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North-Holland Mathematical Library Board ofAdvisory Editors:
M. Artin, H. Bass, J. Eells, W. Feit, P. J. Freyd, F. W. Gehring, H. Halberstam, L. V. Hormander, M. Kac, J. H. B. Kemperman, H. A. Lauwerier, W. A. J. Luxemburg, F. P. Peterson, I. M. Singer and A. C. Zaanen
VOLUME 28
NORTH-HOLLAND PUBLISHING COMPANY AMS T E RDAM. NEW YORK . OXFORD
Graded Ring Theory c. NASTASESCU
University of Bucharest Bucharest, Rumania and
F. VAN OYSTAEYEN University of Antwerp Wilrijk, Belgium
d
NORTH-HOLLAND PUBLISHING COMPANY A M S T E R D A M . NEW YORK . OXFORD
Worth-Holland Publishing Company, 1982 All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 4 4 4 8 6 4 8 9 ~
Published by: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM . NEW Y O R K . O X F O R D
Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52, VANDERBILT A V E N U E NEW YORK, N.Y. 10017
PRINTED IN T H E NETHERLANDS
FOREWORD This book is aimed to be a ‘technical’ book on graded rings. By ‘technical’ we mean that the book should supply a kit of tools of quite general applicability, enabling the reader to build up his own further study of non-commutativerings graded by an arbitrary group. The body of the book, Chapter A, contains: categorical properties of graded modules, localization of graded rings and modules, Jacobson radicals of graded rings, the structure thedry for simple objects in the graded sense, chain conditions, Krull dimension of graded modules, homogenization, homological dimension, primary decomposition, and more .....
One of the advantages of the generality of Chapter A is that it allows direct applications of these results to the theory of group rings, twisted and skew group rings and crossed products. With this in mind we have taken care to point out on several occasions how certain techniques may be specified to the case of strongly graded rings. We tried to write Chapter A in such a way that it becomes suitable for an advanced course in ring theory or general algebra, we strove to make it as selfcontained as possible and we included several problems and exercises.
.
Other chapters may be viewed as an attempt to show how the general techniques of Chapter A can be applied in some particular cases, e.g. the case where the gradation is of type Z. In compiling the material for Chapters B and C we have been guided by our own research interests. Chapter 6 deals with commutative graded rings of type 2 and we focus on two main topics: artihmeticallygradeddomains, and secondly, local conditions for Noetherian rings. In Chapter C we derive some structural results relating to the graded properties of the rings considered. The following classes of graded rings receive special attention: fully bounded Noetherian rings, birational extensions of commutative rings, rings satisfying polynomial identities, and Von Neumann regular rings. Here the basic idea i s to derive results of ungraded nature from graded information. Some of these sections lead naturally to the study of sheaves over the projective
vi
Foreword
spectrum Proj(R) of a positively graded ring, but we did not go into these topics here. We refer to [125] for a noncommutative treatment of projective geometry, i.e. the geometry of graded P.I. algebras. Chapter D presents the theory of filtered rings and modules and we focus in particular on the gradation associated to a filtration. In the sections entitled 'Comments, References, Exercises' we included references to interesting related topics that have not been treated in this book.
A remark about references: only references to a chapter different from the chapter containing the reference contain the chapter's indication A, B, C or D; references within the same chapter only mention the section numbers.
ACKNOWLEDGEMENT The authors thank the department of mathematics of the co-author's institution for i t s hospitality while much of this work has been conceived and written. Francine did a nice job typing this book and we thank her for the quality of her work as well as for her patience.
