanische Ringe.
(1977).
[8] Well, A., Foundations of algebraic geometry. Amer. Math. Soc., New York 1946.
Department of Mathematics University of Stockholm Box 6701 S-113 85 STOCKHOLM (SWEDEN)
YET ANOTHER PROOF OF A RESULT BY 0 G 0 ~
Crister Lech
As a further illustration
of the ideas in [2] we shall give a summary
proof of the following theorem of Ogoma ([3])-
THEOREM (0goma). There exists a non-catenary,
normal, Noetherian,
local
domain of dimension 3.
The proof of Ogoma has been simplified by Heitmann ([I]). Both Ogoma and Heitmann apply a method of Rotthaus (cf [2]). Though different respect,
in this
our proof will have certain basic features in common with theirs.
Proof. We shall use the following notation: k is a countable field,
S = k[[X,ZI,Z2,Z3]]/(ZIZ2,ZIZ3);
X,Zl,Z2,Z 3 are the uatural
images of X,ZI,Z2,Z 5 in S;
M_= (X,Zl,Z2,Z3) , P_= (Zl,Z2,Z3) , P1 = ( X ' Z l ) '
P2 = ( x ' z 2 ' z 3 )
(: Spec(S).
A local subring R of S will exhibit the truth of the assertion if it satisfies the conditions
(1)
k[x]
(2)
R-->S
(3)
-Pi = ( R f l ~ i ) s
(4)
R0Z=(O).
In f a c t ,
~
listed below.
R; is flat and unramified;
( i = ~,2)~
i t follows from ( 1 ) ,
domain having
S
as completion.
for i= 1,2, Pi = R ~ P i '
(2) and (4) that R is a Noetherian local In particular, R has dimension 3. Putting,
we have --i P =pi S by (3). We conclude that ht(P2) = I
as the primary decomposition xR = x S D R
= P I ~ P2 ~ R
=PI~P2
must be irredun-
dant, and further that coht(P2)= coht(P2)= I. Hence R is non-catenary.
249
(4),
The only singular prime ideals in S are M and ~. In view of therefore
true for every ~
Spec(R) - I R ~ M )
it is
that ~S has a regular minimal
prime ideal. Thus every prime ideal in R is regular,
except the maximal
one. As depth(R)= depth(S)= 2, R is normal by the Serre criterion.
To show the existence
of a ring R satisfying
the conditions,
make slight changes in the proof of the implication in [2]. The construction
we need only
~(i) -(iii)
~
should start with k[x] rather than the prime
ring of S; the r~le of ~ should be taken over by {~}; generators and ~2 should be incorporated by transcendental multiple
adjunctions,
as each generator can be modified by a our proof.
The method of Rotthaus has the advantage R/~ has a very simple structure rings are indeed essentially the examples
To attain this property, that ~ = ( R ~ ) S of non-zero
of providing rings R such that
for all ~
Spec(R) -{(0)}.
finitely generated
These factor
over a field. As a
of Ogoma and Heitmann are pseudo-geometric. we could sharpen condition
for those ~
Spec(S)
ideals in S generated
that this stronger condition ~S prime for ~ S p e o ( R )
of ~I
in R. (The latter task can be performed
of x2.) This finishes
consequence,
(~)'
(3) by demanding
that appear as minimal prime ideals
by elements
in R. It is not hard to see
could also be satisfied.
Then we would have
- {(0)}. If k is chosen to be of Qharacteristic
O,
this gives the result.
REFERENCES
[1] Heitmann,
R., A non-catenary,
normal,
local domain. Rocky Mountain J.
of Math. 12, 145 -148 (1982). [2] Lech, C., A method for constructing
bad Noetherian
local rings. These
Proceedings. [3] Ogoma, T., Non-catenary Math.
6, 147 -163
pseudo-geometric
normal rings. Japan. J. of
(1980).
Department of Mathematics, University of Stockholm, Box 6701, S-113 85 STOCKHOLM (SWEDEN)
MODELE MINIMAL RELATIF DES FEUILLETAGES
pag~ Da~
LEHI{ANN
I. I n t ~ o d u ~ o n . Soit
V
d'un feuilletage
une vari~t~ connexe paracompacte F
(codimension
Ib : ~b(F) l'inclusion naturelle de la sous-alggbre pour
F
(c'est-~-dire
vecteurs
X
qui vgrifient
tangent aux feuilles de
Un point de base et
~DR(V)
x°
Cm
(dimension
n)
munie
diff~rentielle
des formes
~
basiques
q).
On notera
~
~DR(V)
ix~O = 0 F)
et
LXW = 0
une fois choisi dans
d'une augmentation,
Ib
pour tout champ de
dans l'alg~bre de de Rham. V
permettant
admet un "modUle minimal
de munir
~(F)
relatif"
(~b(F) 0 *,],D,¢) rendant commutatif les graduges
le diagrarmne
suivant de morphismes
de
~R-alg~bres diff~rentiel-
:
{2DR(F)
+
~F
(-~,d)
JF
gDR(V)
+
%
~b(F)
(~b(F)@~ 'D)
251
o~
(~,d)
d~signe une
~-alg~bre minimale au sens de Sullivan
~5] ,
~(F)
@
d~signe le produit tensoriel gradu~ en tant qu'alggbre gradu~e, mais avec une diffgrentielle l'id~al de de base
D
"tordue"
~b(F) @ ~
Xo,
~
(~ u c ~ ) ,
D(| @ u) - I @ d ~
engendr~ par l'id~al maximal de
est un morphisme de
~b(F)-alg~bres
induisant un isomorphisme en cohomologie, relle dans que
V
de la feuille
(~b(F) @ ~ , D )
finie ~
V
~b(F)
(~b(F) 0 ~,D)
relatif au point graduges
d@signe l'inclusion natu-
Rappelons
~b(F)-isomorphisme
pros une fois
D'autre part,
x . o
appartient
diff~rentielles
JF : F + V
contenant
est bien d~finie ~
~b(F)-homotopie
morphisme,
F
u
(ef. S. Halperin pros,
et
~
[5])
bien d~-
fix~ dans sa classe d'iso-
~tant connexe, on a l e
Lemme I. (i)
@~(F)
(ii) (iii) (iv)
Ib
est cohomologiquement
connexe,
est injectif en cohomologie de dimension n'a pas de g~ngrateur en degr~
on peut choisir
(~b(F) 0 ~ , D )
degrg (v)
~(F)
@ ~
O,
dans sa classe de
de fa~on que, pour tout gl~ment tienne ~ l'id~al
I,
u
de
~,
~b(F)-isomorphisme
D(lOu) - 1 @ d ~ u
appar-
engendr~ par les formes basiques de
> O,
la classe de
~b(F)-isomorphie
point de base
de
(~b(F) @ ~ , D )
ne d~pend pas du
x . o
Ce lemme sera dgmontr~ au § suivant.
Dans le cas o3 le feuilletage
F
triviale de base une vari~t~
W
f~ : ~DR(W) ÷ ~DR(V),
au moins si
H~(F,~)
et si
et
H~(F,~)
ou
est eompacte par exemple) mologie, de sorte que
-
H~(W,~) -
(~,d)
est une fibration
de dimension
~F :
q,
~I(W)
f : V + W
l'inclusion
Ib
localement
devient
op~re de fa~on nilpotente sur
est de dimension finie en chaque degr~ (si ~
÷ ~DR (F)
induit un isomorphisme en coho-
est le modgle minimal rgel de la fibre
dule de la fibre est ~gal ~ la fibre du module" Dans le cas ggn~ral d'un feuilletage,
(cf. S. Halperin ~F
V
F : "le mo-
~5]).
n'a en g~n~ral plus aucune rai-
son d'induire un isomorphisme en cohomologie, ne serait ce que.parce que les feuilles F
n'ont plus ngcessairement
routes le m~me type d'homotopie.
On se propose de d~montrer le :
252
Th~or~me : Supposons : (i)
V
est compacte, orientable, de dimension
la cohomologie
(ii)
H~(F)_ de l'alg~bre
Poincarg pour la dimension chaque espace H~(~)
(iii)
H~(F)
q
n,
~b(F)
v@rifie la dualit~ de
@gale g la codimension de
F,
et
est de dimension finie,
est de dimension cohomologique finie
(Hi(~) = 0
pour
i
I
suffisamment grand). Alors : H~(~)
(i)
vgrifie la dualit~ de Poincar~ pour la dimension
des feuilles de
p = n-q
F,
Ii existe une application naturelle injective
(ii)
¢I o8
H~(V,~)
faisceau
: H~(7)
÷
o H (V,~b)
d6signe la cohomologie de ~
. 2,
induit un isomorphisme en cohomologie,
{%(F) ÷ ~DR(V)
~b
.... ~D~(V)
°7-
\
~b(F)
de faqon que
a'.[ b = Ib.a.
255 ne faisant intervenir aucun point de base dans du module minimal
%(F)
@ ~
~
ne d~pendant pas du point de base
÷
~DR(V)
x
V, de
on en d~duit une construction 1b
tel que
a'.P = ~.(a 0 ]~),
d'o~ la partie (v). O ~
4. D ~ m o ~ t g o n
du ~or~me.
Gottlieb a d~montr~ (E3J)
que si, dens un espace fibrg, la base et
l'espace total v~rifient tous deux la dualit~ de Poincar~ en cohomologie avec dimensions respectives
q
et
n = p+q,
et si la fibre a une dimension cohomolo-
gique finie, alors cette fibre v~rifie aussi la dualitg de Poincar~ en cohomologie pour la dimension
p.
La conclusion (i) du th~or~me est une transcription alg~bri-
que de ce r~sultat, qui g~n~ralise un th~or~me de F~lix-Halperin cas oO la pseudo-homotopie (Thomas (~b(F)
~9])
de la "base" est de dimension
[2] (relatif au
I) : elle se d~montre
par rgcurrence sur la dimension de la pseudo-homotopie
en l'occurrenee),
d'o~ la conclusion
de la "base"
(i) du th~or~me.
La filtration
o ?) d~finit une suite spectrale 0 ~ s ~ p)
~r,s
convergeant vers
El-r,s = ~(F) 0
Notons
H~(V,~)
cohomologie
~u]
et
# 0 e HP(~)
De la commutativit~
o7
tandis que
r,s
~i
HS(~),
u 6 zP(~)
~r
de support inclus dens le rectangle
suites spectrales multiplicatives On a alors :
= %
:
(et
v e ~(F) et
[v~
~r,s ~
~
induit un homomorphisme
+ Er,s i
E2-r,s = H~(F) O H S ( 7 )
si
H~(F) = 0)
des ~l~ments induisant des classes de # 0 e H~(F).
~ , o 0 ~l~°'P
.....
HP(v,a~(F))
~'P Hn(v,~)
de
"
du diagramme
f~(F) e HP(~])
(0 ~ r ~ q,
,
Hn(v,~)
256
et du fait que
Ev]. Eu] # 0
dans
E~'P,
~°'P(10
on d~duit
[u~) # 0 e HP(v,~)
I
Puisque
H~(~)
et puisque llapplication
v~rifie la dualit~ de Poincar~ avec H~(~)
H~(V,
÷
)
induite par
[u] # 0 e H P ( ~ )
,0~ ~l
est un homomor-
phisme d'alg~bres, on en dgduit la conclusion (ii) du th~or~me.
]° )
Supposons v~rifi~es simultan~ment les 2 conditions suivantes (c'est,
par exemple, le cas si si
F
F est une fibration de Seifert ggn~ralis@e,c'est-g-dire
a toutes ses feuilles compactes et est localement stable E6]) : (o)
V
est compacte orientable,
(i)
F
est minimalisable,
(ii)
dim ~o,p = 1 -2
D'apr~s Kamber et Tondeur E6], de Poincarg ,
Soit
X
une
~(F)
p-forme sur
V
volume associg ~ une m~trique riemannienne sur doX = O,
X
d~finit
EX]o ~ E? 'p,
[xJ! = O e E o,p 2 •
[X]l e E~ 'p, portionnel ~
~X]I.
L'inclusion
homomorphisme d'alg~bres .~ compacte, puisque iF(X)
"~
D'apr~s
v~rifie automatiquement la dualitg induisant sur chaque feuille V
E~,
rendant
X
d~finit donc
~ ' P ( | ~ [u~)
est pro-
induit, pour toute feuille
~
3F : H~(V'~ ) ÷ H~(F'~)
et
~-
jF(I_X]o
le
minimale. Puisque
diX = O ,
Par consequent, JF : F ~ V
F
F
F
un
) ~ 0
si F est ~o,p JF o ~1 = ~F '
est une forme volume. De la relation
on
d~duit donc la
Pro~os~on 1.Si
V
~F : H ~ ( ~ )
est compacte orientable, si ~
H~(F'~)
F
est minimalisable et si
est injectif pour toute feuille
F
dim E~ 'p = 1,
compacte.
2 °) De faGon plus triviale encore, mais souvent v~rifi~e en pratique, on a la
Propos~ion 2.i
Si
~
v~rifie la dualit~ de Poincar~ en dimension
p,
s'il est possible
257
de choisir
u 6 zP(~)
de telle fa~on que compacte
F
dans
[u] # O e H P ( ~ )
~(] @ u)
et
~ : ~b(F) 8 4 ~
induise une forme volume
~F(U)
÷ ~DR(V)
pour une feuille
particuli~re, alors : ~F : H ~ ( ~ )
÷
H (F,~)
est injectif pour cette feuille compacte.
3 °) Lorsque les hypotheses du th~orgme sont v~rifi~es, l'application
n'est pas n~cessairement surjective, conm~e le prouve l'exemple des droites de pente irrationnelle
~
sur le tore, chaque fois que
II est ais~ de voir que, pour tout nombre est le module minimal de
S !, Si
que les constantes tandis que la forme ferm~e dx-~ dy
~
~(F)
~
~st un nombre de Liouville :
rationnel ou non, ~ =
est irrationnel,
~(F)
F. Si
H~(r,~)
~
(Al(x),dx=O)
ne contlent
ne contient que les formes
d~finissant
E1Kacimi a alors d~montr~ que
~,
~-proportionnelles
est un nombre de Liouville,
a une dimension infinie ~I].
4 =) On pourrait chercher ~ interpreter
~
comme le module de la feuille
g~n~rique (rev~tement eo~mlun ~ toutes les feuilles) lorsque celle-ei existe (cf. Haefliger [4~) : il n'en est rien, puisque pour les droites de pente irrationnelle sur ie tore, la feuille ggn~rique est
~, alors que
~
est le module de
S I.
5 = ) Les r~sultats exposes sont ~galement valables si, au lieu de prendre toutes les formes basiques du feuilletage, on considgre seulement celles appartenant ~ une certaine sous-alg~bre diff~rentielle gradu~e de associ~e g certains types de Le module ~
~(F),
g~n~ralement
F-structures transverses.
d~pend alors de la structure en question. De plus, ~ l'~tude
du module minimal relatif, se greffent des questions d'irrationnalit~ de morphismes entre
~-alg~bres diff~rentielles gradu~es admettant des
Q-structures donn~es,
qui mesurent, en quelque sorte, en quoi ces feuilletages different de fibrations d'oO une th~orie de "l'homotopie irrationnelle" (cf. ~7]).
258
REFERENCES
[|]
EL KACIMI-ALAOUI A.
Cohomologie f e u i l l e t ~ e - Ex~nples de c a l c u l s ,
-
Th~se de 3~me cycle, Universit~ de Lille I, 26 juin ]980. E2]
[3]
FELIX Y. - HALPERIN S. -
GOTTLIEB D.
-
L.S.-category, (Transactions of American Mathematical Society,
]983).
Poinca~£ d u a l i t y and f i b r a t i o ~ , (Proceedings of the American Mathematical Society 76.1.79).
~4~
HAEFLIGER A.
-
Groupoldes d'holonomie ~t c l a ~ s i f i a ~ , (A para~tre dans les comptes rendus des journ~es de Toulouse 1982 sur la g~omgtrie transverse. Ast~risque).
E5]
HALPERIN S.
-
E6]
KAMBER F. et TONDEUR P. - Foliations and megric~, (Differential Geometry ) Birkhauser - |983).
E~
LEHMANN D.
-
L e c t u r ~ on minimal mode£s, Pub. I.R.M.A. Lille I, Vol. 3, fasc. 4, 1977.
S t r u ~ w t ~ de M a u r ~ - C a ~ n ~ Fo-structur~, I - f e u i l l e t a g e s de Ma~er-Cartan ( P r e p r i n t ) . II - espaces classifiangs,
Ast@risque,
E8]
RI~@fLER D. et SULLIVAN D. - Currewgs, flows and diffeomorphisms, (Topology 14 - 1975).
E9~
THOMAS
~0~
J.C.
-
Communication priv~e.
VAISMAN I .
-
Vari~t~S riemanniennes f e u i l l ~ e s ,
116, 1984, 134-148.
(Czechosl. Math. Jal 21 - 197|).
E.R.A au C.N.R.S. 07 590 UNIVERSITE DES SCIENCES ET TECHNIQUES DE LILLE U.E.R. DE MATHEMATIQUES PURES ET APPLIQUEES 59655 - VILLENEUVE D'ASCQ CEDEX (France)
LUSTERNIK - SCHNIRELMANN
CATEGORY
Jean-Michel
The introduction important Thomas,
results,
of L.- S. category
LEMAIRE
F@lix's and Halperin's
fundamental
but some seasoning
and also Jean-Claude
Lie algebra of a finite complex.
to the leading theme of this conference, The following
INTRODUCTION
(Nice]
due to Yves FElix and Steven Halperin,
local algebra as well.
material,
AN
into rational homotopy theory has lead to
on the structure of the rational homotopy
According
:
such results
are of interest in
notes are meant to serve as an introduction
paper [FH] : they therefore contain
of geometry which m a y make reading
They also provide an opportunity
to thank Jan-Erik Roos for his kind invitation
1983, which was a most pleasant and profitable
Definition A ~ X
fines
oat
X
is categorical
Definition
be a topological
rical covering
exists,
1.3 Examples
:
oat
space with base point
X
admits a finite covering
of
X
one sets
X = O
cat S n ~ I
* E X . A subset X , homotopio
h : X
h(A) ~ *
X , the L. - S. category of
there exists a covering
one.
if there exists a continuous map
such that
1.2. If
Symposium
AKO ELEMENTARY PROPERTIES.
1.1. Let
to the identity,
no original
[FH] easier.
to give this set of lectures in the Nordic Summer School and Research
§ I. - BASIC D E F I N I T I O N S
to
by
by categorical subsets,
X , to be the least number
n + I
categorical
subsets.
cat X ~
iff X for all
is contractible. n ~ I
.
n
one de-
such that
If no finite catego-
260
This definition originates
in the work (1934) of the two eponymous authors,
proved that any smooth function on a compact manifold of category least
n + I
admits at
critical points. Actually their definition required the categorical
sets to be closed, topic,
n
who
but only contractible
in
but the homotopy need not extend to
X
(i.e.
A ~
-X
must be null-homo-
X ). Later Fox (1941) modified the de-
finition by requiring the sets to be open, and proved that category so defined is a homotopy invariant,
while it is not if one insists that the sets be contractible
in
themselves.
The definition
adopted here is due to George Whitehead
to be equivalent to Fox's for
One can reformulate space of the
(1954) and can be shown
cw-oomplexes.
this definition
as follows
T nI X ~ X n+ I
: let
be the sub-
(n + 1)-fold product which consists of those sequences
(Xo, xl, ..., Xn)
such that
x.i = *
for some
i . Clearly the following definition
is equivalent to (1.2).
Definition
(1.4] . The category
fold diagonal
A : X
• X n+1
eat X
of a space
factors through
X
A
OY
of basE~-point
be the loop space on
preserving maps
Theorem
[1.5) (Ew]
I°f class
s n .
[SX,
). if
iff the
(n + 1)-
up to homotopy
: let
SX
be the suspension
Y . Recall that the set of homotopy classes
Y] ~
oat
s n
Tn "~I X
G. Whitehead made the following crucial observation X , and
is
X n+l
~
of
X
n TI X
Ix,
OY]
is a group. Then
x s n , the group
~8X, Y]
This result shows that category is some kind of "homotopical
: is nilpotent
nilpotency".
shall see that the rational category of a l-connected space actually is the
We
261
"homotopical
nilpotency"
some elementary
of its Sullivan m i n i m a l model.
topological
the homotopy type of
Before we briefly collect
facts. From now on we assume that spaces in sight have
cw-complexes
of finite type, with a base point. Details can
be found in [W],
J Lemma
1.B : If
ticular,
in
X
cat
is a homotopy
Lemma
I.? : Let
then
uo U u I U
Proof
: Since
H*(X n+1,
pairs).
is a homotopy retract
R
be a ring,
. Let
cat
Now
f
exists
; R)
. If
cat X s n ,
.
u° x u I x
, the cross-product ~
~
(X n+1,
T~)
o" . x u n
be the inclusion
lies
(of
Proof
h
x u I x ... with
and
jh ~ A , where
j * k* = 0
x Un) : Tn1 X ~
j
-X
is
the
in the long exact sequence of the
.
Let
f : X
be a map,
-Y
and
Cf - Y U f
C×
be the m a p p i n g
cone
. One has cat
(Of)
~cat
X +
: By (1.6) one may assume that
Cf = Y U CX
and both
Y
and
Then one easily sees that if A o, A I, ... , A n , CX
J ~oplication
1.9 :
Proof : CP(n) fore
m
Uu n
A* k* = h * j * k *
T~]
Lemma 1 . 8 .
then
cat X ~ cat Y . In par-
uo, u l, o.., u n 6 ~ ( X
k : (xn+1
IU
X s n , there
(X n + l ,
of
Y , then
Then
inclusion.
pair
and
u i E H (X, * ; R)
uoUu But i f
invarianto
..o U u n = 0
T nI X ; R]
of
cat ~ ( n )
~ n
CX
I
f
.
is a closed cofibration.
have the h o m o t o p y
Ao' ~I'
... , A u
is a categorical
Then
extension property
is a categorical
covering
of
YU
CX
in
Cf
covering of .
. Y ,
•
cat OP(n] I n .
is the mapping
cone of the Hopf map
by (I .8) and induction
the Chern class of the canonical
on
line bundle,
S 2n-I
n o Now if one has
.OP(n-
c E H2(O~(n]
cn ~ 0
and
I)
, there-
; Z)
cat ~(n)
is >- n
262
by ( I , ? )
.
"
One may p r o v e is
along
n . Incidentally,
cat X -< n
if
X
cat X < n + I
t h e same l i n e s
that
the category
of a product
of
n
spheres
from (1.5) on we have seen several good reasons to set
can be covered by
n + I
categorical subsets,
instead of
which was the original convention.
Lemma (1.10). Let homotopic. Then
F
i
cat
E
P~ B
be a fibration sequence such that
B . Let
H : B x I
, 8
p-1(A]
with
~o = idE
that
F has the h .e.p. in
that
k~id
E and
pP = H(p x I )
k(F)
Observe that if
i
. Hence
is categorical in
be a homotopy with
HI(A ) = * . By the homotopy lifting property of
and
is null-
E s cat B .
Proof : Plainly it suffices to prove that is categorical in
i
K,~I(p-I(A))
is null-homotopic,
= *
E : let
and
by exactness of the homotopy exact sequence.
A
and - E
one may assume
k :E
~ E be
k.~ I ~ id E .
• ~.(B)
TT.(p) : ~.(E)
if
~ : E x I
= F . W.n.l.g.
E , and thus is categorical in
= * E E . Then
H ° = id B
p , there exists
~I(p-I(A))
E
such
E
is injective
The converse is not true in general,
but it is true for rational spaces : this is [FH]'s first theorem, the mapping theorem, which we now discuss.
§ 2. - THE MAPPING THEOREM
We begin with recollecting some rational homotopy theory. Let
is
S
be a l-connected space. The ~urewicz homomorphism
a Lie algebra
map, w h e r e t h e
Lie
structure
is
given I
bracket, and on
H.(~S)
by
Milnor and Moore asserts that
(2.1)
h.
: "rr.(~S)
®Q
[a, b] = ab - (h
on
~.(OS)
Samelson
. A fundamental result of
induces an isomorphism of
~ ~ PH.(C~S ; Q)
by t h e
I
I ) lal Iblba
Lie algebras
263
where
P
stands for the (Lie subalgebra
We can choose a retraction
r i : H i(08
of
h.l
; G]
Hi(~s
r ." ~
i :
; ~i(f~)
--K(~i(~ ] ®Q,
: ~S
The latter induces an isomorphism
(2.2)
, for each
elements.
~ ' z (OS) ®
which can be viewed as a c l a s s i n
ri
of) primitive
on
~ ~
~. ® Q
, o r as a map
i) . Thus the product map
m
K(L(~
®Q)
® O, i)
i=I where t h e p r o d u c t i s g i v e n t h e weak t o p o l o g y , homotopy groups and t h e r e f o r e
is a rational
l - c o n n e c t e d space has t h e r a t i o n a l
i n d u c e s an isomorphism on r a t i o n a l
e q u i v a l e n c e . Thus any l o o p space on a
homotopy t y p e o f a p r o d u c t o f E i l e n b e r g - M a c Lane
spaces. We can now prove the mapping Yves F@lix during
Theorem
Proof Q-vector of
p
TT.(p)
p : E
*
-- B
, hence
is a surjective
Then
-~ F
This w i ! i
cat(E)
s cat(8)
follow from the existence
i -- * . To construct
sequence
for
s , we observe
groups
fibre
~.(j)
i ~
so that
#.(j)
maps
U.
± : F
±js
: ~.(~8)
s
~ E of and
. ft.(F]
v e c t o r spaces, by t h e e x a c t n e s s o f t h e homotopy s e -
quence. W r i t e
%(m)
spaces,
are
of a section
p : indeed
that
rational
.
to prove that the homotopy
in the fibration
map o f r a t i o n a l
see [FL].
space is a space whose homotopy
By Io10, it suffices
j : ~8
simple proof was found by
be a map between simply connected
is injective.
is null-homotopic.