1
CHAPTER A: SOME GENERAL TECHNIQUES IN THE THEORY I: Graded Rings and Modules
OF GRADED RINGS
I . 1. Graded Rings and the Category of Graded Podules
Let G be a m u l t i p li cat i v e group with i d en t i t y element e . A ring R i s said to be a graded ring of type C- i f there i s a family of R , say {Ru, u c G } , such t h a t R
=
8
oCG
o f additive subgroups
R u a n d RoRT c RUT f o r a l l
where we denoted by RoRT the s e t of a l l f i n i t e sums of products r
C
U,T
G,
with
c Ru, rITc RT. The elements of U Ro ar e called homogeneous elements
of R, a nonzero r
oEG
6
Ru i s said t o be homogeneous of degree o, and we write
deg r = u. Any nonzero r
C
R has a unique expression as a sum o f homogeneous
elements, r = 2 ro, where r o i s nonzem f o r a f i n i t e number of u in G . of G
The nonzero elements ro in the decomposition of r are called the homogeneous components of r. Let R be a graded ring of type G. I f then t h e ring S = R with gradation
TI
: G
+
Sa = 8CICN
ring o f type H . This r i n g wi l l be denoted by
H i s an epimorphism of groups
'(a)
R
Ro,
f o r a l l a C H , i s a graded If
(H)'
e
morphism of groups, the ring R ( H ) with gradation ( R ( H ) ) u 0
c H i s a graded ring of type H. I n case G = Z , H
i s denoted by R ( d ) i . e . R(d) =
@
ncZ
=
: H =
+
F i s a mono-
Re(u)for a l l
{nd, n C Z } then R ( H )
Rnd.
I f G i s commutative and R i s a graded ring o f type G then the ring S = R
with gradation defined by S graded ring.
o
= R
-1 f o r a l l u
u
c
G , i s called the opposite
1.1.1. Proposition I f R i s a graded ring of type G then Re i s a subring of R and 1 c Re.
2
A. Some General Techniques in the Theory of Graded Rings
Proof : Let
From ReRe c Re i t follows immediately t h a t Re i s a s u b r i n g of R .
1 =
z r
u ~ G
be the homogeneous decomposition of 1
and h, F RT, then h rUiT
u
=
, with rUXT c RUT.
1.X = 2 r X UCG
-
e R.
Pick
T
e
G
Consequently
c7
0 holds f o r a l l u # e in 6 . I t follows t h a t rUX = 0 f o r a l l
# e i n G and f o r a l l X e R . Therefore 1 = re e Re.
1 . 1 . 2 . Examples
1. Let G be a group, A a r i n g . The group r i n g R = A [ G I i s a graded
r i n g o f type G where Ro = A.
for all
u F
G.
2. Any r i n g R may be considered a s a graded r i n c o f type G , f o r any
group G, by p u t t i n g Re
=
R , Ro
0 for
=
u
# e i n G . Such a r i n g i s
s a i d t o be t r i v i a l l y G-Graded.
3. Let R be a graded r i n g of type G . Let R" be the o p p o s i t e r i n g f o r R i . e . R" has t h e same underlying a d d i t i v e group a s R b u t m u l t i p l i c a t i o n i n R" i s defined by the r u l e x =
O
y
=
yx f o r x , y R".
Putting ( R 0 ) d
R -1 makes R" i n t o a graded ring o f type G. 0
4. Let R be any r i n g and l e t
9 :
R
--f
The r i n g o f t w i s t e d polynomials S a variable X t o
r
6
R
R an i n j e c t i v e r i n g homomorphism =
R [ x , , ~ ] i s obtained by adding
R and d e f i n e m u l t i p l i c a t i o n by ,o(r)X
=
Xr f o r a l l
and extend i t t o polynomialsin X ( w r i t i n g c o e f f i c i e n t s on t h e
l e f t hand s i d e o f the powers of X ) . Obviously S i s graded of type i
w i t h Si
=
0 if i
c
0 and Si = {a X i , a
c R
}
i f i z 0. I f
automorphism of R then i t i s c l e a r t h a t R I X , X-',,O
'0
i s an
1 may be c o n s t r u c t e d
i n a s i m i l a r way. 5. Let k be a f i e l d , X an i n d e t e r m i n a t e , and l e t k ( X ) be the f i e l d o f r a t i o n a l f u n c t i o n s i n X over k . Although k ( X ) can only be graded of type 2 i n t h e t r i v i a l way, i t i s very well p o s s i b l e t o d e f i n e nont r i v i a l g r a d a t i o n s of type Z/nZ on k ( X ) , f o r every n e N. Indeed;
3
1. Graded Rings and Modules
I t i s e a s i l y seen t h a t , i f i a n d j have the same image mod n then
k ( X ) : = k ( X ) ? , hence the gradation i s well defined. Straightforward J
1
v e r i f i c a t i o n learns t h a t k ( X ) i s graded of type Z/nZ.
For a ring R we write R-mod f o r the category of l e f t R-modules whereas mod-R denotes the category of r i g h t R-modules.
c R-mod i s said t o be a graded
Let R be a graded ring o f type G , an En l e f t R-module i f there i s a family [Ma,
F! such t h a t M = Q Mu ot G
h(M) =
U
OF G
OF
G} of additive subgroups of
for a l l
and ROYT c Ma-,
T
0,
F G . Elements of
Ma are called homogeneous elements of M. I f m # 0 i s i n \IO
f o r some o t G then m i s homogeneous of degree
o
a n d we write : deg m = a.
Any nonzero m F M has a unique decomposition m = c
o6G
f i n i t e number of the mu a r e zero.
mO where a l l b u t a
Unless otherwise s t a t e d module will mean l e f t module. A submodule f l of M i s a graded submodule i f N =
(N
Q
n
Mu),
or equivalently, i f f o r any
O G
x c N the homogeneous components of x are again i n H .
1.1.3. Example. Let R be a graded ring, M a graded R-module and l e t N be a submodule of M. Let ( N )
9 I t i s easily verified that ( N )
are graded submodules of M.
be the submodule of M generated by N
4
i s maximal amongst submodules of N which
Consider graded R-modules M and P I . An R-linear f : M
a graded morphism of degree Graded morphisms of degree
T, T
T
+
F G, i f f(F?,) c Mu-,
N i s said t o be
for all a
C
build an additive subgroup H O M R ( M , N ) T
HomR(M,N). I t i s c l e a r t h a t HOMR(M,N) = 8 H O M R ( M , N ) T abelian group of type G.
n h(M).
TFG
i s a graded
G. of
4
A. Some General Techniques in the Theory of Graded Rings
Composition of a graded morphism f : En morphism g : N degree
m.
+
P of degree
T
+
N of
degree
o
C C- with a graded
G y i e l d s a graded morphism g o f of
c
I f N i s a submodule of En then P/N may be made i n t o a graded
module by putting : ( t 4 / N ) 0 R-linear projection M
--f
=
Mn
t
N/N.
With t h i s d e f i n i t i o n , the canonical
M/N i s a graded morphism of degree e ( l e f t a n d
r i g h t ! ) . The category R-grG consists of graded R-modules of type G and the morphisms a r e taken t o be the graded morphisms of degree e . I n the sequel we s h a l l write R-gr i f there i s no ambiguity possible about G. The category R-gr possesses d i r e c t sums a n d products. Furthermore i f (Mu, f a $ ; a , p c I ) i s some inductive system, resp. projective system
of objects in R-gr, then the module li+m Mg c R-mod, resp. lim Ma c R-mod a
a
may be graded by putting ( 1 L m M a ) u a
for a l l
u
=
l i m (Ma)4, resp. ( l i m Enna)o = lim(Va)a a
a
a
c G.
If M, N c R-gr t h e n we write HomR-gr(rVI,N) f o r the graded morphisms o f degree e from
M t o N . For any f c HomR-gr(M,N),
b o t h Ker f a n d Coker f
a r e i n R-gr. Indeed, since Im f i s a graded submodule of N i t follows
t h a t Coker f = N/Im f =
@
( N O + Im f)/Imf i s the cokernel o f f i n R-gr.