: the following
; for a generalization,
: Recall that a rational spaces.
the fibre ij -
the conference
2.3. Let
such that
theorem
:, U. e Ker ~ . ( j ) isomorphically
on to
~.(F)
. Then,
by 2.2,
264
~8-- 0 i~I and
the restriction of
j
is the required section
K(Ui, i) x 0
K(Ker ~ i ( j ) ,
i)
i~I to
0 K(U i, i) i~I •
s
is a homotopy equivalence, whose inverse
We shall see in the next section that L. - S. category "localizes" well, that is, if
X
is the localisation of the homotopy type of
0
cat(Xo] s cat(X)
. Setting
X
at all primes, then
Cato(X ) , the rational category of
X , to be
cat
..(Xo] ,
we can reformulate the mapping theorem as follows :
I (1.11') : Let
p : E
• B
be a map such that
spaces are l-connected. Then
Cato(E ) s Cato(B)
is injective and the
~.(p) ® Q .
We conclude this section with another result of [FH], which we derive from the mapping theorem. The following concept, due to D. Gottlieb, was brought to the attention of rational homotopy theorists by H. Baues : Definition (2.4). The Gottlieb group as follows :
~ : Si
- X
Gi(X ]
represents an element in (id, ~) : X v Si
extends to
Gi(X )
GL(X)
~i(X)
defined
i f the map
- X
X x Si .
We leave to the reader to check that indeed
is the subgroup of
is the image of
at the base point and
~ (X) 0
~i(ev]
Gi(X )
, where
actually is a subgroup of ev : So(X )
is the group of self-maps of
- X X
~i(X] :
is the evaluation
homotopic to the
identity.
Theorem (2.5) ([FH] Thin I I I ) . n
.
Let
X
be a l-connected space of finite category
Then
Ca)
V i ,
G2i(X] ® Q - 0
co
(b)
E
dim 0 G2i+1(X) ® Q < n .
i-1 Tn other words, the groups
Gi(X )
are torsion except at most
n
of them which
265 occur in odd dimensions
Proof of 2.5.
: moreover the sum of their ranks is at most
Let us prove (b) first. Let
ft. : S
2r.+I !
n .
~ X , i = I ..... s
1
represent linearly independant elements in
easy
Godd(X ) ® Q . An
induction on
s
shows that
" (~1 .....
~
: (
" ~ s2ri +I
)v
s 2r.+1 V S 1 i=1
:
I-IS s2ri+ I I ] i¢=~I
extends to the product
to
~s ]
=X
Indeed, assume that
g
(~I'
....
~J)
can be extended
J
s2rj +I
X . Thena further extension to
~I~ S 2ri+1
is
3 given by the diagram S 2ri+1 ) v S 2rj+1
~j v id
XvS
2r.+I 3
(id, ~;) d_ X g /
I s2ri+l
/
F
~.xid 3
/
"
X x S 2rj+Iz"
Localizing we obtain a map of rational spaces s
s2ri+1 0
0
'~ich is injective on homotopy groups. By 1.9, the category of a product of (rational] spheres is
s , therefore
s ~ cat(Xo) = Cato(X ) ~ n
s
by the mapping
theorem. A proof of Ca] along the same lines is a little more involved : we need dames's reduced product construction. Let duced product (Z)~
i=o
zi/
be a connected,
pointed cw-complex
; the re-
is the quotient space
((~I'
"
Concatenation gives *~
Z
"'~j-1'
(Z)~
*' ~
'
3+I .... zi] ~ (z I ..... zj_1, zj ..... zi] )
the structure of a topological monoid with unit
( ) . dames's theorem asserts that the canonical map
multiplicative homotopy equivalence
(Z)~
Z
~ - ~ SZ . Now if
- 0 SZ ~ : mr
extends to a ,X
266
represents an element in : ( m r )~
Gr(X ) , one easily sees that
extends to
~ X , Localizing we obtain a map
Sr
sr+l
~
o
If
~
O
r - 2s , ~ S 2s+i ~ K(Q, 2s)
0
and
~
is injective on homotopy groups iff o
r e p r e s e n t s a non-zero element i n
G2s(X ] ® ~ = G2s[Xo)
is a polynomial algebra on one generator of degree cat[K(O,
2s]) = ~
G2s(X ) ® 0
2s
. But
H (K(O, 2s)
;
and therefore
by lemma 1.7. Thus the existence of a non-zero element in
would contradict
the mapping theorem.
Theorem 2.5 is a key ingredient
I
in the proof of [FH] Thm. IV, which says that
the rational homotopy groups of a finite complex are either zero for large enough degrees or grow exponentially.
§ 3, - SPACES OF CATEGQR,Y,,,I AND THE HOMOTOPY SUSPENSION DIAGRAM By definition
1.4 , a space has category
~ I
if
the diagonal can be factored
*
on
IX, Y]
through the wedge up to homotopy A
X
~XxX
h''-
"~Xv The map
h
defines a natural composition
X
h ,, X v X
X
law
(f'g]-
by
f * g ~ (f,g) o h
Y
which admits the trivial map as unit.
One says that theorem
h
is a co-H.spaoe structure on
(I.5) says that
this is why
~ (X)
[SX, Y]
is abelian for
n
Suspensions structure map
is abelian if
X
X . Observe that Whitehead's is a co-H.space
: incidentally
n ~ 2 "
are canonical
examples of co-H.spaces,
; but there are examples of co-H.spaces
with the "pinching" map as which do not have the homotopy
267
type of suspensions
: the simplest
example i s
one may show~ u s i n g t h e Hopf i n v a r i a n t , ture
that
(which extends the standard one on
only if
~ ~ 0(6)
S 3 U ~ e ? , where this
S 3)
~ E ~ 6 [38) "
= Z/12Z :
space a d m i t s a c o - H . s p a c e s t r u c -
iff
~ ~ 0(2)
. This is a torsion phenomenon of course,
, but is a suspension and over the rationals
things are much simpler :
Theqrem
(3.1)
(I. 8ernstein).
Every simply connected space of category
I has
the rational homotopy type of a wedge of spheres.
Tnis result will follow as an easy exercice from the characterization
of rational
category on the Sullivan model that we will discuss in the last section. We now give a proof which avoids models because it leads to interesting
side comments. We need
the
B o t t - S a m e l s o n theorem ( 3 . 2 ) .
The P o n t r y a g i n a l g e b r a
H.(O SX ; k)
, k
a field,
i s i s o m o r p h i c t o t h e t e n s o r a l g e b r a g e n e r a t e d by t h e graded v e c t o r space ~ ( X ; k)
.
We can now prove (3.1) for suspensions (2.1) and the Bott-Samelson
theorem that the Lie algebra
t o t h e f r e e L i e a l g e b r a g e n e r a t e d by ~.(X
: it follows from the Milnor-Moore
~.(X
; Q)
~(~
SX) ® Q
; we choose a b a s i s
theorem
is isomorphic
(x)
of
; Q) , and representatives
x
The family of maps
(x~)
: S
~OSX
.
defines a map
l×L x:
VS
~SX
lx t+1 whose adjoint groups
: V S
---~SX
induces an isomorphism
: since both spaces are simply connected,
valence.
on rational homotopy
this is a rational homotopy
•
We now introduce the "homology suspension diagram"
equi-
268
OX * DM
,
~SOX
X
(3.3) f
OX * ~X
VoH ~ X v X
f
c
j -XxX
in which is the evaluation map A
is t h e diagonal
j
is the inclusion
H
is the mopf map
V = (~ v ~) o ~
~(t, X) - k(t)
H(k, t, ~) = (t, X.p)
where
~ : SOX
~ SOX
v SOX
is the pinch map.
Theorem (3.4). The diagram (3.3) is homotopy commutative, the rows are fibration sequences and the right-hand square is homotopy-cartesian.
By homotopy-cartesian,
we mean that if one replaces
A
or
j
(or both) by a
fibration, the pullback square is homotopy equivalent to the given square.
S k e t c h of proof of 3 . 4
(a) the right hand square is homotopy commutative A~(t, m) = (m(t), m(t)) jr(t, m) =
(w(2t), ~o(0))
t S~
1 I
(m(1), w(2t - I)) t > 5
The required homotopy is a "simplicial approximation" of the diagonal in
I x I
(b) the right-hand square is homotopy catesian : a standard way to replace by a fibration is to consider the evaluation map at the ends
269
x [°'10
. × x x
Composition with the inclusion of constant paths ¢
is a homotopy equivalence.
those paths in contractible,
X
Now the pullback is
g : X
it is not hard to construct a homotopy
* OX
A , and
E-X
and
E+X
are
equivalence
~ ,, E-X UOX E+X
(c] it remains to show that the homotopy fibre of DX
is
E-X U ~ X E+X , that is, the set of
which start or end at the base point. Since
S~ X
the join
- X [0'I]
j
has the homotopy type of
: one may consider the fibre square
E-X x OX U OX
x OX
DX
x E+X
= E-X x E+X
(~(1), ~(o)) XvX
in which
E-X x E+X
r
is contractible,
OX
* F~X
J
•
and construct
• E - X x OX U OX
We leave the details to the reader (see [Si]).
a
XxX
weak equivalence
x E+X ,
•
Now, in the homotopy cartesian square
SDX
X
(~.s) X v X ~'\,,,-
the existence of
h
J
• X x X
is equivalent to the existence of a homotopy section
~
of
Thus : I Proposition suspension,
3.6. A space has category •
~ I
iff it is a homotopy retract of a
270
The proof of (3.1] is achieved if we observe that a retract of a free Lie algebra is free - in fact, any subalgebra
(3.7) RemarkS
: we call diagram
of a free Lie algebra is free.
(3.3) the homology suspension diagram because the
Serre exact homology sequence for the fibration G. Whitehead's
OX ~ P~
exact sequence for the homology suspension
= ~(0
On the other hand, if we apply the functor
(3.3),
diagram
L ~ ~(×)
setting
- S OX
•X
is
(see [W]).
.) ® Q t o t h e whole
we have
=~(SOX) ~ L{U-L)
where
UL
denotes the augmentation
ideal of the enveloping algebra of
is the free Lie algebra functor from vector spaces to Lie algebras,
Finally,
L , and
L
and
the map
2(~) ~(sn×)
.~(×)
:
is surjective,
because
O~
has a section
~IX : FiX
O8(OX)
. We therefore get
a diagram with exact rows
L(U'L ® U'E)
~ E(O-£)
O
=
O
- L(~'C ®~J"l~) - - - " L
II
l
Lt L
in which the right-hand square is a pullback. Lie algebra over
Proposition
Q
3.9.
occurs as
Let
L
~(×)
for some
'-'
J-L
l
x L
~
O
-g
Since by Quillen's theorem any graded X , we can conclude
be a graded connected
J : LEL
L
Lie algebra over
Q , and
.LxL
be the canonical map, represented
by the unit matrix.
Ker j
are free with minimal generating vector
and
j-1(a(L))
spaces isomorphic to
of UL ® ~
L~L and
UL
respectively,
Then the Lie subalgebras
u
271
Let us conclude this section with observing that the analogous statement holds for
discrete
groups
- and can be d e r i v e d
from Gruschka% theorem
: as a c o n s e q u e n c e ,
the fundamental group of any co-H,spaoe (connected) is free,
§ 4 . - .,THE GENERALIZED SUSPENSION DIAGRA¥~ THE GANEA FILTRATION,
AND L . - S .
CATEGORY
FOR DG ALGEBRAS. We wish to generalize the homotopy cartesian square (3.5), to g e t
X(n)
n
X
I~ n TI X r
I~ j
To achieve this, we may replace either back
xn+1
&
or
j
by a fibration and take the pull-
X(n) , whose homotopy type is then well-defined, We will then have, by general
homotopy theoretic nonsense :
I (4.1)
cat
X ~ n
iff
X
is a homotopy retract of
Of course, we must try to describe Again we may first replace
&
X(n]
X(n)
to make (4.1) of any significance.
by the evaluation map at integral points
e : X[O'n]
r Xn+1
. (X[O],
X
~(1) . . . . .
X(n))
which is a fibration. Then
X(n)
Note that each piece
.
n U i=e
{X ( X [ O ' n ]
{X I X(i) ~ ~}
} X(i)
= *}
is contractible : thus
equipped with a standard categorical covering,
X(n)
comes
272
Exercise :
cat[X(cat
X)) = c a t X .
Moreover the intersection more-than-formal
space, say
o f two p i e c e s has t h e h o ~ o t o p y t y p e o f
a n a l o g y between t h e s t r u c t u r e
here
a deformation retract. Indeed associative
H.space
OX
and t h a t
(n + 1}
of a projective
affine spaces given by
the intersection of two affine charts admits X(n)
is the
n-th
SI
as
projective space of the homotopy
, and on t h e o t h e r hand one checks t h a t
GP(n)
~ K(Z,2)(n),
0 K [Z, 2) ~ S 1
Another approach to finn
X(n)
C P (n) , together with its covering by
homogenous coordinates :
with
of
~£ . There i s a
X(n)
through a construction
Pi : Ei
~ B , i = 1,2
, due £o
inspired
W. G i l b e r t ,
is
to convert
by t h e Whitney sum o f v e c t o r
be two fibrations with fibers
be the projection of @he mapping cylinder of
Pi
on
~1 x ~ 2
into
a libra-
bundles. Let
F i , and let
~i : Zi
~8
B . Let
Pl ~ P2 : Zl x E2 UElXE 2 E 1 x Z2 = E I ~ E 2
be the restriction of
j
- B x B
' and
Pl @ P2 : EI#~E2
-B
be defined by the pullback square
EI
iE2 Eli Pl @ P2
Pl &~ P2
&
B
~ BxB
one may check the following
Proposition 4,2 (a]
P l ~ P2
and
Pl ~ P2
are Hurewicz fibrations with fibre
F 1 ~ F2
(the
join of the two fibres] (b)
Let
~ ~ E+X
~X
i s a homotopy e q u i v a l e n c e
be t h e p a t h space f i b r a t i o n
with fibre
, Then t h e r e
273
E+~!E+X
~
~
>#< E+X
such that the above triangle commutes (c] If lence
p : E
...=. B
E U F CF
up to homotopy
is a fibration with fibre
, ~ E ~ E+B
PIE ~ P , F I C F
= *
F , there is a homotopy
equiva-
such that the triangle
E U F CF
where
"n'>)&(n+1)~xn+1
~
" E ~ E+B
, c~mutes
up to homotopy.
=l
From this we readily deduce the
Theorem 4 . 3 .
In the diagram
nx -Pv ~ ( ~ ) P
is a differential ideal,
. This algebra represents a rational
X[n] , whose minimal model is a model of
~/-~+I.
:
Theorem 4.4 ([FH] VIII and IX)
(a] The Canna space X[n]
X(n)
has t h e r a t i o n a l
and a wedge of spheres
~n(X) c
.X[n]
~(X)
homotopy t y p e o f t h e wedge of
, such that the composition
v ~n(X) - - ~ X ( n )
c
,X(n
+ I)
is null-h~notopic (rationally]. (b) eatoX ~ n
iff
~
is a retract of the minimal model of
~/~+I
|
275
To prove 4.4, F@lix and Halperin first translate the pullback square
x(n)
-x t°'n]
Xc
into
J
. ~
-
, Xn + l
to obtain a (neither free nor minimal) model of
DG-algebras
show that this model is quasi-isomorphic
X(n]
. Next they
~/~+1
to the direct product of
and a
trivial algebra. This trivial algebra is huge in general
X(1) = S e X
while
~/~
is the trivial
the reduced homology of of
Hem(V, Q) = s~(X)
~I(X) in
rational homotopy type of
: for instance if
algebra
is isomorphic
~.(SOX) X[I]
O ®V
CP(~)
(n) ~ CP (n]
is when
, while clearly
discussion
~(X) = ~ ]
CP(~]
of the rational
h~otopy
~ n
(4,4)(b]
i.e.
X~
has the ~(X)
(over the integers
, for all
type
of
This p r o v i d e s an i n t e r n a l
is
K(Q, 2r) : in !)
n .
X(n)
, see It]
.
means that a space has
iff its model is a homotopy retract of a DG-algebra
~ n : compare with (1.5)
of nil-
!
definition
of the L.-S, category of a DG-algebra.
It is an open question whether the model of any space of category quasi-isomorphic
X(1)
, that is, when
I n ] ~ CP (n] £
We conclude with pointing out that Theorem rational category
to a supplement
= s ~=---~) . The only case when
we already mentioned the hemotopy equivalence
Fo~ e f u r t h e ~
potency
, d = 0 , VoV = O ; t h u s
(as vector spaces)
an abelian algebra on a single generator of odd degree, particular,
n = I , one has
to a DG-algebra of nilpotency
and has been checked by F@lix and Thomas for
n
actually is
n : this is obvious for
n = I ,
n - 2 .
REFERENCES, [FH]
Yves FELIX and Stephen HALPERIN, R a t i o n a l L . - S . c a t e g o r y and i t s tions,
T r a n s , A,M.S, 2?3 (1982) p p , 1-38,
applica-
276
EFL]
Yves FELIX and Jean-Michel LEMAIRE, On the mapping theorem for L,-S, category,
[Ga]
Topology, 24, 1985, 41-43.
Tudor GANEA, L . - S . c a t e g o r y and s t r o n g c a t e g o r y ,
t11o J . Math.
11 (1957)
p p . 331-348.
[Gi]
W. GILBERT, Some examples f o r weak c a t e g o r y and c o n i l p o t e n c y ,
Ill.
J . Math.
12 (1958) p p . 421-432.
EJ]
Ioan JAMES, On c a t e g o r y i n t h e sense of L . - S . ,
Topology 17 (I£78)
p p . 331-
348.
ELS]
J e a n - M i c h e l LEMAIRE and F r e n q o i s SIGRIST, Bur l e s i n v a r i a n t s
d'homotopie
rationnelle li@s & la L.-S. cet~gorie, Comm. Math. Helvo 56 (1981),
EW]
103-122.
George ~tHITE~EAD, Elements of Homotopy Theory, Graduate Texts in Math. Springer-Vet lag.
[L]
Jean-Michel LEMAIRE,
8ur le type d'homotopie rationnelle des espaces de
Gan@a, in Homotopie Alg@brique et Alg@bre Locale, Ast@risque n ° 113-t14 (1984),
pp 238-247.
Jean-Michel LEMAIRE Laboratoire de Meth~matiques U.A. eu C.N.R.S. n ° 168 Universit@ de Nice Parc Valrose F - 06034 NICE-CEOEX
S~RIES DE BASS DES MODULES DE SYZYC~IE par
Tousles n~th~riens,
anneaux consid~r~s sont des anneaux commutatifs, locaux,
de m~me corps r~siduel
Soient socie ~
M
Jack LESCOT
(R,m)
k.
un anneau local et
M
un R-module de type fini. On as-
deux s~ries formelles :
la sgrie de Poincar~ : P~(t) =
Z
bi(M)t i,
o5
bi(M) = dim k Tor~(M,k),
la s~rie de Bass : IM(t) =
Consid~rons
~ '~i(M)ti , i~>O
P" :
jective minimale de
M
"''--~ Pn+l --~ Pn -~...--~ Po --~ 0 et soit, pour
n i~me module de syzygie de
Dour
n > O,
~i(syzn(M))
une r~solution pro-
syzn(M) = Im(Pn --+ Pn-1 )
le
bi(syzn(M)) = bi+n(M)
pour
M. II est clair que
i > O. On montre ici comment les IR(t)
i M oil D.(M) = dim k EXtR(k,~).
sont d~termin~s par
M (on note
I~(t)) :
Th~or~me A.-- On a :
l~yzn(M)(t) = (bn_1(M)+...+tn-lbo(M))IR(t)-tn-ll~(t)+(1+t)tn-IIFn(M)I(t),
o__~ IFn(M) I(t) gradu~
d~signe la s~rie de Hilbert d'un certain espace vectoriel
F (M), associ~ g n Soit
E
M.
une enveloDpe injective de
le dual de Matlis de
k
sur
R
et soit
M v = HomR(M,E)
M. Ii existe un produit homologique associ~ ~
M :
R R v ,k) -~ Tor.( R R v ,k), Tor.(M,k) ®R T or~(M
et
F (M) est un sous-espace vectoriel de Tore(MY,k), d~fini ~ l'aide de ce n produit. La situation est simple Dour le module M si le produit est nul.
Dans ee eas
IFn(M) l(t) = I~(t). Ainsi Dour
M = k :
278
Th~or~me B.- Soit n>O
(R,m)
un anneau local non r~gulier alors pour tout
:
l~yzn(k)(t)
= (bn_1(k) + tbn_2(k) +...+ tn-lbo(k))IR(t)
+ t n P~(t).
On utilise ces r~sultats pour montrer que la dimension syzyg~tique introduite par Roos dans
y(R)
[11] est infinie pour la plupart des anneaux qui ne
sont pas de Gorenstein. Afin de mesurer la complexit@ de l'anneau th~or~me A, on pose la question suivante Existe-t-il un entier type fini, on ait :
On montre qu'il e n e s t
o(R)
W(M)
le sous-espace
de
classes d'anneaux.
:
de
Tor$(RV,k)
W(M) = Im s~
o~
associ~ ~
M
engendr~ par les valeurs du s : R--~ R/J
est la pro-
Ce r~sultat est utilis~ pour donner une nouvelle d~monstra-
tion d'une caract~risation Szpiro
M
Vp > o(R) ?
on d~finit le produit homologique
M = R/J, on a
jection canonique.
tel que pour tout module
bien ainsi pour quelques
Dans la premiere section,
produit. Lorsque
vis ~ vis de la formule du
Fo(R)(M ) = Fp(M)
Voici le plan de cet article
et on ~tudie
R
:
des anneaux de Gorenstein,
due g Peskine et
[10]. Dans la deuxi~me section,
on d~montre les th~or~mes A et B, et dans la
troisi~me section, on ~tudie l'existence de D'autres propri~t~s et applications
o(R)
pour quelques
du produit homologique
cas. associ~
un module se trouvent dans [9], papier auquel nous ferons r~f~rence pour quelques d~tails. O. NOTATIONS ET RESULTATS PRELIMINAIRES Soient
(R,m)
O.1. Soit A =
@
p>0 TAl(t) =
A
V
un anneau local,
k
un k-espace vectoriel,
un k-espace vectoriel
son corps r~siduel. on note
p ~ IA Itp
p~O
p
IVI
gradu~. Si pour chaque
la s~rie de Hilbert de
A.
sa dimension. p,
IApl < =
Soit on note
279 Soit
f : M--~ N
un homomorphisme de R-modules, on note
f~ (resp. fp)
l'homomorphisme induit en homologie :
f. : TorR(M,k)--~ T o r R ( N , k ) ( r e s p .
Soit
M
fp : TorR(M,k)--~ P Tor pR(N,k)).
un R-module de type fini. Les modules de syzygie de
M
sont d~-
finis ~ un isomorphisme pros g partir d'une rgsolution projective minimale par
syz°(M) = M e t ,
pour
n > O, par
P.
syzn(M) = Im(Pn--~ Pn_1).
0.2. Duals de Matlis On choisit pour l'anneau siduel Si
k. Si
M
f : M--~ N
R
une enveloppe injective
est un R-module, soit est un
R
M v = HomR(M,E)
homomorphisme, soit
induit entre les duals de Matlis, ainsi si
E
le dual de Matlis de M.
fv : Nv __~ M v
a C Nv
de son corps r~-
l'homomorphisme
fV(a) = aof
(composition
des applications)• Nous ferons un usage constant de l'isomorphisme canonique E ~ R v = HomR(R,E ) dans
E
identifiant les ~l~ments de
E
~ des applications de
R
et vice versa.
La formule de dualit~ de ([3], chap. VI,5.3) montre : Pour tout R-module
M
et pour tout
p 6 ~, il existe des isomorphismes
fonctoriels : TorR(MV,k) ~
P
Comme
kv ~ k
Ext~(k,M) v.
il en r~sulte imm~diatement : V
Pour tout R-module de type fini En particulier
P~ (t) = l~(t).
Rv IR(t ) = P R (t) =P~(t). On notera aussi que P~(t) =l~(t)°
Le probl~me du calcul des s~ries de Bass est ainsi ramen~ g u n
calcul
de s~ries de Poincar~ (pour des modules qui ne sont pas n@cessairement de type fini). Rappelons enfin que de l'anneau
R
TorR(RV,k)
peut ~tre d~finie par
n'est jamais nul et que la profondeur [l] :
Prof R = inf {i I EXtR(k,R) # O} = inf {i i TorR(RV,k) # O}. I
I. LE PRODUIT HOMOLOGIQUE ASSOCIE A UN MODULE I•I. Soient ~valuation, e ( a ® b )
M
un R-module et
8 : M @ R M v --~ E ~ R v, l'homomorphisme
= b(a). On associe ~
e
un ~roduit homologique :
280
R R V ,k)--~ Tor~(RV,k) Tor.(M,k) O R Tor.(M
par composition du produit ext@rieur :
R Tor,(M,k) %
Tore(MY,k) -o Tor.~ R'M ®RMV,k)
et de l'homomorphisme induit en homolo~ie par
0 :
R ® R MV,k) -~ Tor~(RV,k), 8. : Tor.(M
([3], chapitre XI). On note
< ,>
l'application R-bilin@aire correspondante
et on l'appelle le produit homologique associ~ ~
M.