IJCG
I t i s not hard t o verify t h a t R-gr i s an abelian category which s a t i s f i e s Grothendieck's axioms Ab 3, Ab 4 , Ab 3* and Ab 4*, cf [1021. Since a l s o Ab 5 i s e a s i l y v e r i f i e d , we w i l l say t h a t
R-gr i s a Grothendieck
category. The forgetful functor, U : R-gr
-f
R-mod, ( U f o r "ungrading", unusual
perhaps, b u t preferable t o "degrading") associates t o M ungraded R-module.
the
underlying
The forgetful functor U has a r i g h t a d j o i n t which
associates t o M 6 R-mod the graded R-module $4 = 0 where each uc G "M i s a copy of M written tux, x 6 MI, with R-module s t r u c t u r e defined by : r
*
Tx = uc(rx) f o r each r c Ro. If f : ).I
+
N i s R-linear t h e n
5
1. Graded Rings and Modules
Gf :
GM
+
N' i s given by G f ( O x )
= uf(x),
and i t i s c l e a r t h a t Gf has
degree e 6 G. The f u n c t o r M
+
GM, from R-mod
be a d i r e c t sum o f copies o f
G R-gr i s exact. Note t h a t U( M) need n o t
--t
M
s i n c e t h e component
ON,
u
e
C-,
i s not
G an R-submodule o f M ( b u t i t i s an Re-submodule o f course). 1.1.4.
Notation
. b u t we o m i t t h i s f o r R s i n c e i t w i l l always be c l e a r
We w i l l w r i t e U(M) =
from the c o n t e x t whether R i s being considered as a graded r i n g o r n o t . One o f t h e standard problems concerning R-gr t h a t we w i l l be
P as an o b j e c t o f R-gr,
w i t h may be phrased as f o l l o w s : i f F4 has p r o p e r t y i s i t t r u e then t h a t
dealing
has p r o p e r t y P i n R-mod. The converse problem i s
u s u a l l y f a r more e a s i e r t o handle.
I f M 6 R-grG then f o r u C G we d e f i n e t h e 0-suspension M(u) o f Y t o be t h e graded module a f u n c t o r To : R-gr
by p u t t i n g M ( u ) ~ = PITU. So we d e f i n e d
o b t a i n e d from +
R-gr, To(M)
=
M(u)
and t h i s f u n c t o r
may be
c h a r a c t e r i z e d by :
1" TOoT,
=
, for
Tm
all u,-rCG
2" T o T -1= T - 1 , ~ T = Id, f o r a l l 0
0
0
o
c G
U
3" U o Ta = U, f o r a l l
c G
o
I t f o l l o w s t h a t , f o r any
c 6,
CT
Tu d e f i n e s an equivalence o f c a t e g o r i e s .
I n a s t r a i g h t f o r w a r d way one proves t h a t {R(o), u
c
F) i s a s e t o f
generators o f R-gr and t h a t R-gr i s a Grothendieck category w i t h enough i n j e c t i v e o b j e c t s , c f . [ 102 J
.
An F c R-gr i s s a i d t o be o r - f r e e i f F has a b a s i s o f homogeneous elements, o r e q u i v a l e n t l y any
M c
F
8 R(u), where S i s a subset o f G. Since acs R-gr i s isomorphic t o a q u o t i e n t o f a g r - f r e e o b j e c t i n R-or and
A. Some General Techniques in the Theory of Graded Rings
6
s i n c e t h e l a t t e r a r e c e r t a i n l y p r o j e c t i v e as o b j e c t s o f R-gr,
i t follows
t h a t R-gr has enough p r o j e c t i v e o b j e c t s .
1.1.5.
Example. L e t H be a subgroup o f 6 and l e t S cG be a s e c t i o n f o r
G modulo
H i . e . a s e t o f r e p r e s e n t a t i v e s f o r t h e r i g h t H-cosets o f G.