On d~signe par : R~RV ,k ~ le sous-espace vectoriel gradu@ de Tor.~ j image de R M v ,K) ~ T o rR. ( M , k ) % T or.( par le produit associ~ g M, Wp(M) sa composante W(M)
de degr~
p, (Wp(M) = 0
si
D
est dans
2) Comme
R R ! ~ = T o r . + l ( M , k ) --~ T o r . ( s y z ( M ) , k ) , x E TorR(syzl(M),k) P
est surjective, on a
x = 6(x')
et
avec
<x,y> = O,
les modules
= Im(J r-1 __~ jr). De mani~re similaire au
W(co-syzr(N))
c W(N)
(voir [9] pour des dE-
soit un R-module de dimension injective finie r(de type fini)o N est une somme directe de modules injec-
Le r igme module de co-syzygie de tifs tous isomorphes ~ et donc
E
[I]. Par cons6quent
W(E) = W(co-syzr(N)) c W(N)
W(N) = W(E) = Tor~(RV,k).
Ce dernier point permet de donner une nouvelle demonstration d'un r6sultat de Peskine et Szpiro ([10], th~or~me 5.5).
1.9. Th6orgme.- Pour qu'un anneau local noeth~rien il faut et il suffit qu'il existe un id6al R/J
J de R
R
soit de Gorenstein,
tel que le R-module monog~ne
so it de dimension in jective ' finie. Preuve : Rappelons qu'un anneau local noeth~rien
il est de dimension injective
est dit de Gorenstein si
finie [I]. La condition est clairement n~cessaire.
Pour la r~ciproque, notons
s : R --~ R/J
W(R/J) = Im s,v (th~or~me
D'autre part, puisque R/J est un R-module de R v W(R/J) = Tor,(R ,k) (I .8). Par consequent
1.4)
dimension injective finie, s, : Tor ((R/j)V,k) que l'anneau
R
--~ Tor (RV,k)
est un homomorphisme
surjectif.
IIen
rEsulte
est de dimension injective finie, donc qu' il est de Gorenstein.
II. UNE FILTRATION SUR 2.1. D~finition.croissante
la surjection canonique. On sait que
(Fn(M))n6~
Tore(MY,k) Soit sur
M
un R-module,
Tor~(MV,k)
on d~finit une filtration d~-
de la mani~re suivante
•
284
R
Fo(M) = Tor (MY,k) Vx E TorR(M,k),
et pour
j < p,
R v F p (M) = {y ly E Tor.(M ,k)
'
et
<x,y> = 0}. II est clair que les
espaces vectoriels gradu~s de caractgris~e par :
p>O
F (M) sont des sous P et leur intersection F (M) est
TorR(MV,k)
Foo(M) = {y l y E Tor,(M ,k) et
2.2. Th~or~me.- Soient
M
Vx C Tor (M,k), <x,y>}.
un R-module de type fini et
nombres de Betti. Alors la s~rie de Bass de
syzn(M)
(bp(M))pE~
est donn~e pour
ses
n > 0
par la formule : l~yzn(M)(t)=(bn_l(M)+tbn_2(M)+...+tn-lbo(M))IR(t)-t Preuve : Soit pour tout
p C~
P.
n-I M n-I IR(t)+(l+t)t IFn(M)l(t).
une resolution projective minimale de
les suites exactes : O - ~
syzP+1(M) --~ P
M. Consid~rons
--~ syzP(M) --~ O
P et les homomorphismes de connexion associ~s ~ ces suites et aux suites duales :
6
: Tor~+|(syzP(M),k) --~ Tor~(syzp+l(M),k),
6' : Tor~+]((syzP+](M))V,k) --~ Tor~((syzP(M))V,k).
En it~rant ces homomorphismes, on obtient pour
8n
n > O :
R n : Tor +n(M,k) --~ Tor~+(n_l)(syzl(M),k ) --~...--~ Tor~(syz (M),k)
v R v ,k). ~,n : Tor~+n((svzn(M))V,k ) --~ Tor~+(n_ I)((syzn-1 (M)),k)--~. ..--~ Tor~(M
Posons en outre
6°
Tore(MY,k). Notons que morphisme
et
~n
8 'o
les identit~s sur
Tor~(M,k)
sur les composantes de degr~ sup~rieur ou ggal ~
x C Tor~(M,k)
et
et
est un homomorphisme surjectif de degr~ -n (iso-
y E Tor$((syzn(M))V,k)
de la proposition 1.2, on obtient :
n). Soient
des gl~ments homog~nes, ~ partir
<x,~'n(y)> = ± .
La d~monstration du th~or~me se fait en plusieurs ~tapes : Point 1.- On a Soient Si
z
x E Tor~(M,k)
Par suite
F (M) = Im(8'nl. n
un ~l~ment homog~ne de avec
j < n, on a
Tor~((syzn(M))V,k)
et
y = 6'n(z).
<x,y> = <x,6'n(z)> = ± = O.
Im(6 'n) c Fn(M ).
Pour obtenir l'inclusion inverse, nous avons besoin du
285 Lemme.- Soit O -~ N] --~ R n s N --~ O une presentation minimale de R v v N. S i y E Tor,(N ,k), on a s,(y) ~ O si et seulement s i i l existe x E TorR(N,k) o
tel que
<x,y> # O.
Admettons le len~ne pour l'instant. Soit
R v y E Tor (M ,k)
un ~l~ment
homog~ne qui n'est Das dans Im ~,n. On peut trouver un entier r, tr ,r+| tel que y E I m ~ et y ~ Im ~ . En particulier, il existe z E TorR((syzr(M))V,k), ~l~ment homo~gne, tel que pas dans l'imaF~e de
O = <x,~
vr
x E TorR(M,k)
Preuve du lemme : Soit x E ToroR(N,k)
tel que
~r(x) = u
(z)> = ± # O. Par consequent
tel que
n'est pas dans
F (M). n
y E TorR(NV,k). Supposons qu'il existe
<x,y> # O. Comme
surjective, on peut trouver
alors y
So : T°rR(Rn'k)o --~ T°rR(N'k)o
x' E TorR(Rn,k)
tel que
est
s,(x') = x. On a
v * y)" # O. R~ci<x',s,(y)> = <s.(x'),y> = <x,y> # O, (1.2), et donc sV( v proquement supposons s.(y) = O. Soient (el) , | = <s.(c~(ei)),y> = # O, (1.2).
Point 2.- Pour chaque de degr~
r
l'homomorphisme
G 'n
induit un isomorphisme
-n ~,n : Fr(syzn(M))/Fr+1(syzn(M)) --~ Fr+n(M ) /Fr+|+n(M).
Comme l'application Tor~((syzn(M))V,k)
an
est dans
est surjective, un ~l~ment homog~ne Fr(syzn(M))
x E Tor~(M,k), j < n+r, on a :
y
de
si et seulement si pour tout
= O
ou de fagon ~quivalente
<x,6'n(y)> = O. On a donc l'~quivalence : y E Fr(syzn(M)) ~=~ ~,n(y) EFr+n(M). D'autre part, le point precedent montre que pour tout
r :
~'n(Fr(syzn(M))) = Fr+n(M) o On en d~duit le r~sultat.
Nous sommes maintenant en mesure de d~montrer le th~or~me.
286
A partir de la lonF~ue suite exacte d'homologie associ~e ~ la suite exacte
0 -~ (syzn-l(M)) v --~ pV -~ (svzn(M)) v O, et en tenant compte du n-I ~ --~ I, on peut ~crire :
point
IR yzn(M)(t) = bn_ I(M)IR(t) - IR yzn-l(M)(t) + (l+t) rF l(syz n-l(M))] (t). Ce qui gtablit la formule pour
n = I. On peut aussi gcrire :
n-1
IRsyzn(M) (t) = bn_l, (M)l~(t)+tlsYZK ~
n-1
(M) (t)-(1+t) IFo(sYz
.
n-1
(M))/Fl(sYz
(M))[(t) •
En utilisant le point 2, on obtient : n-1 (M~t)-(l+t)tn-IjFn_l(M)/Fn(M)l(t).
IRYZn(M)(t) = bn_ I(M) IR(t) + tlRYZ
Ce qui permet d'obtenir la formule par r~currence.
Remarque I.- On peut d~montrer la formule du th~or~me 2.2. en utilisant les r~sultats de compl~mentaire
Foxby
[4]
sans toutefois obtenir une description de terme
(|+t)tn-lIF (M)[(t), ce qui est le but recherch~ ici. n
Remarque 2.- On peut d~finir sur la filtration
Tor~(M,k)
une filtration similaire
F. . Cette nouvelle filtration intervient dans une formule ex-
plicite de la s~rie de Poincar~ des modules de co-syzygie d'un module M [9].
2.3. Exemple.- Soit finie
d,
@ < d+t-n
TorR(MV,k)
F.
se d~crit simplement :
(Voir [9] proposition 1.2.9 pour la d~monstration).
3
IF (M) l(t) = a +...+ a n
pM(t)
un R-module de type fini, de dimension injective
IM(t) = a o + . . . + a d td. La filtration
F (M) = n j Ainsi
M
o
t d-n d-n
et comme d 'autre part
IR(t).IM(t -I) [4], on obtient des formules explicites pour la s~rie
de Bass des modules de syzygie de
M.
Ii est clair que la filtration
F.
est triviale (Vn Fn(M)=TorR(MV,k))
si et seulement si le produit homologiques associ~ ~ dit si
M
est nul, autrement
W(M) = O. Dans ce cas, la formule du th~or~me 2.2 se simplifie et on
obtient : 2.4. Th~orgme.- Soit s~rie de Bass de
syzn(M)
M
un R-module de type fini. Si
est donn~e pour
n > 0
W(M) = O
par la formule :
la
287
l~yzn(M)(t) = Lorsque
R
(bn_1(M)+tbn_2(M)
n'est pas un anneau r ~ u l i e r ,
le th~or~me B de l'introduction Dans
+...+ tn-lbo(M))!R(t) + tnl~(t).
puisque
W(k) = O (1.6) et on obtient
l~(t) = P~(t).
([11], chapitre 7), J.E. Roos introduit la dimension syzyg~tique
d'un anneau local
R :
y(R) = inf {t I tout t i~me module de syzygie de type fini est projectivement ~quivalent ~ un (t+I) i~me module de syzygie}, ou n'existe Das de tel
t. (On dit que
si on Deut trouver
pet
q
anneau de Gorenstein,
on a
des autres cas, on a
y(R) = =,
2.5. Proposition.~d(R) # 1
on a
Preuve
M
tels que
et
N
y(R) = ~
s'il
sont ~rojectivement
M@R p ~ N@Rq).
Si l'anneau
~quivalents R
est un
y(R) = dim R. Nous allons voir que dans la plupart
Soit
([l]], probl~me 3.8, p. 249).
R
un anneau local de profondeur
d. Si
y(R) = ~.
: Si
y(R) < ~
et
q > p ~ y(R), alors on v~rifie facilement que
tout pleme module de syzy~ie de type fini est projectivement ~quivalent ~ un i~me module syzygie. Soit M = syzr(k), r > d on va montrer que M n'est
q
pas projectivement
~quivalent ~ un r+2 igme module de syzy~ie.
n'est pas r~ulier,
le th~orgme B permet d'~crire
Comme l'anneau R
:
IRM~RP(t) /IR(t) = (P+br_l(k)+tbr_2(k) +...+ tr-lbo(k))+trp~(t)~ /iR(t)
C'est une s~rie formelle g coefficients non nul de
IR(t )
seuls les Soit
r-d
est
l~yz (N)@ Donc si
Rq(t )
/ IR(t)
a au moins ses
Wj(syzr(M))
M = O
on ait en fait
n-l-d
on doit avoir
est
I,
~.
premiers
coefficients
dans
~.
n < r+1. Par consequent on a
y(R) = =.
III. UNE CONSTANTE ASSOCIEE A L'ANNEAU Soit
P~(t)
de la s~rie ci-dessus sont dans
le th~or~me 2.2 on v~rifie que
M~R p = syzn(N)@R q
n~cessairement
~. Comme le premier terme
~d(R)t d, et que le terme constant de
premiers coefficients
n > d, en utilisant n
dans
R
un R-module de type fini, on a vu (corollaire si
j < r. On peut esp~rer que si
W(syzr(M))
de la faGon suivante
:
r
1.3) que
est ehoisi assez grand
= O. On est conduit ~ d~finir une constante
o(R)
288
3.1. D~finition.-
Soit
(R,m)
un anneau local.
o(R) = inf {r I pour tout R-module S'il n'existe En utilisant de comparer tions,
pas de tel entier,
M de type fini, W(syzr(M)) on pose
W(syzr(M))
de
M
et de
= 0, et, Fr(M)
aussi ~tre d~finie
par
du th~or~me
syzr(M))
= F(M),
= 0}.
a(R) = =.
le point 2 de la demonstration
les filtrations
Posons
2.2 (qui permet
on constate
sont ~quivalentes.
que les asserDonc o(R) peut
:
o(R) = inf {r I pour tout R-module
M de type fini, Fr(M ) = F ( M ) } .
Ainsi
de l'anneau
o(R)
mule du th~or~me Question Avant pri~t~s
mesure
la complexit~
2.2. Comme
W(R) ~ 0
on a toujours
: Pour un anneau local
de donner des exemples
simples
de
R
(R,m),
vis ~ vis de la foro(R) ~
I.
a-t-on toujours
o(R) < ~ ?
o~ il en est bien ainsi, notons
deux pro-
o(R)
a) o(R) > prof R = d. En effet, jective b) Soit
il existe des modules d. On a alors
x
un ~l~ment
(R) < o(R/xR) On obtient
M
W(syzd(M))
de l'id~al
de type fini,
de dimension
pro-
= W(R) # O.
m,
x
non diviseur
de z~ro alors
+ 1
ce r~sultat
des modules
en comparant
de syzygie
sur
les modules
de syzygie
R/xR (pour les details
sur
R
voir [9],
1.3.7).
3.2. Proposition.s : R ~-~ R/a
la projection
l'application Preuve (thEor~me a. Donc si
V
s.
canonique.
: L'application
1.4). Tout W(R/a)
l'application
(R,m)
un anneau local de socle Alors
o(R) = |
!
; notons
si et seulement
si
est nulle. s~
rleme-module
= O
o(R) = I. R~ciDroquement injectif.
Soit
on a soit
f : R/~--~ R n,
Par consequent
R/a
est nulle si et seulement de syzygie
M
avec
W(M) = 0 (corollaire xl,...,x n
si
W(R/a)
= 0
r > 0, est annulE par
1.5), et par suite
un syst~me
g~nErateur
de
m
;
f(y) = (yxl,...,yx n)
est un homomorphisme
est un premier module
de syzygie.
Si
289 o(R) = I, on a
W(R/a)
II existe Soit
lier.
des anneaux
(R,m)
en outre que
= O.
R
R
n'est pas de la forme
l.ll). Consid~rant
I~R/m2(t) R --
la suite exacte
on en d~duit que la condition IR(t)
= (Im21-1m/m21t+t2).p~(t)._ condition
le produit
d, alors Preuve
Tor~(RV,k).
o~
:
est satisfaite
par B~gvad
est un anneau r~gu-
)
R v --~ ( m 2 ) V - - ~ O ,
si et seulement
d'anneaux
v~rifiant
avec
sa s~rie de Bass qui est calcul~e
(R,m)
On suppose
si
cette der-
[2].
des anneaux artiniens
Soit
2
([2] ou [8] exemple
sv
O--~ (R/m2) v
o(R) > 1 : il suffit
la condition
dans
pr~c~dente
[7] ou [8].
un anneau local de Gorenstein
de dimen-
o(R) = d+|.
: En effet la seule composante
non nulle de
On en d~duit que pour tout R-module
la profondeur
(B,b)
fibr~ de deux anneaux v~rifiant
3.3. Proposition.sion
B/b 3
L'existence
a ~t~ d~montr~e
On notera qu'il existe
m_3 = O, de socle
= (Im/m21-t).P~(t)
s~ = O
nitre
et de consid~rer
o(R) = I :
un anneau local tel que
Dans ces conditions
de prendre
tels que
de
R
est
d
3.4. proposition.-
M,
on a n~cessairement
Soit
(R,m)
Tor$(RV,k) ~
Fd+I(M)
est
= F (M). Comme
o(R) = d+].
un anneau de Golod,
n = Im/m21. Alors
o(R) ~ n+2. (Pour la d~finition Preuve on montre
: En utilisant
Soient
phisme de R-modules. O < p < l+Im/m21. Soient de
M
(R,m) Pour que
([9],
[5] ou [6]).
de Ghione et Gulliksen
([5] th~or~me I)
f, = O
f : M --~ N
un homomor-
il faut et il suffit que
de type fini et
tenu du point
Fn+2(M ) = F=o(M)
l'homomorphisme
O --~ (syzr(M))V
un anneau de Golod et
fp = O
pour
1.3.|O).
un R-module
M . Compte
pour ~tablir que r > n+2,
l'argument
voir
le
Lemme.-
minimale
des anneaux de Golod,
P.
une r~solution
I de la d~monstration
projective
du th~or~me
2.2,
il suffit de montrer que pour tout
de connexion
associ~
s r'v ~ pV --* (s y z r+! (M) )v --* 0 r
g la suite exacte est surjectif,
ou ce qui re-
290
vient au m@me que
s: 'v = O. Le corollaire 1.3 montre en particulier que
Wo,q(syzr(M)) = =0 si
q < r. Par consequent, pour tout
x E Tor~(syzr(M),k)
et tout
y E Tor~((syzr(M))V,k), q < r, on a < x , y > = O. En utilisant le r,v s = 0 q q < r donc finalement que s: 'v = O en utilisant le lemme precedent.
lemme de la d~monstration du th~or~me 2, on peut conclure que pour
Bibliographic [I] H. BASS.- On the ubiquity of Gorenstein rings, Math. Z., 82 (1963), 8-28. [2] R. B~GVAD.- Gorenstein rin~s with transcendental Poincar@ series, Math. Seand., 53, 1983, 5-15. [3] H. CARTAN, S. EILENBER~.- Homological Al~ebra, Princeton Univ. Press, Princeton, N.J., 1956. [4] H.B. FOXBY.-
Isomorphisms between complexes with applications to the
homological theory of modules, Math. Scand., 40 (1977), 5-19. [5] F. GHIONE, T.H. GULLIKSEN.- Some reduction formulas for the Poincar@ series of modules, Atti. Accad. naz. Lincei
LVIII Ser., Rend., CI.
Sci. fis. mat. natur., 58 (1975), 82-91. [6] T.H. GULLIKSEN, G.L. LEVlN.- Homology of local rings, Queen's papers in pure and applied Mathematics, n ° 20, Queen's Univ., Kingston, Ontario, (1969). [7] J. LESCOT.- La s@rie de Bass d'un produit fibr@ d'anneaux locaux, Comptes-rendus, Acad. Sci., Paris, 293, S~rie A (1981), 569-571. [8] J. LESCOT.- La s@rie de Bass d'un produit fibr@ d'anneaux locaux, S@minaire d'Alg~bre P. Dubreil
et M.P. Malliavin (1982), Lecture
Notes in Mathematics 1029, 218-239, Springer 1983. [9] J. LESCOT.- Produit homolo~ique associ~ g u n
module et applications,
Pr@publication n ° 14 (1983), D~D. de Math. et de M~canique, Univ. de CAEN. [lO] C. PESKINE, L. SZPIRO.- Dimension projective finie et cohomologie locale, Inst. Hautes Etudes Sci. Publ. Math., Paris, n ° 42 (1973), 47-119. [11] J.E. ROOS.- Finiteness conditions in commutative algebra and solution of a problem of Vasconcelos, Commutative Algebra, Durham 1981, Ed. by R. Sharp, London Math. Soc., Lecture Notes, Vol. 72, 1982, 179-203.
Dgpartement de Msthgmatiques, Informatique et M~canique Universit~ de CAEN 14032 CAEN CEDEX - FRANCE
ON THE SUBALGEBRA GENERATED BY THE ONE-DIMENSIONAL ELEMENTS IN THE YONEDA EXT-ALGEBRA by Clas L~JFWALL INTRODUCTION. Let k be a field and R a ring with a ring epimorphism R --> k. Then k is a module over
R
and
E = EXtR(k,k)
is a graded algebra under the Yoneda
product. We study the structure of this algebra in two situations. In the first ease
R
is an augmented algebra over
second case
R = (R,m)
k
(in general non-commutative) and in the
is a local (commutative noetherian) ring with
R/m = k .
We are mainly interested in the second case, but within this theory it is natural to consider certain algebras of the first type. For example the Yoneda algebra itself is an augmented algebra and the homology theory for about the structure of complex for
E
E
gives information
E . Another example is the homology algebra of the Koszul
R . The Yoneda algebra for local rings has been studied by Levin [9]
and SjSdin [15], [18] end Roos [13] . Sj6din determines the structure of
E
when
the local ring is a complete intersection or a Golod ring. In [13] Roos gives an example of a local ring for which
E
is not finitely generated (this answers
negatively a conjecture by Levin [9] ). In this example the subalgebra of
E
generated by the one-dimensional elements plays an important r$1e. The main goal for our work will be to "compute" this subalgebra in the two eases described above and examine to what extent it is a good approximation of the whole Yoneda algebra.
Stmumar~
In part I of the paper we consider rings of the first type, i.e. augmented algebras over a field
k . If
we prove that the subalgebra Yoneda algebra EI
E = EXtR(k,k)
R A
belongs to a certain class of such algebras, generated by the one-dlmensional elements in the
is equal to the free non-commutative algebra on
divided by the two-dimensional relations ker(E I ~ E I
-~
E2). This theorem
and its corollaries may essentially be found in Pride/ [12]. He studies algebras satisfying
A = E
and we prove that this condition is equivalent to the "FrSberg
formula" (see [5] ) being true. When the cube of the augmentation ideal of zero, we prove that where
T(V)
E ~ A 8 T(V)
R
as left A-modules and as right T(V)-modules,
is the free non-commutative algebra on the graded vector space
We also give a formula for %he Hilbert series the Poincar6 series of
R
is
HE
of
V .
E (which is the same as
and is defined by the formal power series
~n>0dimk(En)zn) in terms of
H A . As an application we study the homology algebra
of the Koszul complex for a local ring with imbedding dimension three. Our methods in part I applies to get the rationality of the Poincar~ series of a class of local rings, indeed let where
p
R = k[tl,...,tn]/monomials of degree two + (tl,...,tn)P
is any number > 2
then the Polncar6 series of
R
is rational.
292
In the second part of the paper we study local commutative rings (R,m). We give an equivalent condition for a local ring~nomomorphism ¢: (R,m) --* (S,n) Indeed, if
¢
to be a Golod homomorphlsm in the sense of Levin [8] .
is surjective the condition is as follows:
There is an exact sequence of Hopf algebras k --~ T(V) where
T(V)
--* EXts(k,k ) --~ EXtR(k,k)
--~ k
is the free non-commutative algebra on
V = ~>2Vi
and
2
Vi+ I = Ext~(S,k)
for
i~I .
We prove that for some rings there is a differential which is a free
R
is two-homogeneous or
are fulfilled~i.e. then
E
is a
R = GrR
such that m
= 0
"semi-tensor-product"
~7]
HomR(U,k) = A . This is the
and if both these conditions
(with respect to the m-filtration) and m h = 0, (see Smith
non-commutative algebra. If furthermore (see SjSdin
U,
R-algebra with divided powers~containing the Koszul complex
and contained in a minimal resolution case when
R-algebra
) we prove that
PR
A
~9]
) of
A
and a free
is nilpotent as Hopf-algebra
is rational.
Part I. Non-commutative algebras
Notations and basic facts
I.
k
is always a commutative field.
2.
A graded vector space
IVil = dimkV i < ~ power series
for all
V = ~.>oVi
is locally finite if
i>0 . The Hilbert series of
Hv(z) = [i>01Vilz i
V
is the formal
293
3.
if
V, W
are graded vector spaces then (V m W) n = ~ i + j=n (Vi ~ Wj)
space with
vector space with
and
is a graded vector
Homk(V,W)
Hom~(V,W)n = ~ i H O m k ( V i , W i _ n )
is a graded vector space with h.
V m W
V~=n H°mk(Vn'k)
is a graded
, especially -
We use strict sign convention. This means that when we in a defining
formula replace
a,b
by
b-a , we must multiply by (-I) deg(a)'deg(b)
For more details about this principle we refer to Gunnar SjSdin If
a
and
h
are bigraded elements with bigrade
we use the sign
(-1)sls2+tlt2
when
formula. The graded commutator for 5.
a ~ b Let
and
UCX
[a,a] = a 2
~
and
or
V
and
{x 6 X;
6.
0
and
b
V c X* , f(x) = 0
V0
(sl,t 1)
is defined by
y
and
s1+t I
(s2,t 2) ,
U0
as
odd or even. {fE X*: f(U) = 0}
then,
(imp) 0 = ker¢ ~ .
and
X
then, (U~V) 0
=
U0 + V0
and
U 00 = U ,
f C V} .
There is a natural map V~
.
ab - (-1)SlS2+tlt2 ba ,
means the set
for all
~6]
are interchanged in a
according to
X--~
are suhspaces of
(U + V) 0 = U 0 ~ V 0
where if
[a,b]
he a linear map
(ker@) 0 = im~~ U
a
be graded vector spaces. Define
=(X/U) R. Let
If
V ~-- HOmk(V,k)
V~
--,
(v ~ v)*
defined by
f~g(x~) = (-1)stf(x)g(y)
for f E(v*) s
and
g ~ (V*) t
294
This map is a monomorphlsm, indeed suppose and
gl,...,g n
0 + [10 II (V~n) s ~
instead of
Ri+ j .
H~Ig(x,y) = ~i>_0Hii/ii+1(y)x i
is short for
a E T(V) . If
Ri'R~C
is called the a u ~ e n t a t i o n
is locally finite if
non-commutative algebra on
for
such that
R = ~.>0Ri
I(R) (or just
s
y~V
but this contradicts the minimality of
e(1) = I . The kernel of
R
is linearly independent of
for i = 2,...,n . It follows that
a graded vector space R
(where gi is the dual of the map gi: V ÷ k )
T(V) ( )
and
dp(a)
we write
respectively (Xl,...,x n)
is equal to
I .
is also a bigraded vector space and it is an algebra
with the following definition (cf. no.6):
295
Let
f,gET(V) ~
be homogeneous
elements of degree
(n,s)
and
(m,t)
respectively..Then f'g(x ~ y) = (-1)n'm + s-t f(x).g(y) where
x ~(vSn) s
T(V)
and
y 6 (vSm) t .
is also a Hopf-algebra
to be primitive.
If
V
(see
~(M) = T(M)
T(M)
as bigraded
= (X 2 ..... Xk)
and
x'y :
) by requiring the elements of
is locally finite, the dual of
Hopf-algebra and we denote it is possible to define
~I]
modules.
Let
If
R
(Y,dy)
of
B.
Y:
(Yl . . . . .
Yl ) '
8 (xi+ 1 ..... x k)
~(M)
,
dp(x)
even.
inductively
(using the formula
is any ring, we define the Yoneda 10roduct (A,B,C
in the following way(see
PB
'
(i). (j) Y ) •
ExtR(B,C)
and
Xk)
(XI,X'y) + (ml)dp(Yl)'~P(X)(y1,x'y)
This defines the structure of
9.
= (x~ . . . . .
y = (Y2 ..... y£) . Then
= (x1,~.x(n-1))
(x+y) (n) = Li+j=n x
is also a
T(V W) . This is even a Hopf-F-algebra and it -.over a commutative ring j (ef. ~ar~ ~j ): for any graded module M ~
Ax = ~0 0"Y
d
making
(Y,d)
to a
-~
"-~
Yn+l,j
j
(e.g.
-~
if
Ynj"
X . = 0 nj
Rs+l ~ Xn-l,t-1
Yn-l,j
-~
for almost
all
--~
) " Then for each
"'" n
for each
j
j
= ~n(-1)nIHnj(Y)l
gives the formula,
"lemma"
following Let
X n , n>O_ ~ are
t
differential
HR(Z).HX(-1,z ) = l-LH(y)(-1,z) This
and
and
is a homogeneous
~n(-1)nlYnjl which
algebra
such that, "'"
is finite
finite
R
is due to J6rgen
proved in Lemaire
be a connected
a minimal Tor~j(k,k)
resolution = 0
for
Backelin. ~]
~0~
a finitely generated R0-module (or
I 0 = 0)
M = 0 .
M.l = 0
Ro-module (or
Mn
R0
I = ~.>OIi
for
i_0 .
is two-homogeneous if there is a decomposition as graded I = V ~
12
such that
V2~I 3 =
i.s two-homogeneous then as a k-space
0
R = k ~
(I 2 is short for
V ~
I-I).
V21~ 13
(and conversely).
Proposition I. I V
(R = k ~ " ~ n %
with
(;~
If
R
V I I V2 ~ with as above
%
is two-homogeneous then the subalgebra generated by I3) is isomorphic to G V @ V
and
%
is two-homogeneous.
T ( V ) / ~ where, as a k-space,
c.ELi.> " 3v @i . Conversely
T(V)/~
.
299
Proof.
Consider the morphism
is in the kernel, then
x2
T(V)
--~ R . If
~i>2xi , x~E~ V ~
must be in the kernel since
Hence the kernel has the stated property. Conversely two-homogeneous if indeed let I , V
I
in
=
~]~
~$
with
. If
~ ~ q2~ ~
y = z = x . Hence
I ,
R = k ~
V 8 V
--* V 2
where
~
(V~
V~
V ~
13
Let
and let
I , V y ~ V2
C ~ - > 3 V~i the images of and
y$~
z E Is and
~ = 0 .
be the multiplleation map
im(¢*)~V ~
(I2) 0
V# .
(V @ V)K" by the map given in Notations no.6).
H(T(I~,d ~) . Let
I . We have
with
and
Z
be two-homogeneous with augmentation
R
From no.t0 in Notations we have that
algebra is equal to
V~
and
is certainly
then,
is embedded in
~
T(V)/~
y - z ~ ~ , which implies that
is the two-sided ideal generated by
Proof.
I 8 I
V2 ~
T(V)
, then there is
Theorem 1.1 (cf. [12,Theorem 2.5] ) Ideal
g~ C V 8 V
be the augmentation ideal of
T(V)/02
such that
~
V 2 ~ I s = {0} .
m
EXtR(k,k)
as a bigraded
be the multiplication map
Ext~(k,k) = ker(m@) = V ~t (we identify in the sequel
as a subspaee of
I~). Hence,
(Ext~(k,k)) 2 = V~ @ VM/im(m*)~V* 8 V ~ =
V~8
V~/im(@~)~V*8
V*
and generally (Ext~(k,k)) n = (V~)Sn/im(dW)~(V~) 8n Since
dW(V ~) = 0
we have 6~ C. im(d ~) ~ (V~) 8n . We will now prove the other
inclusion, i~(dW)r~(V~ @n = { f ( ( V ~ 8n : ~ g ~ < i < n _ 2 ( V
8i ~ V 2 8 vS(n-i'2)) * and f = @ o d}
this set is contained in, ~00Un
~ U
(-1,0) . The complex
Proposition
(U,d)
R @ Kn_ 2 Kn ~
= 0 .
is a bigraded R-module and
d: U
--* U
is of degree
has the following two properties
PI.
~: Un/IU n
P2.
Z U ~ 12U + BU n n n
1.2
m@1 ~
and from the definition of
(1@¢81)(1818in_1)(1@in) Put
2
K = ~>0~ n
--~
and
IUn_i/I2Un_ I for
is mono for
n~1
n>1 . --
[Ext IR(k ,k )]
are isomorphic as bigrade@
vector spaces. Proof.
K
is a s~hspaoe of
V ~n
~a
K~
~
(V~)~n/(K) ° . It fonows
from no.5 in Notations that (Kn)0 = Hence
~(V*) @i @ (ker ¢)0 @ (V~@(n-i-2) K 9~= T(V~)/~im ~ )
from Theorem 1.1 .
= ~(V*)@i @ im ~* @ ( V ~ 8(n-i-2)
as bigraded vector spaces. The statement now follows
304
Proposition
1.3.
Suppose
R
is two-homogeneous,
=kllvliv Consider the following
Suppose
U = R ~k X
(b)
~XCV~X
(c)
ZU
(U,d)
C
12U
and
identity map,~This
n
+ BU
X
(U,d)
of R-modules.
is a graded vector
for
n
(U',d')
n>1 --
s~isfy
f: (U,d)
--*
homomorphism
are graded algebras algebras~
where
for a complex
space and
X0 = k ,
,
homomorphism
unique
properties
(a)
n
lil 3
such that
.
these
conditions.
(U',d')
such that
is an isomorphism. (R ~ X,d)
Then there
and
fo: R
Moreover
(R ~ X',d')
--* if
is a R
X
is the and
X'
are differential
i.e. d(u°v)
(and the same formula
= du.v + (-I) deg(u)
for
u-dv
d' ), then the homomorphism
f: U
--~
U'
is an
algebra homomorphism.
Proof.
Suppose
f. : 1
we get a commutative
X. 1
--~
X: 1
is defined
--'~ X
Because
--~
n
V ~ Xn_ I
X~ n
--~
--*
V ~ X' n-1
an isomorphism.
If
(using the fact that
fn-1
(a) and (b)
--*
Xn_ 2
[ 1~fn-2 V
~ X' n-2
f : X n n
--~
we extend the maps
f (or the "five-lemma")
is an algebra homomorphism,
d': X' n
is an algebra homomorphism.
of
V2
d'
and hence
By linearity
U'. The uniqueness
--*
i 1~fn-1
of (c) the rows are exact,
defined by the diagram. f: U
, From
d
--*
d' 0
i~n
diagram d
0
for
--~ Q.E.D.
V ~ X' n-1
X' n
is uniquely
{f.} i
to a map
shows that
f
is
then it is easy to see
is injective)
that also
f
n
305
Theorem 1 . 2 V
locally
Suppose
finite
and
R R
is
two-homogeneous,
generated
by
V
R = k jj_ V j_[ V 2 jJ_ 13 ,
as
an algebra~
then
the
following
equivalent:
are
(i) R
is a homogeneous Koszul algebra,
(ii) Ext1(k,k) (iii) R
generates
is a homogeneous pre-Koszul algebra and
(iv) R
ExtR'P'q(k,k)
= 0
for
n ~ p,
is a homogeneous pre-Koszul algebra and all matric Massey products of
EXtR(k,k)
(v) R
are zero,
is a homogeneous pre-Koszul algebra and satisfies the
"FrSberg formula"
(vi)
EXtR(k,k),
the complex
(vii) R
PR(X,y)'HR(-x,y)
(U,d)
= I ,
defined above is acyelic,
is a homogeneous pre-Koszul algebra,
R = T(V)/(~ 2)
and
R
has distibutive associated lattices in the sense of Backelin [3]
§2. Proof.
We first prove that (ii) implies that
R
is a homogeneous
pre-Koszul algebra. We know from Proposition
1.1 that
(~ = ~ 2
C If.
of
j~ ~ 3
with
U I --~ U 0 , i.e.
we get T(V)
Henee
Z = ker(R 8 V
EXtR(k'k) ~- K2 = ~ 2 --~ R ~ V
g~/V~
~2
" Now
" Hence
and
~3
-~
~>_3
R) .
+ ~V (i)
we refer to Priddy [3]
Let
Z
From Proposition
Z/VZ ~- ~ 2
--~ g~V = and
Z/VZ (ii)
-~
~
-~ ~ 2
--*
Z
be the kernel 1.2 and (ii)
" The natural map
--~ 0
" It follows that
is generated
are equivalent by definition and from Corollary 1.1
it follows that (ii) and (iii) are equivalent.
Backelin
V ~i
where
induces sm exact sequence 0
by
@~2 C,V 8 V
R = T(V)/~
For the proof of (i) (iv)
[12] and for the proof of (vi) (vii) we refer to
306
(ii)--> (vi): Suppose that
Hi(U) = 0
for
i~p-1 . Suppose
We want to prove that d: h + 1
[Ext~(k,k)] i = Ext~(k,k) nO
~where x,y,z are the variables ) for the homological degree, PR (x'y'z'u) = ~r>O H (x,y,z)u r , ) t h e tensor degree and the pure -- Qr L degree respectively.
Definition.
(Y,d')
R , but first we
, hence n
dg1(Cn1+l @ Cn2 ~ ... ~ Cnk) ~ V 2 and hence g1(Cn1+1 8 Cn2 8 ... 8 Cnk) E VY I and therefore f o gl(Cn1+1 ~ Cn2 @ ... ~ Cnk) = 0 .
. Let
f(HomR(Y1,k)
--* YO " We must . Suppose gl
on
Yn+1 8 k
Yn+1 ~ k that are
=
310
An element in
K
8 C ni+I
xfK I
and
Y(Knl
8 ... 8 C n2
may be written as
8 Cn2 8 ... 8 C
. Define
(_1)n + deg(x)deg(y) then
~'gl = (-1)ngod'
g1(x 8 y)
E = A 8 T(C ~)
is a subalgebra of
E
and
is defined by the following: Let x@(K
8 T(C))
Theorem 1.3.
E = A 8 T(C ~)
I
and
R
x 8 y~K
8 T(C)
g(Homk((T(C))n,k)
such that
•
y £ (T(C)) n
g(y)'x .
be a graded augmented algebra with augmentation ideal
13 = 0 . Suppose
V = I/l 2
is locally finite. Let
be the map induced by multiplication PR(X,y,z,u)
.
as right T(C~')-modules
T(C@)'A): Let
then m [(x 8 y) = (-I) n'm + deg(x)deg(y)
Let
f(x)g(y)
as left A-module. We can also prove
(however nothing is said about the product
and
as
and
Thus we have proved that T(C ~)
where
g(y)-x(K I ,
f 0 g1(x 8 y) = (-I) n + deg(x)deg(y)
that
x 8 y
nk
: XHA(Xyu,z)/(1
in
@: V 8 V
--~ V 2
R . Then,
+ x - HA(xy,z)(1
- Hv(Z)Xy + Hv2(Z)X2y2))
where A = [Ext~(k,k)]
and
where
Proof.
PB(x,y,z,u)
= T(V~)/(im¢ ~)
is defined On page 19.
It follows from above that n,n+p Qr
~
I~Z nl.=n-r
K
r
~C
nl
8 ... ® C
np
Hence, HQr(X,y,z)
= %~0(xy)rHKr(Z)(Hc(xY,z))PyP
= (xy) r HKr(Z)/(I
- YHc(XY,Z)).
According to Corollary 1.4 , XyHc(~Zy,z) + I = HK(xY,z)(I and if we use the fact that
H K = HA
- Hv(Z)Xy + Hv2(Z)X2y2)
we obtain the formula for
PR(X,y,z,u)
.
311 Remark.
If we put
Poincar@ series for Application. Let
(R,m)
z = 0 R0
and
y = I
and
u = I
in
(2)
we get the
obtained in [101 .
(This application is due to Gerson Levin.)
be a commutative local ring with
Im/m21 = 3 , which is not an
artinian complete intersection. Wiebe [20] has proved that in this case (HI(K))3 = 0 , where
K
augmented algebra with
is the Koszul complex. Hence
H(K)
is a graded
(I(H(K))) 3 = 0 , and the results above may be applied.
From Avramov [2] we get
PR(X) = (1 + x ) 3 P H ( K ) ( x , t , x , 1 ) Hence the problem of rationality for where
PR
.
is reduced to that of
HA(X,X)
A = ~[Ext!(K)(k'k)] ~ J
Theorem 1.h.
Let
(R,m)
be a local ring with imbedding dimension
which is not an artinian complete inter~tion. Then
(3)
H(K) = T(V)/(g~ 2 + V @3)
Suppose
S = T(V)/(~2)
where
~2 ~V
Let ~ V
K
3 ,
be the Koszul complex.
and
V = I(H(K))/(I(H(K))) 2 .
is a homogeneous Koszul algebra then,
(~)
PR(X) = x(1 * x)3~I . Xms(-X,X) - ~H(K)(-x,~. Proof.
By Theorem 1.1
A = [Ext~(K)(k,k) ] = [Ext~(k,k)]
, and by Theorem 1.2
HA(X,y)Hs(-X,y) = I . Hence by Theorem 1.3
PH(K)(X,I,x,1) and
(4)
Remark. three if
follows from
= xl(t + X)Hs(-~,~) - F~(K)(-x,x ~
(3) •
Levin [9] has proved the rationality of (HI(K))2 = 0 . His formula is the same as
"S = H(K)"). A ~ a m o v
has proved the rationality of
and three relations", and again the formula for
PR
PR (h) PR
in imbedding dimension (in this case for "three generators
is given by
(h) .
312 A Poincar$ series.
We end Part I with a theorem, that is a generalization of the method above to compute the Poincar@ series for satisfy the condition
Theorem 1.5. (&~2 C V 8 V
V
to algebras that not necessarily
13 = 0 .
Suppose and
H(K)
S = T(V)/(~ 2)
is a homogeneous Koszul algebra
is locally finite). Let
p~3
and
R = T ( V ) / ( ~ 2 + V~ ) then PR(-x,y,z,u)
= x p-2 .
(Hs(Xyu,z))-I/(xP-2
I + (Hs(XY,Z))-IHR(XY,Z))
and particularly
PR(-X,l,z,1)
Proof.
Since
as above
R
= x P - 2 / ( ( x p-2 _ 1)Hs(X,Z ) + HR(X,Z))
.
is two-homogeneous we may construct the complex
[see pp. 12-13). Put
Cn+ I = Hn(U)
(Cn = 0
for
(U,d)
n0
and
d -'* V 8 Ki_ 1
i = n+p-2 . Hence
--
for
d d --~ ... --~ V p-I @ Kn_ 1 C
has tensor degree
--~ Cn
--~ 0
n+p-2 . Exactly as
n
p = 3
we get a minimal resolution of the form
EXtR(k,k) = A ~ T(C ~)
as left A-modules where
U @ T(C)
and
A = [Ext~(k,k)] = Exts(k,k)
As in the proof of Theorem 1.3 we get PR(x,y,z,u) = We have
On+ I = Hn,p_1(U)
HA(XyU,Z)/(1 - yp-2Hc(XY,z))
where
n+p-1
is the tensor degree, hence by (I)
we get HA(X'Z)HR(-X'Z) = ~n~l xn+p-I(-I)P-IHCn+I (z)
+
I •
Hence xP-2Hc(-X,Z) = I - HA(-X,Z)HR(X,Z) We now put this into the formula above for
PR
.
and use the fact that
HA(-X,Z) = (Hs(x,z)) -I , which follows from Theorem 1.2 o
313 1..5.
Corollary
Let for
p>3
R = k[x 1 ..... where
deg(x i) = 0
x.x. i j
for
and
1I
u~mU
(c), appendix
X' = X<S: dS = s>
Proof.
is called an
property holds du~m2U
(cf. condition
U
a basis
The Koszul
for
There is a free extension
U = K I
I
i
) such that
m/m 2
U
complex
is obtained by
.
of the Koszul U
complex
is an S-R-algebra
and
7U C m2U + BU .
Proof. Ui
Put
UI = K
and suppose
is an S-R-algebra
represent
a basis
and
~.uic 0
Ui
m2U i + BU I
for z.ui/(m2U i + BU i) . l
is constructed for
j
for . Let
i>I
such that
S1"" "" "Sn
~ 0 .
315
Define
U i+I
as
U i+I
ui<s1 ,...,Sn: dS~ = s 0.> . It is clear that
is an
S-R-algebra (Proposition 2.1) and z.ui+I~: m2U i+I + BU i+I j Finally, define
U
as the union of all
Proposition 2.2.
for
j
Ui .
Let
U
be an S-R-algebra. It is possible to imbed
in a minimal resolution
Y
of
(i.e.
Y = U ~R F , F
d(u~f)
= duff
Proof. ~(yi) and
is a graded free R-module, F 0 = R , and
y0 = U
for
F i. = 0 0
~
for
and suppose and
j>i
i>0
and
yi
yi = U ~R Fi ' where
is graded free, F i 0=
Fi
R
= (du).y
- ~.~y
for
u~ U
and
y ~ yi
is considered as a U-module in a natural way). Choose a free R-module such that
Fi+ I
~
•
N
"
.yi
F i + 1 ~ k ~ Zi Y1/(mZ'Y~l + Ba Choose a map
a:Fi+1
~
I Define
d
j~
on
Z'Yla
i
Fi+i/mFi+ I yi+1 = yi
) .
__~ ~.yil such that the following diagram commutes
Fi+1
Put
is constructed such that
and
d(u.y)
(yi
k , which is a free differential U-module
- ~.df).
Put
= 0
U
~
ziYi/(mZi Yi + BiYi) •
U ~ Fi+ I = U ~ F i+I , where
U 8 Fi+ I
(5)
F i+I = F i
I]
Fi+ I .
by the "derivation" formula,
d ( u ~ f) = d u ~ f - Z - ~ ( f )
Then
d2 = 0
and for
ueU
and
y = u' 8 f ~ y i + 1
d(u-y) = d(uu' ~ f) = d(uu') ~ f - ~ ' . ~ ( f ) = (du)-y
- ~-dy
.
we get
= (du)'y - ~.(du' ~ f) + ~'~''~(f) =
316
B.Y i+I = im(~) + B.Y i . From (5) we get
We have
1
i
im(~) + B.Y I + mZ.Y i = ~.yi . i
i
i
Hence by Nakayamas lemma ~.yi = im(e) + B.Y I = B.Y i+I 1
Hence
Hi(Y i+I) = 0
Hence
yi+1
i
and also
follows that
an
S-R-algebra).
Y
for
Z Y C mY . We have
jI
r>i
and
,
~ u r ~ F 011Ur_ ~ ~ F IL[... ]IUr_ i ~ F i • We write and
x = [0
u E ~ 2 U ' + ZU' .
Combining these facts we get that (6) is true also for
EExt~rR(k,k) ] = Ho~,(U',k)= ~x~:~le.
Ho~(U,k)
~t is not always tr~e that
U' . Hence
: [Ext~(k,k)]
[~x~(~,~]
. Q.E.D.
: [~Xt~r~(~,~]
,
as the following example shows:
R = k[x,y]l(x 2 + y3,~) is a complete intersection and
~>~R(, ~R R<Sr+1'''''Sn >
i=r+1,...~n
. It follows from Proposition
K<SI,...,S r : dS i = si> is an S-R-algebra.
k
2.1 that
329
(iv).
Let
X
be any R-algebra that has "trivially"
equal to zero, i.e. there is a set
{s.} 1
~X , such that
i
s.-s. = 0 l j
for all
all Massey products
of cycles representing
and
j . Let
a basis for
s E {s.} . We claim: l
Y = X<S: dS = s> has also trivially
all Massey products
zero. Indeed, suppose
y = x 0 + xiS + x2S(2) + ... + XnS(n) is
a homogeneous
to
X'n = ~risi
cycle
in
where
Y . We g e t
riE R , so
dx
y
= 0
n
x
, hence
is homologous
n
is homologous
to
x0 + xlS + "'" + x~-Is(n-1) + x's(n)n Since d(x~S (n)) = 0 , the same argument may be repeated and we get eventually that
y
is homologous
to
x~ + x~S + ... + x'S (n) n where ~Y
x ~ ( [Rs i
for all
in this manner,
j . If we choose cycles representing
it follows that the product of any two of these is zero.
Thus the claim above is proved. a complex
U
zero. Since
that
has property E.
(v).
If
R
ZUcm2U
+ BU
U
Golod, we may construct
has trivially
all Massey
it follows as in the proof of Theorem 2.2
This follows from (iii) and (iv) (for a proof, see [6] ). Q.E.D.
Remark.
The condition
(i) of the theorem is equivalent to
and two-homogeneous.
Indeed if
and two-homogeneous.
Conversely suppose
then
is trivially
as in Corollary 2.2 such that
products R
a basis for
R
is a quotient of
R
k ~t I ..... t n ~
By definition of a two-homogeneous tl,...=t n
satisfies R
(i) then clearly
R
complete R
is complete
is complete and two-homogeneous, , since
R
is an algebra over
k
@
algebra we may choose the variables
in such a way thai the relations between them are of the given form.
S30
It is possible to prove (i) of the theorem by using the complex (R 8 K, d) defined on page 12-13 for two-homogeneous algebras. In the commutative ease this complex has additional structure. K
is defined there as a suhspace of
T(V)
on page 5, it is fairly easy to
and, using the definition of
see that
K
T(V)
is a sub-Hopf-F-algebra of
T(V) (where the elements of
V
have
degree one). This also follows from [16, Theorem 2, page 17] , since the dual of the map
K
--~ T(V)
this map is equal to
is the natural map
W(f)
T(V ~)
--~ T(Ve)/(im@q)
for a map of graded Lie-algebras where
and W(')
is
the universal enveloping algebra functor. Moreover it is an almost direct consequence of the definitions on page 5, that the differential on
R 8 K
is compatible with the algebra structure and the structure of divided powers, Proposition 1.3 page lh tells us that this d{fferential algebra structure is unique and also that the differential algebra defined in the proof of (i) is isomorphic to
CorollalV
(R 8 K, d).
2.5.
Suppose
by linear forms in
R = k 6 1 ..... tn~ / ~ + ~
{ti-tj}
and
~C(tl,...,tn)S
where
g~ is generated
and one of the following
two conditions is fulfilled:
( a)
(t I ..... tn)4
(b)
k~1,...,tn]/( ~ k
such that
~_. is a homogeneous Koszul algebra and there is
(t I .... , t n ) 2 ~
Then there is a graded vector space
C ( t I ..... tn )k+1
V = __/I~>IVi
and a split exact sequence
of Hopf algebras
k
--~
T(V)
--~
EX~tR(k,k )
--~
KExt~(k,k)] ....
i.e.
EXtR(k,k)
is a semi-tensor product of
a
T(V)
and
-'*
k
I
" -rExt~(k'k)7" n
331
Proof.
Suppose (a) is fulfilled. We want to apply Proposition 2.3 (d)
so we must prove that the complex defined above has the lifting property. But this is easy: Let
f @HOmR(Un,k)
= HOmk(Kn,k). Put
f(x I ~ ... ~ Xn+ r) = (-l)nrf(xr+1 ~ ... ~ Xn+r)(X 1 ~ ... and extend
f
to
R 8 Kn+ r
Xr )
by linearity. We get the split exact sequence
and the argument for the fact that the Hopf algebra structure on
T(V)
is
the natural one is the same as in the proof of Theorem 2.2. Suppose (b) is fulfilled.
Let
be the minimal resolution of HU = H(R 8R' U') cycles in
k
be the ring
over
kIEt I ..... t n ~ / ~
(tl,...,tn)kU
[Ext~(k,k~
. Let
U'
R' given by Theorem 1.2. Then
has all Massey products zero(trlvially),
is a Golod homomorphism fact that
R'
representing a basis for
since we may choose R'
HU . Hence
R
(see [8] ) and we may apply Corollary 2.4 and use the =[ExtR,(k,k)]= HomR(U,k)
, which follows from
Theorem 1.1 and 1.2. Q.E.D.
EExt~(k,k~
as enveloping algebra of a graded Lie-algebra.
It has been proved by Andr@ [I] and SjSdin EI6~ , that = universal enveloping algebra of a graded Lie-algebra
EXtR(k,k) = WL = L = ~.>iLi
where
i
L I = Ext~(k,k). Let ~
be the graded Lie-algebra generated by
that
.
W~
= [Ext~(k,k)]
L I . We claim
The following lemma is due to Gunnar SjSdin. Le~mna 2.4. WL
Suppose
generated by
Proof.
~
~
~
L
are graded Lie-algebras,
is equal to
is imbedded in
WL
(PBW) (see e.g. SjSdin [16] ). Let
W~
.
by the theorem of Poincar@-Birkhoff-Witt A
be the subalgebra of
. There is an epimorphism of algebras diagram commutative
then the subalgebra of
W~
--~ A
WL
generated by
which makes the following
332
--~
W~ The map
W~
--* A
--~ A
--~ WL
WL
--~ A is a monomorphism, because of PBW. Hence also
is a monomorphism.