L e t R be a graded r i n g o f t y p e G and M M("'H)
0
=
Mho.
hCH
Then M(''H)
c
F o r any u c S we p u t
R-yG.
i s a graded R(H)-module w i t h g r a d a t i o n :
( M ( " y H ) ) h = MhUfor a l l h c H. I t i s c l e a r t h a t M = any
M
S,
0
+
M(OYH) d e f i n e s a f u n c t o r : R-gr
i f G = Z and H = {nd, n c 7) w i t h d For 0 5 k 1.1.6.
d - 1 we w r i t e R ( k '
5
Remark
tJ
all u
+
{O,l,
=
..., d - l } .
d, = R ( k y ').
I f R i s a graded r i n g o f t y p e 6 t h e n
N i s s a i d t o be graded o f degree
c G.
R(H)-gr. I n p a r t i c u l a r
0 t h e n we may choose S
c a t e g o r y gr-R o f graded r i g h t R-modules. f :
+
0 M(OYH and f o r ocs
Note t h a t t h e s i d e on w h i c h
T
I f V, N T,
T
c
c 6,
we can d e f i n e t h e gr-R t h e n an R- i n e a r i f f(MU)
c
F?Tu f o r
i s w r i t t e n i s d e t e r m i n e d by
t h e f a c t t h a t f s h o u l d be r i g h t R - l i n e a r . The o-suspension o f M s h o u l d t h e n be d e f i n e d as f o l l o w s (u)M i s d e t e r m i n d e d by g r a d a t i o n ( ( u ) M ) ~ = FIUT; a g a i n , t h e s i d e on w h i c h
0
c
c
gr-R
mod-R w i t h
a c t s i s determined
by t h e f a c t t h a t ( u ) M s h o u l d now be a graded r i g h t R-module ! Consequently
if f i s b o t h a l e f t and r i g h t y a d e d morphism o f t w o - s i d e d graded Rmodules t h e n f must have c e n t r a l degree i . e .
deg f c Z ( G ) .
7
I. Graded Rings and Modules
1 . 2 . Elementary Properties of Graded Modules
Throughout t h i s section R i s a y-aded ring of type G f o r some fixed g r o u p G , so we omit notational references t o P i n w h a t follows.
1.2.1. Lemma. Let M; N, P
c R-or and consider the following commutative
diagram of R-morphisms :
where f has degree e . I f g ( r e s p . h ) has degree e then there e x i s t s a graded morphism h ' ( r e s p . 4 ' ) of d e y e e e , such t h a t f f
=
9 o h ' (resp.
g' o h).
=
Proof
We give the proof in case deg g = e .
I _
Pick x c Mo. From h ( x )
=
h ( x ) u i t follows t h a t h ' : M
2
+
N may be
UCC-
defined by putting h ' ( x ) : h ( x ) o . 1 . 2 . 2 . Corollary.
Consider P 6 R-gr; then P i s a projective
in R-gr i f a n d only i f
p
i s a projective R-module.
I f P i s a projective in R-gr then i t i s a d i r e c t summand of a
Proof
gr-free object F. Therefore suppose
p
f
p entails o g
=
g' : P
F
+
f
--t
P
i s projective in R-mod. Conversely,
i s projective i n R-mod. Since P c R-gr there i s a or-free
object F c R-or such t h a t F Of
object
+
f
P
+
0 i s exact in R-gr. P r o j e c t i v i t y
the existence o f an R-linear map
a
:
p+5
such t h a t
l p . According t o Lemma 1.2.1. there e x i s t s a graded morphism -
F
of degree e such
+o,
t h a t f 0 g ' = I,,. Thus the exact sequence
s p l i t s and therefore P i s a projective object of R-gr.
We will sometimes r e f e r t o projective objects of R-gr as gr-projective graded modules. So by the above corollary gr-projective i s nothing b u t
A. Some General Techniques in the Theory of Graded Rings
8
graded p l u s p r o j e c t i d e . 1.2.3.