Q.E.D.
We know, by PBW, that the Hilbert series of
has the form
(I + x)60(I + xS)62..../(I - x2)61(I - x~)63-... where
6i = dimk~i+ I .
Definition.
The natural numbers
Definition.
(cf. Sj6din [17~ )
nilpotent,
i.e.
6i = 0
are called the subdeviations
[Ext~(k,k)3
= W~
for sufficiently large
tde fined above l Since ~ Y ' i s generated by
Theorem 2.5.
{6i}i>0
Let (R,m)
~I
' W~
~
is
i .
is nilpotent if
be a local ring and
6 0 = dimk(m/m2)
is nilpotent if
R
of
K
6.1 = 0
i .
for some
its Koszul complex. Then
, 61 = dimk(ZIK/(m2K + BK))
and
82 = dimk~Z2K/(m2K + BK)) Proof.
We will use Lemma 2.1. Consider the minimal algebra resolution
obtained by killing cycles (see [6~ ). Let H(Y/mZY)
--~ Y/mY
By Lemma 2.1, the algebra cokernel of
¢
Y
be the natural map
•
~
is equal to the dual of
~Ext~(k,k)~
We are going to compute the dimension of the ideal generated by im(¢) in degrees ~3 • Choose a basis {[i,...,~nI}
is a basis for
{[1,...,Se }
for
ZIK/(m~K + BK)
ZIK/BK and
such that
Sn1+1,...,selE mZK . Put
333
E = K<Si: dS i = s.l , i = 1,...,si> . Choose a basis for
{~I'''''~E2}
Z2E/(m2E + BE)
and
for
Z2E/BE
such that
is a basis
{~1'''''Un 2- }
Un2+l , .... uc2E m2E . Put
F = E . We have of
F. = Y. J
for
j~3 • Since
is an S-R-algebra, there are no elements
K
K/m~K in im(@). Suppose y = ~riS i + x
is an element of degree two in
r.~R i
with Y
for
such that
i=I,...,~ I
and
x~K
d y E m 2Y • Then
~risiE m2K + BK which implies that ~1~i!nlrisiE m2K + BK and hence
ri6 m
for
dx6 m2K
i=l,...,n I , it follows that
We have proved that im(@) is generated by z = [riU i + [xiS i + x 0
with
riE R
SnI+I,...,ScI for
hence
x{m~
.
in degree two. Suppose
i=1,...,E 2
and
xi~ K for i=0,..,E I.
We get [riui6 m~E + BE and as above this implies that
d(~xiS i~ hence
dxi(~ m2K , hence
im((~) is generated by
+
r.~ m i
for
i=1,...n 2 . It fellows that
Xo)~ m2E
x i @ mK
and finally also
Unz+1,... ,UE
Xo~ mK . We have proved that
in degree three. Thus we have proved that 2
a basis for the algebra cokernel of of degree _
x'6 m2K + BK
be the minimal algebra resolution obtained by killing cycles.
be the subalgebra of
Y
generated by variables of degree O
has characteristic
in the ring degree,
j
is a derivation
i.e.
j(x i) c x i-deg(j) Let
S
be a variable of degree
it is enough to show that But we may choose
s
in
j(s)
i+I . In order to extend is a boundary
V ~ X i , hence
and this is a boundary since
Z.U i c J
and hence we may choose the extension the ring degree
m2U i + BU i of
(since this construction
formula on page 41). Q.E.D.
j(s)
j
j
to
Ul<S: dS = s> ,
(see [6, Lemma 1.3.2 page 16~). is a cycle in for
V @ X i-deg(j)
~Q'
is a h o m o m o r p h i s m
a local
a local
ring Q ' . T h e n we have a natural e x t e n s i o n h of h m a p p i n g g Q' o b t a i n e d by m a p p i n g the i n d e t e r m i n a t e s X I , X 2 , . . . into t h e m s e l v e s . g F u r t h e r , i f I is the kernel of h , t h e n Ig=IQg is tile kernel of hg. In
Qg into
particular,taking
Q'
= Q/l,we
obtain
(Q/l)g As a c o n s e q u e n c e , Next,suppose isomorphism pg,bOth
being
the
the
Note used
in the
Qg and
last
and
the
g
.
to be d i m ( Q / l ) , d i m l ideal
of Q.Then
localisation
as Q - a l g e b r a s . T h i s the
fact
that
consisting ideal
= INQg=
least
I ChQN
to
paragraph,we isomorphisms
M be a f i n i t e l y
have
are
have
will
of QN
that
isomorphism a natural
ideal
htp.
I be an ideal
if N is
I ~Q
this
prime
htpg-Let
= Io).
large Then,for
(QN)g
with
Qg in
is true
will
be
be d e n o t e d
isomorphisms
generated
that
identified
N for which
I and
the natural also
implies
the
is n o e t h e r i a n .
of QN(SO we
integer of
Qg
= diml. g there is a natural
of Qg at
of e l e m e n t s
(IN)g(here
of d e f i n i t i o n
in a d d i t i o n
(Qg)N.BOth
Next,let
of
a basis
way.).The index
that
(Qp)g
by I N the
enough,I
natural
termed
use
is d e f i n e d
p is a prime
considered
I has
enough. D e n o t e N large
that
between
Now we make of Q g . T h e n
if diml
an i s o m o r p h i s m
Qg/l
by i(I).
between
isomorphism
Qg and
(QN)g
between
as Q - a l g e b r a s .
Q-module. Then
we
define
M N and
343
M
a s , r e s p e c t i v e l y , M ~ Q Q N and M ~ Q Q g . W e can g as f u n c t o r s f r o m the c a t e g o r y of f i n i t e l y respectively,the
generated simply
categories
Qg-modules.These
reflects
the fact
of
finitely
functors that
M - - - > M N and M - - - > M
generated
generated
are
QN and
consider
both
Qg are
Q-modules
QN-mOdules
faithful
and
faithfully
g
to, and
finitely
exact.This
flat
extensions
of Q. Nowsuppose L ( ( Q / m ) N) Hence by
both
Hilbert
the
ideal
that
M is of
have
value
functions
passage
finite l,the
f r o m M to M N or M
a consequence,it
and Qg. We
simply
m-primary
ideal
We give THEOREM
1.3.
associated We
.To be p r e c i s e , i f
g
of M
recall
of M we
see
P'.We
can
that
of M
take
in X I , X 2 , . . . are
). g preserved
are
I is an m - p r i m a r y
that
result
is P.By
g
,and
that
u to be of
the
form
coefficients
define
P is
contains
the
of
the c o n t e n t
by the
for all By
the
of
prime
and
prime
the
P ranses
associated
over
the
of M if M c o n t a i n s
u to be a s u i t a b l e
an
element
c(u)
w h o s e a n n i h i l a t o r is g u(X) d e n o t e s a p o l y n o m i a l
of
of
these
sub-module c(f)
f(X).Then
of
of M
f to be
a classical
in the a n n i h i l a t o r c(f).This P'
prime,is
= Pg,where of
the
P' of
that
i(1)
u.
P'
P = P ' ~ Q.
individual
the a n n i h i l a t o r
associated that
implies
of
w i t h M. w(j'k)
n.< m. J-- J c o n s t r u c t T as a p r o d u c t
We n o w
sends
I.j for
j,so
determined
for
:
( u ( j , l ) .... , u ( j , n j ) ) , ( w ( j , l ) , . . . , w ( j , m j ) )
k = 1 ..... of an
suitable
to an m i~x m
Q,for define
T 2-
.We
can
u(j,k)
=
nj.Since
write
j (k,i)w(j,i)
the
elements
u(j,k)(k=l
l.,it follows that,for each j,the J nj. xn.j s u b - m a t r i x with determinant rows matrix
.....
nj)
nlxm. matrix J J a u n i t of Q.
of
an m . x m . i d e n t i t y m a t r i x , w e can J J Aj = ( a j ( k , i ) ) w h i c h has d e t e r m i n a n t
each j.With this extension -I T2 ,by f i r s t d e f i n i n g
of
the
definition
of
a
a.(k,i) J
345
~ T2-1(X(Mj+i) for
j = i .... ,s,i
This
can
now
= 1 ..... mj
be e x t e n d e d
and
to an
T2-1( ~X(Mj+i)w(j
T 2-
(X(r))=X(r)
automorphism
,i)) =
if
r>Ms+ 1 • satisfies
which
~aj(k,i)X(Mj+k)w(i,j)
= as
°
=l~'#j(k,i)X(Mj+k)
~X(M.+k)u(j,k) 3
required. In
the
following
corollaries,we
will
denote
the
ideal
( X l , . . ° , x s)
by X ( ~ ) . Corollary on
the
i) To w i t h i n
set
Corollary of
I and ii)
x I ......
Corollary
not
The
iii) with
ideals
is
of
P ranges
X(1)
follows
we w i l l
now
an i n d e p e n d e n t of _I is
is c l e a r l y prime
ideal
2.2.
Let
(Xl,...,Xs)= P be
i
I and
ideals
a sub-set of over
ring
of
X(i)
p is
not
on
only
the
choice
all
ii)
least prime
only
ring
prime
prime
must
pg. H e n c e
of Qg. T h i s follows
at m o s t
case.We
property that
s and
further
elements
the
dimension
term
a prime
if
independent set
over
of
set
of
ideals
general
X(!),and
prime
elements
suppose
that
of Q , a n d of ~. p=Pp-~Q is
ideal
of Q c o n t a i n e d
in P,
ideals of
dim(Q
ideal
P
to
o
Qg m i n i m a l
over
X(1)
whose
(Qg)p/p(Qg)p, s,whose
then
maximal
R is ideal
a oregular is g e n e r a t e d
of X l , . . . , x
P is
Qg
of
Q is p, rin~
ideal
is e n o u g h
ideal
an
of d i m e n s i o n
images
= d. S i n c e
of
_l=(ll,''',Is)
general
Then
prime
the
a chain
one ideal
of
= dimQ.
minimal
If R d e n o t e s
minimal
height
independent
with
It
have
minimal
of Q.
X(!),then
situation
ideals
set
X(1)
only
i)Clearly,any
dim(Qg)
of
parameters
an
over
= d,
the
bY the
a set
X l , . . . ,x s .It
intersection
.local
that
independent
of
+ dimP
ideals
a particular
of
ideal
the
with
a set
at
prime
ideals
choice
in
be
a]l
the
a good
idea]
Q/p
P is
iv
if
of
prime
concerned
an
of
prime
dim
iii
be
say
prime
ii
be
depends
on I__
set
equality
a good
a minimal
since
on
set
of m i n i m @ l only
I = ( I I ' ' ' ' ' I s ) be
let
the
d e s c r i b e . We w i l ]
a local
Let is
only
either
set
we will
minimal
d-s,with
P of
the
htP
LEMMA
a Q-a1$ebra,Qg/X(1)
of X l , . . . , X s .
depends
over
depends
independent
ideals
at m o s t
or
P~Q
In w h a t
Xl'''''Xs
as
choice
Q rNX(!)
ideal
If
the
which
the
xs
associated set
isomorphism
on
of
in R. s -Q contained in P is c o n t a i n e d
show
that
a good
dim(Qg/pg)
prime
~ P I c" "'" ~ P d contained
/pg)
is
at
ideal
=d°It of
in
is at m o s t
height
s,there
p. d exists
=m with P =P.Then P is a g s o in P and so m e e t s Q in p. H e n e it
least
d.
346
We will
prove
automorphism that ients
iii)
and
of Qg over
iv)
Q,we
t o g e t h e r . By
may
assume
applying
that
X(!)
a suitable
is
standard,and
hence
x. is a l i n e a r form in the i n d e t e r m i n a t e s X I , X 2 , . . . w h o s e c o e f f i c l g e n e r a t e l i , t h e i n d e t e r m i n a t e s e n t e r i n g in d i f f e r e n t x i b e i n g
distinct.
P = PNQg. H e n c e
Choose
htPN=htP
b e l o n g to Q N . T h e n , d e n o t i n g P ~ Q N by PN' s = s and d i m P N = d i m P = d - s , i . e . P N is good. F u r t h e r ,
we can
Q by Q/p
and
write
N so
replace R N for
,replace Qg
that
the
Xl,...,x
assume
1ocalisation
by QN
to o b t a i n
that
Q is a d o m a i n . F i n a l l y
of QN at P N . T h e n
iii),iv),with
the
we will
R =(RN)g.Hence
assumption
we
that
can
Q is a
domain•
ring
We
can
of
fractions
construct
R N in
Q . T h e n , i f F denotes the F [ X I , . . . , X N] latter
in X I , . . . , X N over
there and
prime
over
is
the
F,and
one
ideal
of
fractions
prime
s and
is c l e a r l y
the
of height
so
regular
generated
by
the
we
over
take
the
elements
of
localisation
of
(Xl,...,Xs).But
linear
s. S i n c e
a localisation
is
the
ideal
by e l e m e n t s
of Qg m i n i m a l
stage
of n o n - z e r o
independent
) is g e n e r a t e d
R N is
first
set
of Q,R N is
over
s linearly
so is
the
to the
minimal
s p r i m e ideal
of height
stages. For
respect
ideal
proved. F u r t h e r
ideal
maximal
by
(Xl,...,x
is only
iii)
field
at a p r i m e
is g e n e r a t e d
minimal
two
of QN with
Of
forms
any
prime
QN,it
X(!)
idea~of
follows
meeting
the
Xl,...,x s Qg
that
Q in zero,
of F [ X I , . . . , X N ] at a
of h e i g h t images
s. F u r t h e r
its
of x I, • ..,x s. This
proves
iv).
THEOREM and
2.3.Let
let X(1)
I = (Ii,...,I ( X l , . . . , x s) be
~(or
the
ideal
over
X(1),and
I i , . . . , I t are
they let
iii) iv)
that
in p but
prime
ideal
prime
only
prime
ring
R = (Qg)p/p(Qg)p
dimension
s-t,
whose
As form,which
where,as
. . . . .
in the we will
ideal
minimal
local
e~ (xt+ l
prime
maximal
X(!)
of Q
elements
of
of Qg m i n i m a l
so n u m b e r e d
over
pgiS
that
minimal
meeting
over
Q i___n_np,
(Xl,...,xt)contained
is a r e g u l a r
ideal
ideals
ideal
t,and
over
minimal
of
are n o t . T h e n
of Q of height
ideal
set
of g e n e r a l
I i , . . . , I s are
It+l'''''Is
is the
the
set
P be a good
p =P~Q.Suppose
contained
(x I .... ,xt), P is the only pg
) be an i n d e p e n d e n t an i n d e p e n d e n t
generate).Let
i) p is a good
ii)
s
local
is g e n e r a t e d
by
ring the
in P,
of images
xs),
proof make
of
the
lemma
we will
take
X l , . . . , x s in s t a n d a r d
explicit,and write ~j xj = ~~=uI( j , k ) X ( N j + k )
earlier,Nj=nl+...+nj_l.Let
Ns+I=N
j= and Nt+ l
=
N
v
1 ..... s.
347
Write the
same
I
!
QN,PN
Q'
= Q/p,and
meanings
"N-
as in the
and
Q'
in the
Q~/P~
is the
in P,and
N-s+t.
Q'
Further
two
follows
the
residue
transcendence
I
QN/PN,Of
the
htp+
dimQ' that
a set
it have
dimension
we have
proved
dimension
of
We
IsQ N in
Q~.
Further,x*
ideals.
ideal
set
define
minimal
1emma
over
first
of
(yl,...,ys),where all
contained
I t + l , . . . , I s are
in the
lemma,are
not
linearly
by Y t + l , . . . , y s and
t(E'/F')
and
the
of E'
second
over
F'
so is
> d-t.These
two
i.e.,htp
inequalities
ideal.Further hence
pg must
to the
local
rings
obtain
~ d i m ( Q NI/ P ~ ) + N
the
ring
imply
pg c o n t a i n s all
minimal
be minimal
~
t
= d-s+N equality
and
X l , . . . , x t which
prime
over
ideals
over
( X l , . . . , x t) and
p~,
Q~ = Qg/(X I ..... x t) and write
images
of ideals
is a minimal a minimal the
t+l''''' ideals
x* s _I*
is
= dimP
=
of Qg'QN'
prime prime
images a
ideal
I*)s
htP*
=
Q~
modulo
of Q~,
for
( X l , . . . , x t)
and
has
of x* .. x* which, t+l'" ' s' I* of I IQN,...,
t+l''''' of a
(I~+ l, ....
= d-s,and
ideal
I*
sub-set
have
implies
sets
that
meets
of g e n e r a t o r s
P is
QN'
(Xl,...,Xs).If
the
in PN' the
only .Let
s
set
is
s-t,so
t+ parameters
of
an
in
independent
that
contains
PN'
must
properly
contains
it,and
it must
intersection
contain
of PI with
at
in QN'
P*
prime
and
least
Q properly
.Finally
ideal
is
set a
of
good
dimension
one
the
contains
ideals p.
over ideal
Q contains
PN,in
so has of
prime
of PI with
so is either
we c o n s i d e r
minimal
P,. be a n o t h e r
intersection
QN'
that
of Q'
fractions
fractions
over
since
= s+N-s+t
with
implies
of
Ii,...,I t are
degree
i n e q u a l i t y . We
prime
of
ideal
is g e n e r a t e d
to P'PN
intersection be P,or
field
extension
fields
of Q * . . H e n c e we can apply the lemma. S t a t e m e n t iv) is g iii) follows if we o b s e r v e that the minimal prime ideals
( X l , . . . , x s) which
the
have
of Q'/P' is i s o m o r p h i c to k(X 1 ...,X N) N N ' ' N over k,the residue field of Q'.We now
of
of
( X l , . . . , x t) all
ii).The
the prime
zero,and
P"
elements
Finally,dimP*
immediate,and of
degree
is clearly
general
the
field
denote
"*".Then
d-t.P*
that
all
parameters,and
introduce
are
prime
defined.We
generated
be the
non-zero,and,as
j d-t..Hence
in turn
Q$,so
QN,PN
i).
QN,/(Xl,...,xt). by a d j o i n i n g
is
dimQ'+N-s+t
= dimQ/p
of
we
E'
that
dimension
p is a good
is part
Now
F',E'
N ~ htP N + t(E'/F')
dimQ'+t(E'/F')= whence
a finitely
transcendence
applications,the !
and
F.It
s-t.Henee,the
so has
hence
T of Qg. Let
P/pg
as a l r e a d y
of x.z in F ' [ X l , . . . , X s ] . B u t , a s in p , y l , . . . , y t are
over
height
of
P" is a m i n i m a l
in p , y t + l , . . . , y s are
independent
and
ideal
being
ideal.Let
r e s p e c t i v e l y . Then
image hence
contained
make
prime
lemma,N
zero
F'[Xl,...,XN]/P",where
has
the
is a l o c a l i s a t i o n
N
PN' meets
of Q'
Yi
for
similarly. Now O'/P'
and
P'
of Qg, p,its
which
case
0,j
(rl,...,rj-1,...rd).Then
y l , . . . , y d of Q a j o i n t exists
independ-
introduce
R = ( r l , . . . , r d) be a set of d i n t e g e r s . T h e n , b y r r2 rd I 1 !I 2 ...I d r _ o R = (rl, ... , r d ) w i t h r J r.j ,r.1 for i # j
unrestricted,such
Since
-1
tj
positive,zero,
d integers
a general
J
r °j s u c h
the
consisting
denote
R(!),the
n. We
divide
contain
with
uR(l we
(Pl)g,...,(Ps)g
Ig is not
this
prime set
and
S2
ljtjR(1).
pl,...,Ps,When by
of
l.t j j R(1) --
by ! g = ( l l Q g , . . . , I d Q g ) . W e
associated
set
contained
) into
g consider and in
this any
will
still
two
classes,
R(1)_ , t h e n set
will
ideal
the now
in S 2.
the m o d u l e n (unR(l_g):Xjtj )/unR(l_g) by
by M
(lj tj )N for
some
integer
N depending
on
n. But
349
M
is a f i n i t e l y n d e g r e e R is z e r o
) - m o d u l e and h e n c e any e l e m e n t of M of g m if the j d e g r e e r. is s u f f i c i e n t l y large. 3 N o w let B d e n o t e the i d e a l of R(I ) c o n s i s t i n g of all f i n i t e sums R --g R ~.a(R)T w i t h a(R) c o n t a i n e d in x.Q ~ I . T h e n B has a f i n i t e b a s i s
consisting we
can
and
of
find
hence
generated
elements
an
R(I
th
of
integer
the
form
q such
that
B = x j t j R ( I g ) : U ~.
Now
~b.T w i t h b. in Q .It f o l l o w s that J O q g the e l e m e n t s b.u all b e l o n g to R(I ) R 3 --g s u p p o s e that zT b e l o n g s to B. T h e n
uq.zT R = x.t.W where
W is
Then,by that
a homogeneous
the
first
r.-q-i
part
rj
of
the
is s u f f i c i e n t l y
03
>r~,W ~uqR(l ) and J --g
3
element
3
of R(Ig)_ w h o s e
proof,W
will
large.Hence
hence
j
th
belong
we
can
degree
to u R ( l
find
is
g such
r?
r.-q-l.3
) providing that,if
a
zT R b e l o n g s
o
to x j t j R ( l _ g ) . T h e r e f o r e , i f
r.>r.. 3
3
X j Q g fN(Ig) R = xj(l_g) R(j)
THEOREM
2.5.
If
X l , . . . , x d is
I = (ll,...,Id),then
observation disposes
prove
this
that,if
of
the
( 0 : I i . . . I d)
case
d=0
= 0,which
suppose
that
satisfying d'
= dimQ
this
result
I i . . . I d is
the
and
hence result
condition
this
restriction
for
J =
I_7 (O:(I
.
Id,)q).Then
ring
Q'/J,and
Consider
the
that has
most
d. T h e n all
can
nilpotent,when
the
is
for
we
show Q of
I .
is
restriction
non-zero-divisors.
all
local
then
d'
N'.
of
their
all
1emma to
The
of
Vo,VN which
given
is
due
First
we
introduce
some
of
fractions
F and
residue
field
of v is v d e g r e e t(K
If
an v
elements
algebraically
that
let
fractions
Q'
of
inequality
that
dimQ'>O,it is
We
turn
the
now
v is
form
the
Q',so to
the
p
n+l.Hence
hence
of
have
we
ca,
have
Q/p,implying
thus
reduced
inductive
to
Q'QN
to
in
let
E be
E,~O of
a more
and
k. We
m -valuations g For this
general
form
Samuel[12],vol.ll,
(Q,m,k,d)
a finitely
on
r-l,and
hypothesis.
respectively.
Zariski
as
r to
between
Q and write
to
the
an
m-valuation of
g ~",a
being
finite,
Vg(f)
to
v
to
local
be
a
local
generated
>0
on
m. T h e n
t(v/Q)
.N,where and
Min
the
v(a
on
for
t(E/F).We
~
d+t(E'/F)~
s is can
bounded
now
Q with
~ runs
take
the
the
over
have
E'
be
are
v
apply
the
the
obtain
dimQ'+s above
that
First,let monomials
by
d+t(E/F)-l,proving
Q is we
f be in
belonging
a unique
K
v on
now
and
L(v) = 0.Then
a
of
in
s = t(v/Q).
Q,Qg. Suppose
follows.
now
centre
Q,Q'
rings
images
Q [ Z l , . ..,z s I p . L e t
t(E'/F)~
coefficients
),We
the
their
of
we
as
ring
that
that
rings
Qg
O
pair
that
finite,and
v
be
p and
such s v o v e r k. Let p be
be
follows
t(v/Q)
extension
in that
> t(v/Q)
in
d+t(E/F) Since
by
relationship
on
extension
Zl,...,z
independent
Q[Zl'''''Zs]'and of
n such
/k).
d+t(E/F)-I
field
find
d = dimQ,then
Choose
dimension
l N'
in
z e r o . We
follows
restrictions the
ideals
= 0
field
are
going
Q not
with
extension
LEMMA
the
the
require
Abhyankar.
domain
x in
al,...,aN,
statement We
j
the
~la c Q + m )-n ~ a . v . ( c Q + m n+l) 0, t~l i vi( m "~=i i i = d e s c r i b e d . C l e a r l y we c a n r e p l a c e c by
= aivi(x) for
among
that
v . < c Q * m n) = n v . ( m ) J 3 equations r e m a i n t r u e if n is
~a.v.(c) L~ i I c choosen
where
independent.)
p minimal
c be any e l e m e n t
v.(c) i
and
linearly
r > l.Choose
a
local
define an
domain
the
general
element
of
XI,X2,...
,the
to
we
extension
Q.Then to
Qg
and
Qg
sum
define
i t s e l f . If
of
352
L(v)
# 0,then
LEMMA
3.3.
we
Let
use
Q be
the
a local
such
that
L(v)
= 0 and
such
that
L(w)
= 0.Then
Suppose contains
an
greater and
can
the
this
~sa
).Then
power
of
the
implies
that
w
an
replacing
Q.
m-valuation
on
extension
least
of
hence
Q such
X l is
an
of
not
v
equal
to
Q
to Qg
the
algebraic
> v
over
to,and
O
v to
Q by
a polynomial
ideal
v
QN
(f).Multiplying
g
such
the
QN+I
hence,
of
exists
maximal in
that
QN+l. Replacing
there
wl(f)
coefficients
such
extension
wN
that
that of
integer
general
extension
with
Q/L(v)
v be
be
w(f)
the
generator
f(Xl)
let
the
.B w i t h
N = 0 and in
with
.
w N is
general
that
a polynomial
and
= 0.Let
w = v
coefficients
a suitable
But
not
domain
t(v/Q)
f =
w(a
assume
X 1 with
find
Min
definition
g f a l s e . Let N be
is
element
than
W N + 1 is
QN,We in
this
same
of
f by
Ov,We
that
v
can
(f(Xl))=0.
g f i e l d of
residue
O
,and V
hence an
over
k,modulo
THEOREM N such We P is
3.4. that
Let w
find
= P~QN'
the
field
images
N in
maximal
ideal
m g -valuation
of
O
on
Qg. T h. e n.
s t a g e s . Let
elements may
extension
of
assume
w
,and
fractions
of
QN
let
Q'
be
extension
same
that
of
of
w N on
contains
hence
that
w is
K
w
is
the
we
of
can
find
replacing Next,we
over of
not
Q
can
.Let
on
Q'
N'
by
fractions
integer w
such
N>N'
to QN" that ,where
such
that
in O
these
.Hence
g of
an
QN,/PN,
find
contained k
.
w N of
g QN[Zl,...,Zs]
a valuation
field
t. h e.r e . e. x i.s t .s
restriction
elements
of
w localisation
the
the
Then,by
L(w)=0.
basis
t ( W N / Q ' ) = 0 and
general
of
P=L(w).Then
QN''
that
a transcendence
Z l , . . . ,z s and
as
an
seneral
two
by
of
be
the
,we
form
WN.Then
w
is
generated
PN'
of
the
m -valuation. g
whose w elements be
at w
the is
Q' and g
centre
the
this
is
the
Qg.
4.Mixe d Multiplicities. DEFINITION.Let d ideals
of
Q be
a
Q subject
independent
set
m g -primary.(This ideals).Let
M
multiplicity
of
local to
the
general
condition be
to
and
elements states
a finitely
e(~;M)
ring,
is
definition till,i.e.,if inductively
clearly
of
I is
an
Q-module.
( I i , . . . , I d)
be
X l , . . . , x d is X(I)
=
( X l , . . . , x d)
independent Then
a set
we
of
an
set
define
i_~s
of the
be
independent
e(xl,...,Xd;M) d=O,then
l,then
that
senerated
I =
that,if
of
e(xl,...,Xd;M" This
let
condition
e(~;M
of
that g
we
) g the
choice
shall
) = L(M),while
use if
of
xl,...,Xd.The
is
that
due
d >0
we
define
to W r i g h t it
by
e(x~..... x~; M~)=e(-
x 2 , . . . , X d- ; M g / X i M g z - - e ( x
-
2 .... X- d ; ( 0 : X l ) M
) g
353
where and
x 2 ..... x d are (O:xl) M
are
the
both
images
of
considered
x 2 ..... x d in Q = Q g / X l % as Q - m o d u l e s . l t
and
is c l e a r
Mg/XlMg
that
e(~;M)
g s an a d d i t i v e and
that
an
we c a n
independent
take
in
for
of
lj=
X l , . . . , x d in
definition
the
the
choose
set
(~2 .... , ~ d ) , w h e r e we
on
is n o n - n e g a t i v e ( s e e
observe is
function
category
~
general
right-hand
side
to r e d u c e
the
in
is
a l-dimensional
local
two
results
elements
of
the
we
can
Q-modules,
theorem to
6).We
QN a n d ( x 2 ' ' ' "
-
set
of
write
also
,~d)
ideals
N of QN = Q N / X l Q N . T h i s
form. T h e n
is
clearest
the
inductive
two
terms
if
e(l;M)
= e(l;M')-e(~;M")
to m u l t i p l i c i t i e s
the
definition
define
x 1 belongs
= M N / X l M N , M '' = ( O : X l ) M N , t h e
referring
convenient.This
can
generated
form
enables where
that
IjQN+XlQN/XlQ standard
= (12' .... Id )'M'
us
finitely
example,Nothcott[5],p308 N such
e(!;M) where
of
proof terms
of of
b e c a u s e , if
Hilbert
x is
ring, then
which
hold
t hat
a)
e( I I I 2 ; M )
b)
if
functions
large
case,and
ring
to
the
n. We
which
of of
we
d=l,
more
an ideal
Ig
recall
and
use
I of
hence
without
will
the
QN.This case
is o f t e n
element
is a r e d u c t i o n
for
in
the
results
a general
XQg
= L ( I n M / I n+iM)
over
certain
on
in
we
proof the
next
lemma = e(il;M)+e(12;M)
11,12
have
e(ll;M) LEMMA
4.1.
a)~et
independent
sets
~"
is
an
I = ( I i , . . . , I d)
and
let !"
independent
I'
integral
= (ll,...,Id_l,l
closure,then
~)
b__~e
= (ll,...,Id_l,ldl~).Then
set,and e(!";M ) = e(!;M)+e(~';M
b)Let
same
= e(12;M)
of
d ideals,and
the
I = ( l l , . . . , l d)
_I* = ( l l * , . . . , I d * ) , w h e r e
lj*
be
is
an
the
)
independent
integral
set
closure
of of
ideals,and
let
l..Then3
e(!*;M ) = e(!;M) In b o t h case,we equal if
must to
and
I.
cases have and
in
only
if it
is m - p r i m a r y
,the
simply
a) For
b),we the
for
necessarily same
can
reduce
tile proof
ht(XlQg+...+Xd_iQg) the
case
d=l
is m - p r i m a r y . first
= d-I
a single Since
statement
of
the a)
to the and
ideal
d=l,ln
the
first
dim(Qg/XlQg+...+Xd_iQg) is
product
case
an of
follows,while
independent two
set
m-primary
the
second
ideals is
above.
time,using Xd_l(~)
we
integral
replace symmetry
the of
ideals e(~;M)
I i , . . . , I d by in
I 1 * ,.. . ,Id*
I 1 .... , I d , N O t e
XlQg+...+Xd_iQg,then
Id*Qg+Xd_l(~)/Xd_l(~)
the
closure
of
that
can
integral
closure,so
we
that,if
b)
above..
at
a
is not
IdQg+Xd_l(!)/Xd_l(~),but apply
one
we w r i t e
has
the
354
We some
now
come
to
preliminary
consisting that
of
the
d-I
ideals
(I1,...,Id_l,J)
m-primary elements
of ! ~ a n d
concern
is
is w i t h
of
is
ideal).Let
Xl,...,Xd_l,y
main
theorem
explanation.
We
Q,and
also
let
y be
we
an
a general
J
range
of
over
example,J
of
general
will
set
independent
element
set
multiplicity
paper,which
independent
let
1 be
independent
the
the
an
m-primary(for
Xl,...,Xd_
an
of
fix
!
=
all
ideals
could set
of
J such
require
(Ii,...,Id_
be
I)
such
any
general
that
elements
of ~ , J . O u r
function
e(!,J;M ) = e(xl,...,Xd_l,Y;M) considered Q-modules ideal
as
an
additive
and
as
a finction
(x I .... ,Xd_ I) In
general
of
function of
on
J.As
av(!;M)
is
all
m-valuations
theorem
set
of
the
coefficients
a
v. L e m m a
(l;M),and v -We n o w
of
ali
note We
to
states
ring
the
4.2.
where
the
negative
play of
There
is
integer
We
start
p342,Theorem
the
a
an v all
over equal the
18),which
parameters
to
we
some
being
over
have
all
the
first
term
does
is
l-dimensional
to
Northcott([3],Theorem
the
case
the
for
save
we
the
determines
over p = Q
Qp
a good
and
Mp
Q(P),M(P)
the
ideal.
ring
J(P)
of
set
P,and
prime
the
and
JQg+P/P lemma
for a finite
function
write
e(I,J;M)
on
not
save
uniquely
also
idea]
zero
=
all
(Qg)p
the
Q(P).These
following
proves
e(!,J;M).
for
all
Q,and
of
for
the
partition
can
write
it
the
av(!;M)
a finite
formula
to
form
is
a non-
number
of
v.
multiplicities([5], Xl,...,Xd_lly in
the
of
the
form
~e(J(P);Q(P))e(IQp;Mp) prime
ideals
involve
1-dimensional
the
by by
av(!;M)v(J) m-valuations
apply
is
role.The
expression
e(J(P);Q(P)) In
X(!)
equation
P range
also
2.3,p
denote
Xl,...,Xd_l,Y.We
a
generated
by
on ! , M , d e f i n e d
equation
X(I).We
theorem
for
good
an
I for
n o t a t i o n . Let
/PM and g g temporary
e(!,J;M)
Q(P)
given
above
of
associativity
the
ring
the
also
formula
exists
with
M
only
of
The
we
depending for
notation,denote
will
set
that
zero
further
of
Q(P)-module
sum
is
ideals
of
( M g ) p . We
existence
LEMMA
some prime
convenience
will
integer
that
a consequence
Qg/P,the
notations
implies
that
(l;M),and c a n be u s e d to d e f i n e v -d e r i v e m a n y of its p r o p e r t i e s .
module
finitely
denote
a
minimal
that,as
the
Q,which
introduce
good
will,for
and
3.1
of
will
Gav(!;M)v(J)
a non-negative v of
category
Qg.
terms,the
e(!,J;M) where
the
earlier,we
local
P minimal now
over
consider
X(1).Note
the
first
domain. Further
= e(yQ(P);Q(P)).
have
6Cwith
M.We
a formula
the
for
observation
e(yQ(P) ;Q(P)) that
due
term.
355
lengths
and
multiplicities
notation,this with
the
this
ring
runs
integral
of
these
of
w. T h e n
has is K
as
closure
finitely
w
of
many
a discrete is
are
equal
f o l l o w s . Let Q(P) maximal
over
its
of
ideals,and
of
k
K
g
of
ideal all We
is
we
P.The
can
:k
is
an
r(Q(P),w) that
integer
it
is
is
N such
that
w(y)
= v(J),where
determine
choosen
the
same
e(J(P);Q(P))
on
formula
is
whose
limit
Qg
factor
about
which
we
extension
is
the
be
a general the
now
each
field
integer. Now
w
v,we
=
at
residue
]w(y).
to
v is
that
g
a ramification
that
y is
the
m-valuation
a positive
that
w may
w
an
localisation
Northeott's
= ~r(Q(P),w)[K
determines
assume
follows
different
w
factor
note
choose
we
then
valuations
need
can
and
the
our
associated
fractions(recall
its
w
,and
domains).In
valuations
de~te
w
Each
the
field
ring).Let
extension
e(yQ(P);Q(P))
l-dimensional
range
in
valuation
a finite
in
w
general
element
restriction have
turn
an
to w ( y ) . of
of
of
w
wN
JQN.It to
Q.Since
expression
~d(!,v)v(J) V
where
d(l,v)
finite
set
recall
that
is of
a non-negative
v with
limit
different
integer
ideal
P meet
Q
p.
in
equal
Again
to
zero
referring
different
for to
all
save
theorem
p. H e n c e , w e
a
2.3,we
have
derived
the
moment,not
a formula e(!,J;M) where
av(l;M)_
=
~.~a (l;M)v(J) v v --
= d(l,v)e(IQ_ _ p;Mp).
Tile e x p r e s s i o n
we
have
derived
for
a
(I;M) V
particularly We
will
convenient,and
first
v a partition I'(v)
to
is,p).We on
be
LEMMA
of the
will
v.l"(v)
consider the set
also
our
the set
of
I into
sub-sets
Suppose
we
to
theorem
write
now
number
whose
maximal
apply
the
pQp
the
l"(v)
2.3,we
ideal,namely
see and
ideal
have
the
that
is
following
factor
ring
to
depends
and
two
factors.
with
I"(v).We
each
define
to
lemma.
= e(!' (v) ;M(v ~ 1 so
P has
that
only
Qp/pQp
bythe
is
images
e(!Qp;Mp)
Xl,...,Xd_l.Then
= e ( x I ..... X d _ l ; M p ) = e ( x t ~ l
turn
the
associate
It+l,...,Id_l.Then
xlQp+...+xtQ
formula I of
= e(l'(V)Qg;(M we
now I'(v)
ll,...,Id_ of
generated
associativity
e(~Qp;Mp)
ideals
consists
further,the
Xl,...,x t Xt+l,...,Xd_
simplify
I. c o n t a i n e d in the l i m i t i d e a l of v ( t h a t J Q v , M v f o r Q p , M p to i n d i c a t e their dependence
e(IQp;Mp)
11,...,1 t and
to
e(IQp;Mp).We two
is,for
--
is
ideals
= I-I'(v).We
of
this
factor
4.3.
Now
objective
consists
if
refer
we
minimal
a regular
of
the
back
prime
local
ring
Xt+l,...,Xd_l.We
with we
one
!'(v)
now
partition
obtain
.... X d _ ~ ; Q p ~ p Q p ) e ( x
I .... xt ; ( M p ) p Q P
) ) = e ( l ' ( v ) ; M v) Pg
the
other
only
on
factor
the
set
d(!,v).Our l"(v)
and
object not
on
is the
to
showthat
whole
set
~.
356
To
this
lj+p/p.Note have
end we will
belongs
anm
use ! " ( v )
to d e n o t e
t+l .... ,d-l.
ideals
of
(Q/p)g
ideals
contains
of
and
the
set
of
now
calculate
of Q/p
minimal the
form
ideals
p.We
of
consists
the
set
of
ideals
the
P'/pg,where
P'
of height
use
lj+p/p,where
of
the
set
is a prime
d-l.Hence
lemmas
4.2
and
4.3
ideals
of good These
ideal
P'
meet
lj
e(!"(v),J+p/p;M/pM).
(Xt+l,...,Xd_l)+pg/pg.
( X l , . . . , X d _ I) w h i c h
can now
e(!"(v),J+p/p;M/pM)
ideals
= Q/L(v).Consider
over
(Xt+l,..o,Xd_l)+Pg prime
containing
Q/p
set ! " ( v ) . W e
l l , . . . , I d _ I so that ~ " ( v )
for j=
minimal
ring
I b e l o n g s to l ' ( v ) , t h i s ideal will be zero. Henc£ we J r e d u c e d the set ~ to the set ~ " ( v ) . W e now c h a n g e
to the o r i g i n a l
Again,renumber
prime
the
that,if
essentially
notation
consider
lj+p/p
prime
are
the
of Qg w h i c h
ranges
over
Q in a p r i m e
the good
idea]
p'
to c a l c u l a t e
as ~d(~"(v'),v')e(l(v');Mv,/PMv,)V'(J)
where
v'
ranges
l(v')
is
the set
suppose hence the
that
we
the
vector
Next as
that
the
sum
I"(v)
that
take
I(v)
ideals. d(!,v)
d(!"(v),v).This
:k ],taken w g r e s t r i c t i o n to Q/p
then
we
that
have
that and
see
we
that
replacing
is e m p t y , a n d
of
those
d(I,v)
of
fractions the
of Q/p.
above,
valuations
w on
is p r e c i s e l y
the
depends
on
only
same the
set
d(I,v)
O.Then
of
we d e f i n e
such
ideals
Q by Q / p , w e to this
d i f f e r e n t l y . First
that of Q/p
can
apply
case. If we
I an i n d e p e n d e n t summarise
set
this
d(!,v)
L(v)=p
as above.
#O.Now
containing the
of d-i
dim(Q/p)-I
definition
follow
suppose
this
ideals
of
procedure ( I i , . . . , I d _ I)
in the s t a t e m e n t
of
the
lemma.
l"(v)
l"(v)
contains
e(!,J;M)= is the
set
dimp-I
and
ideals~and
above
on
d(l"(v),v)e(l'(v),Mv)V(J) I +p/p of Q/p w h i c h are n o n - z e r o 3 is an i n d e p e n d e n t set of i d e a l s
tile field
preliminary
a lemma,which
in the
m-valuations.
this
E
of i d e a l s
v is a v a l u a t i o n
conclude
definition
L(v)=
set
L(v)=O
that,with
4.4.
occurring
that
approach
= d(I"(v),v).We
LEMMA
of Q / p , a n d
proved
p,and
particular,
it.
v is an m - v a l u a t i o n
case
where
To
we
redefined
l(v)
is d e f i n e d , b y
is v. This
hence
L(v') ~
the d i m e n s i o n
field
over
the
following
that
is s i m p l y the
that
lj ~ L ( v ' ) . I n
whose
Then,by
of Q , d ( ! , v )
that
we note
e ( l ( v ) ; M v / P M v)
be an i n d e p e n d e n t
in
First
term
Q is a d o m a i n
let
= v.
of Q such
such
= M v / P M v over
suggests
suppose
lj+p/p
Mp/pMp
the
and
Now
v'
factor
as we have This
ideals
~r(Q(P),w)[K
= Qg/P
as d(!,v)
the m - v a l u a t i o n s
of
space
consider
Q(P)
over
indicates
formula
of
fractions
discussion,we
belong
that
of~Q/p.
require
one
the v a l u a t i o n s
to a r e s t r i c t e d
.
class
further v of
357
DEFINITION.Let K v of
t(v)
v over
k.
denote
Then
we
t(v)
We
have
equivalent We
already to
now
the
+
that
restricted
to
be
good.
LEMMA
If
in
lemma
We
recall
restrictions condition
that
of
that
that
=
t(v)
the
v is
degree a good
< dim(L(v)).Hence
that
L(v)
is
valuations
has
w
v in
on
Qg
dimension
the
and
in
1emma
v)
lemma
such
I and
this
good
occurring
4.4,d(I"(v),v)e(I'(v),M
the
of
residue
m-valuation
field
if
d-1.
valuations
mg-Valuations Qg/P
that
statements
prove
4.5.
transcendence
say
ht(L(v))
seen two
the
will
is
4.4
that
4.4
can
non-zero,v
are
obtained
P = L(w)
further
definition
the
be
is
good.
as
the
satisfies
residue
the
field
K
'
of
w
is
generated
w
is
to
of
a finite
QN"
by
It
w N and
Now
let
let
F',E'
follows
p=
be a
that
t(E'/F')
R'
and,since
3.4
field
fields
of
the of
O
v
of
Q',R.Then
the
at
dimension of
dimO v +I(E'/F') 1 kR
is
to
the
+n-t-d+1
residue
degree
t is
now
the
collect
state
our
field
N over height
of
4.6.Le___~t ~
Let v be
a good
of
seen
Q',and R
[R] at the c e n t r e of v generated extension of
domain,it can
is
also
apply
= dimR'
+ t ( k R,
/Kv)
+ t ( k R,
/Kv)
of
of
Q/L(v)
of
the
form
J is
any
ideal
K
I
R CR'~
0 and ~ wN transcendence
has
v
p,which the
=(ll,...,Id_
ideals
I contained
proves
lemmas
I)
such in
he
an
that
L(v),and
I +L(v)/L(v)
- -
ideals,and
M in
is the
any
the
hence
kR,
degree
has
d-t-I
result.
and
parts
of
theorem
independent
ht(L(v))=t. let
where
l"(v) I
set Let
be
does
the not
of
ideals
l'(v) set
be of
belong
of
the
Q.
set
ideals to
l'(v).
3
of
Q
such
finitely
that
generated
l,J
is
an
independent
Q-module,then
we
can
set
of
write
form e(~,J;M)
~,d(I"(v),v)e(!'(V)Qv~Mv)V(J)
where
v ranges
over
all
good
to
taken
zero
if
either
as
the
above
3
be
of
and
theorem.
m-valuation
of
e(~,J;M)
already
Q'.Further
= R'.But
k. H e n c e
together
main
THEOREM
Then,if
kN
~
k and We
2.3
over
E = QN/PN
have
of
ring,we
,
transcendence over
O
local
local
let
P
w N of
extension
algebraic
extension
a finitely
a regular
general
we
generated
1ocalisation O
the
w N is
that
equality
i.e. where
is
Q/p. Further
v l-dimensional
a is
for
is
w
K N of
a finitely
be
contains
Q'
d-t
k
that
fractions
of
R'
v l-dimensional.Since
dimension
write
1ocalisation
it
we
theorem
= N+t-d+l,where
a
and
.We n o w c h o o s e N so g consider the r e s t r i c t i o n
residue
localisation
is
QN
of
the
l-dimensional.Let
w. T h e n
of
= Q~PN,and
the
w
extension
from
that
Q ~P
R is
O
elements
hence
that
is
algebraic
is
t(v) = d i m ( L ( v ) ) - l .
m-valuations the
number
on of
Q,and ideals
e(I'(V)Qv;M in
I'(v)
is
v)
is
not
to
358
equal
to h t ( L ( v ) )
ideals
or
v a l u e s , d e f i n e d for contain
d v-I
followin$ Q/L(v) that
if l ' ( v ) Q v is not
of Qv. F u r t h e r , d ( l " , v ) all
containing
I K.
independent
ideals(dv=
additiona]
ideals,and .
.
.
the is
greater
proofs to be
of
terms
in the
lemma
4.2.The
To p r o v e
can,without
is
a domain.
true
of
obtain
to
only
4.2
of Q / L ( v )
d(l",v)_
two i d e a l s
of
ideals
has
set
of
integer which
the
of i d e a l s
of Q / L ( v )
of Q / L ( v ) , s o
of
such that
of
Now l e t
J
of
~,KIK2,J.We
d(!,Kl,V)+d(!,K2,v)
proof
of
to 4 . 5 . T h e has
this
in
of
be
any of
that
m-primary
use
e(~,K1K2,J;M)
of
proof
v I in p l a c e
the
e ( ! ' ( v ) , M v) of
occurring
in
is
the
that
~,Ki,J
d ideals,and
the
= e(~,J,K1K2;M
last
one.
of v. S e c o n d l y , 0 and
symmetry
in
the n u m b e r
L(Vl)=
ideal,so
Q containing
now make
to w h e n
reducing
further
write
generality,assume
ideals
as
is c o n t a i n e d
the e x p r e s s i o n
requiring
fix v , a n d
theorem
statement
the e f f e c t
occurring
statement
,we first
set
e
set
ideals
.
the
to those
loss
independent
of
to be zero
sum
this
we
part
lemmas
taken
ideals
function
KI,K 2 are
of
independent,then d(!,Ki,K2,v)=
The
I" of
set
non-nesative
I is an i n d e p e n d e n t
is an i n d e p e n d e n t
l
I,KIK 2 is also
sets
dim(Q/L(v)).The
property.lf
dv-2
(i =1,2)
an i n d e p e n d e n t
is a f u n c t i o n , t a k i n $
of
the
that
Q
is
an
same
is
function
)
e(~,J,KI;M)+e(~,J,K2;M) e(I,KI,J;M)+e(!,K2,J;M) Now
suppose
(i = 1,2)
in the
M = Q.The
fact
ideals
are
we
replace
can
we
expand
form
that
linearly
the
given
the
terms
in
the
functions
independent
e above
e(!,KIK2,J;M)
theorem v(J)
according
that
further
to lemma
3.1
implies
of V l ( J ) . S i n c e is the
take
of m - p r i m a r y
field
we
that
L(Vl)=O,it
of Q and
that
e((~,Ki)'(Vl);Q) = e((!,KiK~)'(Vl);Q) = l.Hence
that
are
of f r a c t i o n s we
are
left
the e q u a t i o n d(!,KIK2,Vl)= Note We
theorem in the
that
conclude
this
is also paper
to two s p e c i a l
first
m-primary,and First
d(l,v)
d(!,Kl,Vl)+d(!,K2,Vl
symmetric
by a p p l y i n g
situations
and
in the set the
of
formula
re-derive
)-
ideals
given
some
I.
in the
results
last
already
literature.
The
also
Qv
set
e(~,Ki,J;M)
assuming
with
follows
aboveand
on the
by the c o e f f i c i e n t
and
situation hence
we c o n s i d e r assumed
general
that
we
necessarily
consider form
the m u l t i p l i c i t y
an
is w h e n
independent
function
to be m - p r i m a r y . l f , X l , . . . , X d _ 1 , y
elements
of l , J , t h e n , b y
theorem
l l , . . . , I d _ I are set
e(~,J;M)
of
where
ideals
form
of Q.
J is
is an i n d e p e n d e n t
2.5,they
all
a joint
set
of
reduction
359
of
~,J.But
of
the
the
it
was
ideal
set
of
ideals
multiplicity mixed
~,J
function
multiplicity
theorem
4.6.Let
and
v(J)
and
contains
this
true
in
t of
only
as
that
is
the
is
of
this
sense
formula ideals if
reduces
the
being
require
J
J = xQ
Now
hence
on
consider
must an
Qv
is
Teissier's
given
in
L(v).If
these
are
ht(L(v))=t,
the
a primary
good
m-primary,
prime
such
factor
reduces
v is
all
a minimal
of
the
coefficient,then
be
factor
of
formula
m-valuation
consider
this
all
ideal
of
this
is
that
e(!'(V)Qv,Mv).The
to
L ( M v)
artinian
where
Mv
ring. Hence
is
the
=Zd(!,v)L(Mv)V(J)
proper
m-valuations.We
be m - p r i m a r y . H e n c e , i f
providing
that
and
we
can
the
set
of
ideals
an
a generalisation
Q,and
multiplicity
[10].Hence
the
to
the
multiplicity
to
xQ)
is
Teissier
non-zero
refer
that mixed
consider
t=0,i.e.L(v)
m-valuation.
over
to
is
we
with
e(!,J;M) sum
by
the
Ii,...,Id_l.Since
will
,and
implies as
used
paper
a Qv-mOdule,and
formula
this same
m-valuation
We
empty
as
the
the the
possible
a proper
considered
[9]
in
an
dim['Q/L(v))=d.