Corollary. L e t 0
+
N
sumnand o f M i f and o n l y i f
-
M be e x a c t i n R - o r . Then N i s a d i r e c t i s a d i r e c t summand o f
F.
A graded l e f t R-submodule I o f R i s c a l l e d (an homogeneous o r ) a graded l e f t i d e a l o f R. I f I i s two-sided then i t i s c a l l e d a graded i d e a l o f R.
1.2.4.
Lemna. L e t E c R-gr. The f o l l o w i n g statements a r e e q u i v a l e n t :
1. E i s an i n j e c t i v e o b j e c t o f R-gr. 2 . The f u n c t o r HOMR(-, E) i s exact.
3. For every graded l e f t i d e a l I o f R, t h e canonical i n c l u s i o n i : I - R gives r i s e t o a s u r j e c t i v e morphism HOMR(i, l E ) : HOVR(R,E) Proof. 1
_ I _
+
HOMR(I,E)
2. By d e f i n i t i o n E i s i n j e c t i v e i n R-gr i f and o n l y i f t h e
f u n c t o r HomR-gr(-,E)
= HOMR(-, E ) o i s exact. Moreover E i s i n j e c t i v e i n
R-gr i f and o n l y i f E(u) i s i n j e c t i v e f o r every u € G. Since a morphism f : M
+
E o f degree u may be viewed as a morphism o f degree e, M
+
E(o),
t h e f o r e g o i n g remarks prove t h e d e s i r e d equivalence.
2 = 3. Obvious. 3 = 1. The p r o o f i s s i m i l a r t o t h e p r o o f o f Baer’s theorem i n t h e ungraded case. 1.2.5.
C o r o l l a r y . I f E E R-gr i s such t h a t
E is
an i n j e c t i v e R-module
then E i s i n j e c t i v e i n R-gr. I n j e c t i v e o b j e c t s i n R-Gr w i l l be s a i d t o be g r - i n j e c t i v e . 1.2.6.
Remarks
1. I f E i s g r - i n j e c t i v e then E need n o t be i n j e c t i v e . For example, l e t
k be any f i e l d and consider R = k [ X, X-’]
where X i s a v a r i a b l e
commuting w i t h k and w i t h g r a d a t i o n g i v e n by Ri
=
{ a X’,
a
c
k } for
9
1. Graded Rings and Modules
a11 F 2. Check t h a t R
i s i n j e c t i v e i n R - g r b u t n o t i n R-mod
2. I f L t R-gr i s such t h a t ! = i s a f r e e R-module t h e n L need n o t be g r - f r e e . F o r example, t a k e R = Z x Z w i t h t r i v i a l g r a d a t i o n . L e t L be R b u t endowed w i t h t h e f o l l o w i n g g r a d a t i o n : Lo = Z x { O } , and Li = 0 f o r i # 0, 1. O b v i o u s l y
L
The p r o j e c t i v e d i m e n s i o n o f an R-module
If
M
o f C o r o l l a r y 1.2.2. 1.2.7.
101 x Z
w i l l be denoted by p . d i m R ( y . M i n t h e category R-Cr w i l l
We have t h e f o l l o w i n g d i r e c t consequence
:
M c
Corollary. If
1.2.8. Lemna. L e t M
c
R-gr t h e n gr.o.dimR(M)
R-gr,
= p.dimR
(M)
N a graded submodule o f M.If N i s e s s e n t i a l
i n M as a s u b o b j e c t o f P i n R-gr,
Proof. Recall
=
i s not gr-free.
F R-gr t h e n t h e p r o j e c t i v e dimension o f
be denoted by gr.p.dimR(M).
L1
then
I? i s
essential i n
1.
t h a t a s u b o b j e c t o f M i s s a i d t o be an e s s e n t i a l s u b o b j e c t
o r M i s an e s s e n t i 3 1 e x t e n s i o n o f N, i f N i n t e r s e c t s a l l nonzero s u b o b j e c t s o f M non-trivially.