I'(v)
in
function. Next
v be
occurs
is
Q and
set
proved
{Xl,...,Xd_l,y)
take
dimQ/xQ
y=x. N o w
x is
an
= d-l(since consider
note element
XlX
the
that
is
ring
of
we
do
Q,we
can
a general
Q/xQ
and
not take
element
write
-I- X
of
for
I.+xQ/xQ.Then the i m a g e s of X l , . . . , X d _ 1 in ~ / X ~ g J set of g e n e r a l elements of I . H e n c e we h a v e
independent
--X
e(!,xQtM) which
we
denote
by
d(l
,x,M).We
d(!,x,M) This in
formula,and
greater
M=Q,we
consists we
the
l.Let
also
main
of
just
ideal
one
a special and
such
number ideal
case
XQg
of
in
theorem
the
of
ideal
Then of Q
results
Q has will
of
is
definition
adopted
here,e(IQp,M
where
a general
of
this
paper
1 are
are
all
dimension denote we
good
l.lf
will
prime
dim(Q/p)
2.1n
by
treated
equal,and
this I is
suppose
ideals
-~- I and
p,x
a general
pg
and
being so of
e(J+p/p)e(IQp,Mp). ) has
to
is
equal
to
terms
with
be
are
it.
only
interpreted
as
a
minimal
of
Hence
L(v)=p
that,according
I is over
good
element
the Note
that
there
L(v)=p. Further,the
in
case m-primary
minimal
Q with
contribution
formula
M)
formula
ll,...,Id_
above.Hence set
contains
4.2,the
-e(Ix,(0:x)
the
[6].
we
the the
meeting
l-dimensional,and
proof
of
where
m-valuations
P of
the ideals
which
consider
a prime
of
the
case
obtain
~d(!,v)L(Mv)V(X). of
result
the
p be
prime
[S].If
consider
m-primary
finite
in
now
=
fact,most
we
obtain
not
sum
detail
obtain Now
in
e(!x;M/xM)
=
l,is ,by
to to
the
the
the
e(x(Qp)g,Mpg~
P is Now
x
is
a reduction we
consider
formula. These
of
I
the must
element and
g
other be
so
of
l,but,since
this
possible
proper
definition
Qp
reduces
valuations
valuations
is
.Hence
l-dimensional,X(Qp)g to
the
v occurring the
formula
usual in
one.
the
reduces
to
360
e(l,J;M)
=~e(IQp;M
--
the
first
sum
being
second sem being sum
could
be
over
over
reduced
the
ideas
was
written,and
behind
author
this the
Stockholm
were
a lecture
in
the
Research
first the
August
Symposium
final,presented
minimal
over
l,and
Q(note
that
the
on
simplest
like
to
during
of
during
ideals
seems
hospitality
paper
in
not
1 prime
m-valuations
would
organisers
Symposium
version,still
ht
proper
further,but
its
+ Ld(l,v)L(Mv)V(J)
P
the
the
In c o n c l u s i o n , t h e Institut,Aarhus,for
)e(J+p/p)
~
in
thank
the the
April
and
developed
and
Nordic 1983
Summer
for
which
an
above
May
1979,when
the
first
School
the
form).
Matematisk
and
invitation
formed
the
first
basis
to of
draft Research give the
here.
References I.
S.Abhyankar.
On
the
valuations
centered
Amer.J.Math.78(1956) 2.A
.Grothendieck.
Elements
de
Geom~trie
P u b l . M a t h . IHES 3.D.G.Northcott.
A genera]
theory
P roc.G1asgow 4
"
"
No
in
a local
Algebrique.
11(1961) of
one-dimensional
Math. A s s o c i a t i o n
A generalisation
domain.
pp321-348
of
local
2(1956)
a theorem
on
the
rings.
pp159-169. content
of
polynomials. Proc. Cam. P h i l . S o c . 5 5 ( 1 9 5 9 ) 5
"
"
Lectures
on
Cambridse 6
7
D.Rees
"
Degree
"
University
Functions
ppI-7
pp
Asymptotic
"
,Hilbert
Properties 1983)To
Note
be
of
note
Cycles
of
published
General
P.Samuel
no72(1982)
Multiplicity
Commutative D.von
in
the
given
L.M.S
and
Mixed
planes
et
a Cargese
Conditions 1972
7-8(1973) Theory
Proc. L o n d o n . Math. S o c . ( 3 ) and
1981")
London. Math. Soc.29(1984)397-414.
de W h i t n e y . I n " S i n g u l a r i t e s
12 O . Z a r i s k i
degree
Ideals(Lectures
evanescents,sections
Asterisque D.J.Wright
series
Reductions
Multiplicities.Jour.
ii.
and
Algebra;Durham
series
Generalisations
10.B.Teissier.
Functions
(in"Commutative
Math. S o c . ] e c t u r e
in N a g o y a
"
in
70-78.
Lecture 9.
Press,Cambridge(1968)
Proc. Cam. P h i 1 . S o c . 5 7 ( 1 9 6 1 ) Multiplicities
"
multiplicities.
rings.
London
"
pp282-288 and
Local
functions
8
Rings,modules
Algebra.volume
Nostrand
15(1965) II
(Princeton)f960.
pp269-288
COHOMOLOGIE DE HARRISON ET TYPE D'HOMOTOPIE RATIONNELLE
Daniel TANRE ERA C.N.R.S. O~ 590 U n i v ~ s i t ~ des Science~ e~ Techniques de LILLE U.E.R. de Math~matiques Pur~ et Appliqu~es 59655 - VILLENEUVE D'ASCQ CEDEX (France) I II
-
III-
COHOMOLOGIES DE HOSCHSCHILD ET DE HARRISON. THEORIE DE L'OBSTRUCTION D'HALPERIN-STASHEFF. COHOMOLOGIE DE HARRISON ET FORMALITE INTRINSEQUE.
APPENDICE : MODELE'DE L'ESPACE PROJECTIF TRONQUE
¢P(m)/g~(2).
La th~orie de la deformation permet l'Etude des types d'homotopie rationnelle g alg~bre de cohomologie (ou alg~bre de Lie d'homotopie) rationnelle fix~e ~-S].
Halperin et Stasheff LH-S] ont obtenu les premiers r~sultats dans
ce domaine ; rappelons d'abord la terminologle utilisEe : un espace dont le type d'homotopie rationnelle est entigrement dEterminE par la donn~e de son alg~bre de cohomologie (resp. alg~bre de Lie d'homotopie) rationnelle est appelE formel (resp. coformel). Un espace est intrins~quement formel si son alg~bre de cohomologie est rEalis~e par un seul type d'homotopie rationhelle.
Nous montrons ici que les obstructions d'Halperin-Stasheff ~ la formalitE s'interpr~tent comme classe de cohomologie de Harrison. Cette dernigre semble ~tre le cadre le mieux adaptE ~ cette situation ; g partir du rEsultat ci-dessus, elle a permis ~ D. Merle EM~ Stasheff ~ - ~
et FElix EF~
d'unifier les theories d'obstructions d'Halperinet celle introduite par Lemaire et Sigrist ~L-~
dans le cadre des modules de Quillen. La construction du modgle bigradu~ EH-~
est illustrEe par un exemple
cohomologie non bornEe : ¢P(~)/¢~(2). Nous rendons triviale la premigre d~formation possible de ce modgle. La cohomologie Etant non bornEe, il existe une infinit~ de deformations possibles. Seule l'utillsation de la cohomologie de Harrison permet d'obtenir leur triviaIitE. La demonstration compl~te passe par une determination explicite de tout le module bigraduE et par l'interprgtation de ¢I~(~)/~(2)
cormne espace total d'une fibration ; elle fera l'objet d'une publi-
cation ult~rieure. Plus gEnEralement, le rEsultat obtenu concerne les espaces projectifs tronquEs
g~(~)/C~(n)
;
il s'~nonce ETa I~ :
362
: Rationnellement,
Th~or~me
il existe deux espaces
m~me alg~bre de Lie d'homotopie pas coformel
;
E
rationnelle
que
est l'espace coformel associ~.
n
¢~(~)/¢~(n)
~P(~)/¢~(n).
et E ayant n ~P(~)/~(n) n'est
Ils sont tous deux intrinsgque-
ment formels. Ce texte reprend une partie de ma th~se d'Etat soutenue ~ Lille, 26 janvier
1982.
Notations : gradugs signe tion
Nous emploierons
apparalt.
d'objets
le paragraphe
rationnels. gne par
Si
V
a(o)
x
consid~rfis
gradufi, de base
T(V) = T(Xl,...,x n) dual est notfi par
@ V,
le
o.
gradu~e commutative.
sont sur le corps (Xl,...,Xn), libre,
@
des
on d~si-
~(V) = ~(x I .... ,x n)
l'alg~bre tensorielle, la suspension
q,
~ toute permuta-
et appelg signe de Koszul de
l'alg~bre gradu~e commutative
engendrges par
sV : (sV) n = V n+l,
le
Ixl.
D'une mani~re g~n~rale,
les notations
utilis~es
sont celles de ~T~ .
DE HOCHSCIIILD ET DE HARRISON.
Soient
~
un corps et
D~fi~o~.~ entiers
le signe correspondant
est un espace vectoriel
L'espace vectoriel
I - COHOMOLOGIES
est permut~ avec un ~l~ment de degrg
adgc signifie alg~bre diff~rentielle
AV = A(xl,...,Xn)
degr~ d'un gl~ment
usuelles de signe pour les objets
I, les espaces vectoriels
l'alg~bre de Lie libre, V.
p
En particulier,
gradu~s est not~
L'expression Hormis
les conventions
: si un ~l~ment de degr~ (-I) pq
o
le
Un
{l,...,p+q}
V
un
~-espace vectoriel gradu~. est une permutation
(p,q)-mixage
de l'ensemble des
telle que :
~(i) < o(j)
si
D@f~on.-
L'espace gradu~
ou
1 @ i < j ~ p
p+l ~ i < j ~ p+q.
T(V)
est une alg~bre gradu~e commutative
pour le pr0duit mix~ d~fini par : o ... o v _: (v I ~ ... @ Vp) *- (Vp+ I ~ ... ~ v n) = X ~ ( o ) v u (n) o - ] (1)
o~
~
parcourt
~l~ments
v. 1 Soit
pour tout
p),
les
(p,n-p)
mixages,
g(O)
est
l e s i g n e de K o s z u l de
o,
les
s o n t homog~nes.
A
une
connexe,
Ik-alg~bre gradu~e commutative, (A° = ~), et soit
M
un
de type fini,
A-module gradu~.
(dim A p
finie
363
La cohomologie
de Hochschild,
provient du complexe suivant
Hoch(A;M),
de
A
~ coefficients
dans
M
: 11
si
a. e A, ]
a I @ ... @ a
I +
n
HomP(~ A,M) que
est mnni du degr6
est l'ensemble des applications
f(a I O ... @ a n ) = O
si
a.i = I
~ (lajl - ]) j=l
~-lin~aires
;
de degr~
p
telles
•)
(@f) (a] @ ... @ an+ l) = alf(a 2 @ ... @ an+ l) + (-l)V(n)f(al
0 ... @ an).an+ I
n
+
~ (-l)~(J)f(al j=!
avec
~(j) =
zn'P(A;M)
@ ... @ aj.aj+ 1 O ... O an+l) ,
J [ (;ail - 1) i=l
est formg des
Hochn'P(A
;
6-cocycles de
; M) = Zn'P(A
n-I ; M)/6 HomP-l( O A,M)
La cohomologie de Harrison, s'obtient
Harr(A;M),
~ partir d'un sous-complexe
HomP(~ A,M)
HomP(~ A,M)
de
A
~ coefficients dans
du complexe de Hochschild,
est formg des gl~ments de
HomP(~ A,M)
M
dgfini comme suit
s'annulant sur les d~com-
S
p o s a b l e s du p r o d u i t
Zn'P(A;M)
mixfi ; i l
est
stable
pour la difffirentielle
~
;
= Zn'P(A;M) A Hom~(~ A,M)
S
Harrn'P(A
; M) = Zn'P(A S
n, A,M) I
• M)/6 HomP-l( @ )
S
1
Elle est reli@e ~ la cohomologie de Hochschild par : Th~o~m£ l)application
(M. Barr ; ~ B ~ ) . -
naturelle
Harr(A;M) est injective.
Si
~
: +
Hoch(A;M)
est un corps de caract@ristique
O,
Pour terminer ristique
p
(~ droite) complexe
du foncteur
cotangent
II - THEORIE
V,
H(O)
0
÷
A
et Stasheff
suppl~mentaire
sur
et v~rifie
: dV p C
AV
eat un isomorphisme
de Sullivan
[H-~ V,
[Su] d'une alg~bre
d~finissent,
V =
@ V p~O P
(AV) p-1
," H+(AV,d)
de
Ltappendice
illustre
la construction
de
'•
pour un bon
;
p
celle-ci
s'gtend
eat bihomoggne
de
(A,d A)
du mod6le
(D-d)(V n) C
= 0
A. des premiers
g~n&rateurs
du modgle
H(¢~(~)/¢~(2);~).
construisent
le ~
le modgle minimal
Halperin
bigradu~
Si EH-S]
d6riv6
eat le
D'HALPERIN-STASHEFF.
eat le module
bigradu~
partant
~
Harr(A;A)
;
: Ho(AV,d)
(AV,d)
connexe.
d'alggbre
eat faux en caractgpas d'un foncteur
EQu].
une graduation
en graduation
ne provient
Avec un saut d'un degr6,
O : (AV,d) * (A,O)
commutative
choix de
degr~
D(A/~;A)
notons que ce r6sultat
de Harrison
HomA(A;-).
DE L'OBSTRUCTION
Soit gradu~e
ces rappe]s,
et que la cohomologie
eat une adgc,
un module
de Sullivan
bigradu6
@ (AV) m m&n-2
cohomologiquement
connexe,
(non minimal)
si
v e Vo,
et Stasheff
~ : (AV,D) ÷ (A,dA) ,
@ : (AV,d) * (H(A,dA),O).
;
Halperin
Ce module
la classe de cohomologie
v6rifie de
en
:
~(v)
eat 6ga-
~H-S] .
Appelons
p(v). (AV,D)
eat le mod6le
filtr6
Chacun de ces deux mod61es TJ-graduation
et
TJ-filtration
de
(A,dA).
v6rifie
un th6or~me
(pour Tate-Jozefiak)
d'unicit6
lea graduation
et filtration
suppl6mentaires. De6i~O~w~ que
(H(A,dA),O).
d'alg~bre r6alis6e
L'alg6bre
de cohomologie
Soit
de
0.
H
(A,d A)
eat formelle
eat intrins~quement
isomorphe
g
H
si elle a m~me modgle minimal
formelle
sont formelles
si toutes
; autrement
lea adgc
dit,
H
eat
par un seul type d'homotopie. R6sumons
bigradu6
: Une adgc
la thgorie
d'obstructions
~ : (AV,D) + (A,d A)
p : (AV,d) Dgfinissons
÷
(H(A,dA),O)
maintenant
~ la formalit6
un mod61e = (H,O)
une application
filtr6 ;
d'Halperin-Stasheff.
construit
notons
~ partir du
q : H ÷ AV °
une section
365
: Hom
o(Vp_l,H)
d6rivation
{)u7 = rl~/
de
÷ Homl(Vp,H) (AV)gp_1,
sur
de degr6
O,
= O @~dv
d6finie
par
p
d6finit
:
le premier
un 616ment
de
indice
tel que
Homl(v
,H),
D-d
~7 = O
est l'unique
sur
soit non nulle
on note
0 (D)
P Homl(Vp,H)/Im
I.
Si
DD = ~d
tel que
O~
oO
V = e ~ ]ffID~
de
AV
pour construire
Op+1(D).
de
AV
O (D) P tel que
repr6sente D~ = ~d
est formelle.
construit
sur
[H-~ .-
Th~or~me p, (iV,D)
l'obstruction
sur le modgle
g l'existence
d'un automorphisme
(fiV)~p.
Supposons
H(AV,d)
de type fini.
Si O (D) = O pour tout P bigradu6 (fiV,d), H(AV,d)
p
Si Op(D)
et tout module
de degr6,
de 3 unit6s.
la premi6re
d6formation
Elle est donn6e par 4 D3v I = ~ x 3 x 4
o3
~, 8, 7
de
d
baisse
D3
se prolonge
4 ~(z 1) = ~x 3
:
;
;
en
Z 4 , il faut et il suffit que 3 P(z 2) = ~x3x4, on obtient :
4 = pe~(yly 2 + x4z I + x3z 2) = 2~x3x 4 = 2D3v 1
I(P)(v2)
= 2D3v 2
La d6formation
; D3
modgle bigradu~.
peut donc ~tre rendue triviale
d~velopp6e
La difference
considgrent
toutes
par F61ix entre
EF~
sont identiques,
est ~galement
les deux approches
les applications
en compte que celles se prolongeant
les r6sultats
~ = ~ = T.
l(P)(v 3) = 2DBV 3. par un automorphisme.
L i a ~ o n entre les obstructions d ' H a l p e r i n - S t ~ h e ~ f ~ La th~orie
ne prenant
Pour des
la TJ-graduation
D3v 3 = T x~ ,
~(~)(Vl)
et Stasheff
(iV,D)
formelle.
: 4 D3v 2 = ~ x3x 5
;
filtr6
sont des rationnels.
Pour que En posant
possible
= O pour tout
est intrins6quement
Exemple : Illustrons cette th~orle ~ l'a[de de l'appendice. raisons
D
comme le montre
de degrg g
Vp+ 1 .
de F~lix :
men6e
~ partir
du
tient au fait qu'Halperin I de V dans H, F61ix P Au niveau des obstructions,
le eorollaire
page 26 de
~F~.
366
III - COHOMOLOGIE DE HARRISON ET FO~MALITE INTRINSEQUE. Pour la fin de ce paragraphe,
A
est une alg~bre gradu~e commutative,
connexe, de type fini. La liaison entre les obstructions prgcgdentes et la cohomologie de Harrison passe par l'utilisation d'un module particulier : l'alg~bre des cochalnes sur le module de Quillen ou mod~]e FHS (ECHO, EFe ~ , D~crivons-le pour
~Ta~ page 67).
(A,O) :
p : (AZ,d) ÷ (A,O)
est un morphisme d'adgc induisant un isomorphisme en cohomolo-
gie avec : (AZ,d) = (As -I @
L(W),d I + d2),
W @ ~ = s
-I
@
A,
dI
est lin~aire en
Z
et
d2
quadratique. L'injection canonique de l'alg~bre de Lie libre tensorielle
T(W)
L(W)
dans l'alggbre
fournit par dualit~ et d~suspension :
j : s-IT(@ W) ~ s-I @ T(W) ÷ s-I @ IL(W). Si
(Yi)iei
est une base homog~ne de
@ W,
not@
J(s-l(yi @'''@Yi ))' o p ; l'application s-I ~ ~(W)
Yi ...i fournit un syst~me de g~ngrateurs de o p induite j : A ÷ AZ est une section de p (j@ = id). O
Soit
yiYj = k~!li,jl cij (k)Yk,
e.. (k)lj e Q,
la loi d'alggbre de
A.
En dgtaillant la d~finition de l'alg~bre des cochaTnes, on obtient (ETa] page 71) : m-! d2Yi ...i = ~ Yi ...i Yi ..i o m p=O o p p+ l " m
dlYi
;
m-2 i = ~ ! c.. (k)(-l) x)(j) j=O k lij,lj+ll ljlj+l Yio...i._ik ij+2 .... o "'" m " j
(AZ,d2)
est le module bi~radu@ du bouquet de spheres d'homologie
im "
~ A
(ETa] page 66). La TJ-graduation correspond ~ la longueur des crochets par orthogonalit~ :
Z
= S-I(~P+2~(W)) ~. P
Le lien avec la cohomologie de Harrison appara%t dans la :
Proposition.d~signe par
une application lin~aire de degr~ l, on l m+l la d~rivation d'alg~bre associ~e et par ~ e Homs( @ A,A) le com-
pos~
Si
@ Y y = O Yj.
@
Soit
~ : Zm ÷ AZ o
est la diff~rentielle du complexe de Harrison, alors :
367
(Yi 8 . . . .
i)
67(s
2)
7 peut ~tre fitendue 5 Zm+ 1 ssi
3)
~ peut ~tre rendue triviale par un automorphisme de
o
telle que
~
=
@ Y l m +))l = POy(d2+dl)Yio'''lm+l 6~ = 0
; AZ
ssi il existe
~'
~ y% '.
Dgmons~aZZon
:
I) Par dfifinition, on a : 6~(s-l(YioS"''SYim+Â)) = Yio~(s-l(Yil@'''OYim+l)) +
(-1)
lYi ...iml o ~(s-l(Yio@"'@Yim))Yim+l
( =
lyi 1
-I)
o
+ (-I)
.
P~7
m-I j=O
~(J)% -I y(s (YioS...Syi yi @ .... ) j j+1 ~Ylm+l )
]Yi ...i ] o m Yio ° • "imY im+ 1 +
YioYil " " "im+ 1
m-1
(-1)
|
(-])v(J) ~i
c. (k)Yi o ' ' ' l j -"1 keli j ,ij+ll l j i j + I
j=O
k lj+ ' 2" ..im+l~J
= p@ (d2+dl)Yio...im+ 1" .
2)
y
peut ~tre ~tendue g
(d2+dl)-cocycle,
3)
i.e.
Zm+ 1
pey(d2+dl)y i
.
o'''lm+l
ssi Q (d2+dl)y i . . o ''im+l
est un
= 0.
La derni~re proprigt~ se d~duit directement de la comparaison de %
l(y')
et
~%'
;
la construction de
1
et le th~or~me d'Halperin-Stasheff
se transcrivent tels quels au module FHS. De la proposition ci-dessus et du th~or~me d'Halperin-Stasheff, on d~duit directement : Thgo&~me.- Si
Harrm'l(A;A) = 0
pour tout
m > 2,
alors
A
est
intrins~quement formelle. RemaYcque
l'hypoth~se
: La r~ciproque du th~orgme est fausse en g~n~ral. En effet,
Harr%l(A;A)
= 0
signifie que toute application
l m+l
~ e Homs( 0
A,A)
368
prolongeable Or,
en colonne
d2+dl+7
particulier
Zm+ 1
ne donne pas n6cessairement
Si
Harrm'2(A;A)
sont 6quivalentes (i) (ii)
l'alg~bre
Nous laissons
tre la remarque
A
Harrm'l(A;A)
page 21) s'adapte
une diff6rentielle,
= 0
pour
m > 4,
est intrins~quement = 0
pour
la d6monstration
formelle,
au lecteur
De mSme,
; celle faite par F61ix
l'exemple
de l'annexe
(EFt,
1 de EFe] illus-
ci-dessus. Un simple calcul donne la description
de l'alg&bre de cohomologie
Ixi [ = 2i,
les propri6t6s
m > 2.
rationnelle
de
i(x 3, x 4, Xs)/R o3
sauf dans un cas
:
ici sans probl~me.
Appe~d~ee~ : relations
par un automorphisme.
:
Propos~on.suivantes
peut ~tre rendue triviale
R
est l'id6al engendr6
Les premiers g6n6rateurs Zo
Z1
Z2
Z3
Z4
par
x~-
par ggn6rateurs
£P(~)/K]P(2)
et
:
, 2 x3x5, x4x 5 - x~, x~ - x3x 4.
du module bigradu6
s'gcrivent
:
6
dx 3 = 0
8
dx 4 = 0
10
dx 5 = 0
15
dy 1 = x~ - {x 3 x 5}
17
dy 2 = x 4 x 5 - {x~}
19
dY3 = x ~ -
24
dZl = Yl x5 - Y2 x4 + {Y3 x3}
26
dz2 = Y2 x5 - Y3 x4 - { Y l
31
dv I = Yl Y2 + x4 Zl + {x 3 z 2}
33
dv2 = Zl x5 + Yl Y3 + x 4 z 2
35
dv3 = z2 x5 + Y2 Y3 + {x23 z 1 }
38
dWl = Yl Zl - x4 Vl + {x 3 v 27
40
dw2 = Yl z2 - x4 v2 +
40
dw5 = Zl Y2 + Yl z2
{x~ x4}.
-
x23 )
v I x5 x 4
v 2
+
(x 3 v 3}
369
42
dw 3 = v2 x5 - Zl Y3 - x4 v3
42
dw 6 = Y2 z2 + zl Y3 - v2 x5 + {x~ v I}
44
dw 4 = v3 x5 - z2 Y3 - {x~ v2}.
45
du I
wl x4 - vl Yl + {w2 x3}
47
du 2
w2 x4 + Wl x5 - v2 Yl
47
du 7
x4(w5-w2 ) - Vl Y2 + {x3~w3+w6 )}
47
dUlo
49
du 3
x 4 w 3 - v3 Yl + vl Y3 + w2 x5 + (x~ w I + x 3 w 4}
49
du 6
x 4 w 6 + z I z 2 - w 5 x 5 + {x~ w I + x 3 w 4}
49
du 8
v2 Y2 - Y3 Vl + x5 w2 - x5 w5 - x4 w6 + {x3 w4}
51
du 4
x 4 w 4 + v2 Y3 + w3 x5
51
du 9
51
dUll
53
du 5
w 6 x 5 - v2 Y3 - !/2 z~ - {x~ w 2} 2 Y2 v3 + x 5 w 3 + x 5 w 6 - {x3(w5-w2)} 2 v3 Y3 + w 4 x 5 + {x 3 w3}.
Z5
1/2 z 2
I - v2 Yl + w5 x4 + {x3 w3}
Comme annonc~ dans l'introduction, aux techniques
des modules minimaux
lit~ intrins~que
de cette alg~bre.
la cohomologie
(KS-modUles,...)
de Harrison
permet d'gtablir
allige
la forma-
370
BIBLIOGRAPHIE
[B~
Michael BARR
- Harrison homology, Hochschild Journal of Algebra
LCh~
Kuo Tsai CHEN
-
Extension
of
C~
8, (1968), function
homology and Triples, 314-323.
Algebra by Integrals and
Malcev completion of
1' in Math. 23, (1977),
Advances ~
Yves FELIX
- D~nombrement
des types de
la d6formation,
M@moires
181-210.
K-homotopie. SMF, nouvelle
Th6orie de s6rie n ° 3,
(1980). - ModUles bifiltr6s.
LFe I] Yves FELIX
Can. J. Math. 33, n ° 26, (1981),
1448-1458. ~e
~
Yves FELIX
-
Espaces Luminy
Steve HALPERIN,
formels et (~ para~tre
James STASHEFF - Obstructions Advances
D.K.
HARRISON
-
HOCHSCHILD
-
in Math.
Commutative (1962),
G.
~-formels.
Conf6rence Marseille-
SMF). to homotopy
32, (1979),
equivalences,
233-279.
algebras and cohomology T.A.M.S.
104,
191-204.
On the cohomology groups of an associative
algebra.
Ann. of Math. 46, (1945), 58-67. Jean-Michel
LEMAIRE,
Fran@ois
SIGRIST - D6nombrement
rationnelle.
[M£
Pierre MERLE
- Formalit6
C.R.A.S.
des espaces et des applications
Th~se de 3~me cycle, Nice, Daniel QUILLEN
des types d'homotopie
t. 287 A, (1978), Paris.
- On the (co)-homology
of commutative
Proc. Symp. Pure Math.
continues.
(1983).
17, A.M.S.
rings,
Providence
(1970),
65-87.
Is-s]
Michael
SCHLESSINGER,
James STASHEFF - Deformation homotopy
[Su]
Denis SULLIVAN
Infinitesimal
computations
Publ. I.H.E.S. Daniel TANR~
ETa I]
- Homotopie Sullivan. Verlag.
47,
in Topology,
(1977), 269-331.
rationnelle : ModUles de Chen, Quillen, Lecture notes in Math. 1025, (1983), Springer
!
Daniel TANRE
theory and rational
type (~ paraTtre).
- Th~se, Lille
(1982).
COHOMOLOGIE DE L'ESPACE DES SECTIONS D'UN FIBRE ET COHOMOLOGIE DE GELFAND-FUCHS D'UNE VARIETE par Micheline VIGU~E-POIRRIER(*)
R~sum~
Soit
•
F ~+ E ~
sont connexes par arcs, nilpotents
X
un fibr~ nilpotent,
oN les espaces
et ont le type d'homotopie
de C.W. complexes
+ de type fini. On suppose que le type d'homotopie d'homotopie
d'un complexe
k.+1 r 1 V S i=]
de
H (X,@) # O, qu'il existe
oil
sections continues du fibre. r~elle
C > |
cas suivants
tels que si : ou bien,
type d'homotopie dimension
.< n
et
inf(k.) I
n,
>. n. Soit
tel que et
r
N e ~
F
type
et une constante
~ dim Hi(F,(~) ~ C p dans les deux i=O le fibr~ est trivial (i.e. F = F X), ou bien X ale Sd V y
de Gelfand-Fuchs
(o~
Y
est un complexe
>. 2
d'une vari~t~
telle que
simplicial de
AMS
:
M,
des grou-
C ~, compacte,
con-
H+(M,fR) # O, et dont toutes les
sont nulles est ~ croissance
CLASSIFICATION
exponentielle.
55 P 62, 55 R 05, 57 R 32
MOTS CLES : ModUle minimal de Sullivan, fibr~ nilpotent, cohomo]ogie de Gelfand-Fuchs. (~) ERA au CNRS 07 590
a
p ~ N, on a
de dimension
classes de Pontryagin
ale
X
l'espace des
d ~ I). On en d~duit que la suite des dimensions
pes de la cohomologie nexe, nilpotente
r > 2
de dimension
On d~montre qu'il existe
d'un bouquet et
simplicial
n >. I
372
O. Introduction. Dans
[16], Thom ~tudie le type d'homotopie
cations continues d'un espace
X
dans un espace
de l'espace des appli-
F, homotopes ~ une applica-
tion donn~e. Dans commutative,
[]4], Sullivan d~crit une alg~bre diff6rentielle
module de l'espace des sections d'un fibre alg@brique donn~. Dans
l'espace tent
F
[5], Haefliger d~termine
connexes par arcs, tel que (A,dA)
tel que
alg~bre diff~rentielle de l'espaee
le type d'homotopie
rationnelle de
des sections homotopes ~ une section donn~e pour un fibr~ nilpo-
E : E + X. Si on a un tel fibr~
module
gradu~e
H (X,~)
dim A n < ~
~ : E + X
tels que les espaces soient
soit de dimension pour tout
gradu~e commutative
finie, et
X
a un
n, il d~montre qu'une certaine
(ASZ,D)
est un module de Sullivan
F. Tousles
espaces consid~r~s
dans ce papier sont connexes par arcs,
nilpotents et ont le type d'homotopie de C.W. complexes de type fini ; ce qui nous permettra d'utiliser,
de mani~re biunivoque,
Sullivan entre la topologie et l'alg~bre, Nous nous int~resserons la base
X
et la fibre
ale F
le dictionnaire
~tabli par
voir §.I.
~ des fibres nilpotents
F~+ E
E ~ X
type d'homotopie d'un complexe simplicial de dimension
o~ n ~ ]
est n-connexe.
Utilisant les r~sultats de [5] et [18], nous montrerons
T h ~ o r ~ e 3.3.
Soit
X
un espace n i l p o t e n t a y a n t
le type d'homo-
topie rationnelle d'un complexe simplicial de dimension n ~ 1 et tel que + H (X,~) # O. Soit F un espace ayant le type d'homoto~i e rationnelle d'un k.+l
Vr S 1 o~ r ~ 2 e t inf(k i) ~ n. Alors, si F X est i=! l'espace des applications continues de X dans F muni de la topologie
bouquet de spheres
eompacte ouverte, il existe N e ~ P p ~ N, on a ~ dim H:(FX,~) ~ C p 0
et une constante r~elle
C > |
tels que
373
Th~or~me 3.4. Soit un fibr@ nilpotent propri~t~s
suivantes,
I) 1 . |
tels clue si
p >. N,
>. A p.
I. Th~orie du module minimc~ de Sullivan. Nous rappelons bri~vement qui seront n~eessaires
les r~sultats de la th~orie de Sullivan
dan~ la suite. Les d~tails se trouvent dans
[]4],
[9],
[6], E7], [17]. Les alg~bres consid~r~es oO
k = ~
ou
IR, commutatlves
dans le sens suivant
b.a = (-l)Pqa.b. On notera
lal = p
deux alg~bres eommutatives
gradu~es,
par : (a O b)(a'Ob')
le degr~ de
= (-])Ibl'la'laa'
A ° = k. Une alg~bre diff~rentielle A.D.G.C.)
sont des k-alg~bres : si
gradu~es
~ An n~O b e B q, alors
a e A p,
a e A p. Si
la multiplication
A =
dans
A
et
A O B
B
sont
est d~finie
O bb'. Une alg~bre est dire connexe si
gradu~e commutative
(A,d)
(en abr~g~
est une alg~bre 8radu~e commutative munie d'une diff~rentielle
d
374
de degr~
+I
v~rifiant
A.D.G.C.
(M,d)
d(a.b) = (da).b+(-l)]ala.(db).
est un module de
d'A.D.G.C.
: (M,d) ÷ (A,dA)
Une A.D.G.C.
(A,d A)
(A,d A)
On dit qu'une
s'il existe un homomorphisme
induisant un isomorphisme
en cohomologie.
est dite libre s'il existe un espace vectoriel gradu~
V =
@ V n tel que A = AV est le produit tensoriel de l'alg~bre ext~rieure n>,! construite sur @ V 2n+| et de l'alg~bre sym~trique construite sur @ V 2n. n n On d~montre, ~ ] , que toute (A,dA) telle que H°(A,d A) = k possgde un module minimal unique ~ isomorphisme une A.D.G.C.
libre Dans
A(),
(AV,d)
pr~s. Dans le cas oh
caract~ris~e
HI(A)
par le fair que
d(V) C
A~2V.
[14], Sullivan d~finit un foncteur contrevariant,
not~
de la cat~gorie des ensembles simpliciaux dans celle des A.D.G.C.
~. Si
X
est un espace topologique,
cial des simplexes singuliers
de
X
on consid~re
l'int~gration
morphisme d'alg~bres
gradu~es de
H~(A(X))
singuli~re
H~(X,~).
Une A.D.G.C.
(A,d A)
si
est un module de
(A,d A)
Sing X, l'ensemble
et on note encore
A(Sing X). De plus,
A(X)
des formes diff~rentielles
type d'homotopie espace
sur
simpli-
l'alg~bre d~finit un iso-
sur la cohomologie
rationnelle
est appel~e module de l'espace
X
A(X).
Si on se restreint ~ des espaces topologiques
nilpotents
ayant le
d'un C.W. complexe de type fini, on peut associer ~ un tel
X, un ~-espace
gie rationnelle que
X~
ayant m~me homotopie
X. Le foncteur de Sullivan
de categories entre la cat~gorie homotopique sont les Q-espaces), que
= O, c'est
rationnelle et m~me cohomoloA
rationnelle
et la cat~gorie des @-A.D.G.C.
dim Z n < ~ pour tout
induit une ~quivalence (dont les objets
libr~
(AZ,d)
telles
n, et il existe un ensemble bien ordonn~
I
tel
n que
Z =
$
Z
; pour tout
est une fonction croissante d'un espace nilpotent En particulier,
on a :
X
e, il existe de
~ ," d(Za ) C
n
e N A( @
tel que
n
(X),~).
Z e ;
n
ZB). Le module minimal
correspond ~ la d~composition Z n = Hom(H
Z C
de Postnikov de
X.
375 Soit maintenant
F jr-j-+ E _~N÷ X
un fibr~ dont t o u s l e s espaces
sont connexes par arcs. On suppose que
H (X,~)
dimension finie en chaque degr#, et que
HI(X)
sur
H (F). Soit
(B,d B)
espace vectoriel gradu6 me d'A.D.G.C.
~
A(p)
(B,dB) ~
L'inclusion @
i
i
dule minimal de
d
sur
B
du fibr~
,
A(E)
A(j)
÷ (B @ AZ,d)
~. On montre, dans
E
> A(F)
q
q
,~
~],
que
(AZ,~)
sont des morphismes d'A.D.G.C. ~ : (AZ,d) + A(F)
(AZ,d)
(B,dB)~-+ (B @ AZ,d)
n
~
÷ A(F)
l'appli-
est le mo-
est le module minimal de
~ ~ X.
De plus, il existe un ensemble bien ordonn6 Z =
B @ AZ, et un morphis-
tels que le carr# suivant commute :
et la projection
F. On dit que
sont de
X, alors il existe un
induit un isomorphisme en cohomologie. Soit
cation induite par
base
un mod&le de
~ : (B @ AZ,d) + A(E)
H (F,~)
agit de mani~re nilpotente
Z, une diff~rentielle
A(X)
et
~ A(X)
ou bien
I
tel que
@ Za, d ( Z ~ ) C B ® A(8 (B,dB)
e > (A,dA)
alors
D~monstration
en cohomolo~ie,
:
Elle g@n@ralise
de
dB @ I
Dans le cas g@n@ral, tel que
alors
celle de la proposition
q > I
et
tel que D
alors le lemme est vrai, car les diff@rentielles
I
:
en cohomolo~ie.
§.5.5 de E4]. On remarque que s'il existe
tivement aux transpos@es
d'A.D.G.C.
Z~(~ o 8) = Z~(8) o Z (~).
Le,me @.2. S i e induit un isomorphisme
Z*(~) induit un isomorphisme
donn~
d~finit un
= (E~(A @ Z),A) ÷ (Z:(B 8 Z),D).
~(~)
Ii est clair que si on a des morphismes (C,dc)
Z~(e)'
et
Z = Zq A
3 du
et
dZ e B,
sont ~gales respec-
d A @ 1.
rappelons qu'il existe un ensemble bien or-
Z =
8 Z ; pour tout ~, il existe ~el Z ~ et la fonction ~ + n ~ est croissante ~ " d(Z ~ ) C
n
tel que
n
Z C
~o e I
Soit (B,dB) ~
(B @ A(
~
fix@, l a c o n s t r u c t i o n
Z ),d)
est une A.D.G.C.
~+
(~
= ± <doZ , sy A sy'>
^
si
zeZ
;
y,y' e L(s -l H+) ; (si
l'~l~ment ~gal ~
y
mais
l'espace des gan6rateurs Lie libre
L(s -I H+), et
est de degr~
p, alors
est de degr~_ p+|). De plus,
sy s
y
--1
^
H+
~(~
sur lequel est construite
zk)
sy
Z
O
est
correspond
l'alg~bre de
=o.
k>O II s'en suit que l'application tive, oO
Pk
est la projection
de
(A2Z)k/
Z
(A2Z) k
@
,. l
tels
compacte ouverte,
que,
pour tout
dim Hi(FX,~) i=O
il
existe
p ~ N, on a :
>. C p
N
e t une c o n s -
388
D~monstration (H @ AS + Z,D')
: D'apr~s la remarque 2.6., un module de
FX
sera
o~
(-i)
D' S£(z) = (O ~ Id) S£(doZ) -
D'apr~s le lemme 3.2., on a : suffisamment grand. Ii existe donc (H+~ S|z)P/(Im
N
[ 6il Si(z). 2~i~£-I
D'(H + @ S|Z) p = 0
tel que, pour tout
pour
p
p ~ N
D' N H + @ SIZ)P~+ HP(F,~), ce qui implique que
dim HP(F) ~ dim(H + ~ SIz)P - dim(Im D' ~ H + @ SIz)P.
II s'agit de majorer la dimension de On a:
H ~ AS + Z
D"
(Im D' N H + @ S|Z)~
> Im D"
T
T
SIZ
~ Im D~NH + ~ SIZ s
On va montrer que D] est surjective, ~ partir d'un certain degr@. Soit
¢ e H @~S~Z, on peut d~composer !
mani~re suivante : et
!
~ = ¢I +
¢" ~ (H + @ ASZ) ~
~ ¢i + ~'' o~ 2 $i.<m
!
¢i £ SiZ
de la i e {l ..... m}
si
A~2SZ.
On volt que D' ¢"
+
@ A >'2
@ A >'2 SZ
2.'2 SZ + A SZ) i>.2
(H @ ASZ) + = (H+ e S!Z) e C
il est clair que
D'¢" +
~ D'¢'i e 2. O : i )
~ i~2
que
~
tel
~ = D '~I'
dim(Im D' O H + @ SIZ)P ~ dim(Siz)P-l.
On en d~duit qu'il existe
dim HP(F,Q)
N e ~
tel que si
p ~ N, on a :
>, dim(H + 8 SIZ)P - dim(SlZ)P-l.
r k.+1 H + = H+( V S i ), on a : i=l
Comme
r p-k. - ] r p+~ -k. - ] ~ dim(SiZ) l = ~ dim Z 1 i i=l i=l
dim(H + 8 SIZ)P =
r d'o~,
p ~ N, dim HP(F,~)
si
i=l La d~monstration est identique
l'espace hypotheses
du th~or~me
~ celle du th~or~me
Remarque.
Le th~or~me
des sections du th~or~me
dim Hp+al_l_k i(F)
d'un fibr~
[18].
3.3. se g~n~ralise, E H
(A,dA)~-> (A @ AZ,d) -~ (AZ,d o)
propri~t~s
suivantes (I)
o O~
1 ~ d @ n
X et
ale
o~
Soit un fibr~ nilpotent
: il existe
Y
~ X
3.3., et le fibr~ poss~de
Th~or~me 3.4.
n ~ ]
type d'homotopie
est un complexe
@ Q.
3.3, ~ l'aide de cette minoration,
4.1. de
F~
8 ~ - dim Hp+al_l(F)
oO
de mani~re X
et
F
~vidente, v~rifient
les
un module minimal
d = dA @ 1 + I @ d o
F ~+ E
H
> X
ayant les
tel que rationnelle
simplicial
d'un bouquet
nilpotent
Sd V Y
de dimension
@ n.
(2) F a l e type d'homotopie rationnelle d'un bouquet de spheres r k.+I V S I o~ r ~ 2, inf(k.) ~ n. Alors, si F est l'espace des sections i=l -C > | tels continues du fibre, il existe N ¢ ~ et une constante r~elle que si
p ~ N, on a :
~ dim Hi(F,~) i=O
~ C p.
390 Dgmonstration mod~le
(B,d B)
l'alg~bre de
de dimension
commutative
le lemme 2.3• on peut supposer
finie tel que
gradu~e
S d. II est classique
dule de
: D'apr~s
Bp = O
(A(u)/u2,ds
que
(A(u)/u 2)
= O),
si
off
que Y a un
p > n. D'autre part,
lu] = d , est un module
~ (B,dB), not@e
(A,dA),
est un mo-
S d V Y, on a : (~u/u 2)
A = ~ @
+ @ B ,
Si on utilise proposition
u.B
+
= O,
les techniques
2.4., on voit qu'un module
dAU = O,
de Haefliger
de
"F
dA(b)
r~sum~es
= dB(b).
par la
est
m
(AZ 8 A( ~ SiZ) @ ASuZ,D) i=l
o2
(SiZ)1. ~q, on a
montre
qu'il
D'(H + @ S Z) p = O. On v~rifie, U
comme pour le th@orgme (S Z) p dans U
3.3.,
que
pour p assez grand,
[Im D" n (H + ~ S Z)] p+I
est sur~eetive,
Ii
l~'application
D" de
et donc que:
r
si
p >. N,
et on conclut
dim HP(F,~)
comme dans
>. [ dim Np+d_l_k.(F) i=l i [18],
th6or~me
O ~. - dim Hp+d_l(F ) O @
4.1. H
Th~or~me 5.5.
soit un fibr6 nilpotent
r k.+1 I F = V S , r >. 2, q .< inf(ki). Soit i=I tent de dimension n o~ q < n < inf(ki). nue
f : X ÷ Sq
telle que la q
i eme
F ~+ E
o ~ sq o
X
un complexe
simplicial
o2 -nilpo-
On se donne une application
application
induite en homotopie
eonti-
ration-
391
nelle
(fa~:@ ~ ) q
le fibr~
Ho
: ~q(X) @ @ ÷ ~q(S q) @ @
et le fibr~ image r~ciproque
soit
non nulle . Alors, pour
N = f (Ho) : F ÷ E -~ X, la coho-
mologie de l'espace des sections est ~ croissance exponentielle. D~monstration : Soit
A*(-)
le foncteur de Sullivan d~fini de la
cat~gorie des complexes simpliciaux dans celle des A.D.G.C.. On a un morphisme A*(f) : A*(S q) ÷ A*(X). Soit Sq
dimension finie de du fibr~
g
(a
(A(a)/a2,d = O) __mm_+ A~(S q) e s t un g ~ n f i r a t e u r de degr~
un module de
q ) . Un module m i n i m a l
est de la forme :
o
(A(a)/a 2) ÷ ((A(a)/a 2) O AZ,d) ÷ (AZ,do)
Ona:
dz = d Oz + a O ea(Z) L'application
o~
Oa(Z) e AZ.
(A(a)/a 2) A~(f)°m ~ A~(X)
(A(a)/a2)C i ) (h(a)/a 2 @ AU,6) ~ o~
@
est
(f@~ @ IQ)q
un quasi-isomorphisme et est surjective,
~u = ~ u + a @ ~(u) o me d'A.D.G.C, pour tout de
entraine
6 u e AU o
et
u ~ U. On a
Ker 6
A~(X)
@ o i = Am(f) o m. L'hypoth~se que que pour tout
u ¢ U,
par
r(a) = a,
r(u) = O
r o i = Id. Comme dans le lemme 2.3 , soit
engendr~ par les ~l~ments de degr~ en degr~
on a :
~(u) ~ A+U. On d~finit alors un morphis-
r : (h(a)/a 2 @ AU,6) + A(a)/a 2
A(a)/a 2 @ AU
mentaire de
o~
a un module
> n
n . Le passage au quotient
P : (A(a)/a2 @ AU,~) ÷ [(A(a)/a 2 @ AU)/~,~J = (A,d A)
est un isomorphisme en cohomologie. On a :
1
l'id~al
et par un suppl~-
392
A*(S q)
1
( A ( a ) / a 2)
J
(A,dA) =
---+ ( ( A ( a ) / a 2 ) S A Z , d ) ÷ (%Z,d o)
r
((A(a)/a2)OAU,6)
[(A(a)/a2OAU)I,~-
A*(X)
On appelle
j l'inclusion d@duite de
on d~finit un morphisme d'A.D.G.C,
r' : (A,dA) ÷ A(a)/a 2
Un module minimal de base = f.(~o )
est
i ; com ~
(A,dA)
rtl) = O,
tel que r' o
j = Id.
du fibr@ image r@ciproque
:
(A,dA)~+ (A 8 AZ,D) ÷ (AZ,d o)
o~ Dz = doZ + j(a) O ea(Z) = doZ + a O @a(Z)-
Dans ces conditions~ morphismes d'A.D.G.C.
j ® Id
et
les morphismes r' O I d
j
et
r'
s'~tendent en des
rendant commutatifs les diagrammes
suivants :
(A(a)/a2,d = O) :
, (A(a)/a 2 @ AZ,d)
(AZ,d o)
j @ Id (A,dA)
Ir' ( A ( a ) / a 2 , d = O)
..........
.......(A @ A Z , ~
J
r'
@ Id
( A ( a ) / a 2 e AZ,d) --
(AZ,d o)
II (AZ,d o)
393
II est clair que la construction
E
de Haefliger d~crite dans
le th~or~me 2.1
est fonctorielle,
on a donc des morphismes
R = Z~(r ' @ Id)
et
tels que
l=Z~(j ~ Id)
I o R = Id :
(Z'(A 8 Z ) , D ) I ~==g=;
d 'A.D.G.C.
Z~(A(a)/a 2 O Z,d).
R
En particulier, surjective
l'application
; on a donc, pour tout
dim Hn(F,~)
est l'espace des sections du fibr~
on prend
Y
I
en cohomologie est
n e ~ :
~ dim Hn(Fo,~)
Le th~or~me 3.5
induite par
H
oN
r
(resp.
F o)
(resp. Ho).
se d~duit donc du th~or~me 3.4
dans lequel
~gal ~ un point. On est amen~ ~ ~noncer la conjecture
Conjec~e
:
suivante
Soit un fibr~ nilpotent
F ~+ E
]I > X
o__~ X
a
le type d'homotopie d'un complexe simplicial de dimension n >. | e t + r k.+l l H (X,~) # O, F a l e type d'homotopie d'un bouquet de spheres V S i=I o__~ r >. 2 e t inf(ki) > n, alors la suite des hombres de Betti de l'espace des sections du fibr~ est g croissance exponentielle.
Ce r~sultat aurait des applications de la cohomologie de Gelfand-Fuchs Soit soit
LM
M
d'une vari~t~
C~
une vari~t~
int~ressantes
paracompacte
l'alg~bre de Lie des champs de vecteurs
:
de d i m e n s i o n continus sur
resse ~ la c o h o m o l o g i e de I ' A . D . G . C .
C (L M)
continues sur
M, appel~e cohomologie
de Gelfand-Fuchs
le U -fibr~ n
U
:
n
+ EU (2n) n
au-dessus du 2n-squelette BU
n
~ BU (2n) n
du fibr~ tangent de
M
de
n ~ 1, M, on s'int~-
des formes multilin~aires de
M. On consid~re
restriction du fibr~ universel,
de la base
le fibr~ associ~ au-dessus
dans l'~tude
BUn. Soit BU
n
.Vn : EU(2n)n'- ÷ EU~2nn" Xu EUn ÷ n et de fibre EU~2n)t~. Le complexifi~
est classifi~ par une application
f : M + BU . n
394
L'image r~ciproque par On a l e
f
du fibr~
~n
r~sultat suivant d~montr~ par Haefliger
Th@or~me 3.6. Conjecture de Bott [3] : l'espace des sections continues du fibr~ :
On montre que
EU (2n)
pie rationnelle de
S 3 ; si
n ~ 2,
quet d'un nombre fini de spheres en nombre est I'A.D.G.C., non libre :
lhil = 2i-I,
I c i l = 2i,
&l~ments de degr~ Une base de
ale
EU (2)
Yn
ale
type d'homoto-
type d'homotopie d'un bou-
~ 2. Un module de l'espace
l'id~al de
S[c I ..... Cn]
dh i = ci,
o~
engendr~ par les
(voir par exemple,
[12]).
a ~t~ d~crite par Vey [3].
[3] ou [II], un module du fibr~
partir du module de
le type d'homotopie
(E(h I ..... hn) @ Sic I ..... Cn]/l,d)
> 2n, on a d c i = O,
H~(EU(2n),Q) Dans
Iest
est un module de
EU (2n) + E + M.
n = I,
EU (2n)
:
C (LM)
est 2n-connexe, e t a
rationnelle d'un bouquet de spheres. Si
EU (2n)
EU(2n) -> E -~ M.
est un fibr~ :
et du module de
f
EU (2n) + E ÷ M
est donn~
not~
f~ : H • (BUn,~) = R[~ 1 ..... Cn] + fl~(M) o~
Icil = 2i, et
f (c2i_]) = O,
f (c2i) = P i e
ferm~e repr~sentant la classe de Pontryagin
Pie
~4i(M)
est une forme
H4i(M,~).
II est clair, que si toutes les classes de Pontryagin sont nulles, le fibr@
EU (2n) ~ E ÷ M
poss~de un module minimal du type
(a'(M),d M) + (~(~)
o~
d = dM O 1 + 1 0
0 AZ,d) + (AZ,d o)
do.
On d~duit, de la remarque suivant le th~or~me 3.3 suivant :
le r~sultat
395
Th~or~me 3.7. Soit tente, de dimension
~ 2
une vari@t@
et telle que
les classes de Pontryagin de et une constante
M
A > I
M
connexe,
compacte, nilpo-
H+(M,IR) # O. On suppose qua toutes
sont nulles,
tels que, si
I
C~
alors il existe un entier
N
p ~ N, on a
dim H i (C ~ (LM)) > A p .
i=o Le th@or~me 3.7. s'applique en particulier sion ~ 2),
aux spheres
aux groupes de Lie compacts connexes nilpotents,
(de dimen-
et aux produits
finis de telles vari@t@s.
B I B L I OGRAPH .
.
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.
I E .
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17-37.
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[4]
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n°484,
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!
Micheline VIGUE-POIRRIER 37, Parc d'Ardenay F. 91120 Palaiseau