Hence, t h e f a c t t h a t N i s g r - e s s e n t i a l
in
M
means t h a t
f o r each x # 0 i n h(M) t h e r e e x i s t s an a F h(R) such t h a t ax 6 N - t o } . If x
c El i s nonzero t h e n we may decompose x as x
a r e t h e homogeneous components o f x o f degree o i n d u c t i o n on n, e s t a b l i s h i n g t h a t f o r each x t such t h a t a x c
E.
= xo
. The
1
+. . .+xo
n
where t h e xu
p r o o f goes by
t h e r e i s an a t h(R)
For n = 1 t h i s i s obvious from g r - e s s e n t i a l i t y o f N
i n M. P u t y = xu t... t x . S i n c e xo # 0 t h e r e i s an a c h(R) such 1 “n- 1 n
# 0. T h e r e f o r e a x - a x o = a y has a t most n - 1 “n n homogeneous components i n M. Now, i f ay = 0, t h e n a x c N. I f ay # 0 t h e n that,
ax
6 Nand
x
“n
t h e i n d u c t i o n h y p o t h e s i s p r o v i d e s us w i t h a b F h(R) such t h a t b a y c PI and b a y # 0. Thus b a x = b a xo
n
t
b a y t N. B u t b a x = 0 would
10
A. Some General Techniques in the Theory of Graded Rings
e n t a i l b a xu
n
= b a y = 0, c o n t r a d i c t i n g t h e choice o f b, t h e r e f o r e
b a x # O . 1.2.9. M
Remark.
(mi s
c
Let M
R-gr. and l e t N be a small graded submodule of
t h e n o t i o n dual t o e s s e n t i a l ) , t h e n N need n o t be small i n
For example, i n t h e graded r i n g R = k [ X I , where k i s a f i e l d ,
M.
(X) is
small i n t h e graded sense b u t n o t i n R-mod.
f
c
c
Lemna. L e t M, N
1.2.10.
HornR
(tj,fj)
R-gr. The group HOMR(M,N) c o n s i s t s o f a l l
f o r which t h e r e e x i s t a f i n i t e subset o f G, F say, such
that :
(*) f(Mu) c
z
'pcF
Proof.
Noo,
f o r a l l u 6 6.
Suppose t h a t f
i
1. The r i n g R i s s t r o n g l y graded 2. I f 'u
uo c h(U(R)) n Ro t h e n t h e r e i s an isomorphism o f Re-bimodules : l(Re)aR(ua)
Ro,
+
g i v e n by
eu(A)
=
:
Xuu.
Proof. 1. Exactness o f E < R > y i e l d s t h e e x i s t e n c e o f a u n i t uU C G. S i n c e -1 u C Rdl, i t follows t h a t 1 6 R Ro f o r a l l 6 G. U
0-
1
2. Obvious.
Note.
L e t R be a s t r o n g l y Graded r i n o o f t y p e G such t h a t t h e sequence
e < R > is exact, then :
1. F o r e v e r y subgroup H o f G, t h e sequence
E
i s e x a c t .
2. IfH i s a normal subgroup o f G t h e n t h e sequence
E
< R
(GIH) >
si
exact. The f o r e g o i n g r e s u l t s 1.3.23.
may now be combined i n :
Theorem (Crossed P r o d u c t ) L e t R be a graded r i n g o f t y p e G such
t h a t t h e sequence p r o d u c t Re [
E
i s e x a c t and R 2 RJ X,X 1.3.24.
C o r o l l a r y . I f R i s 2 - g r a d e d and i f we suppose t h a t h(U(R))
where q : Ro one
+
Ro i s an automorphism, X i s an i n d e t e r m i n a t e o f degree
and m u l t i p l i c a t i o n i s g i v e n by Xa = m(a)X.
An a r b i t r a r y r i n g S i s s a i d t o have t h e - l e f t I . B . N .
property (Invariant
B a s i s Number) i f f r e e l e f t Re-modules o f d i f f e r e n t r a n k c a n n o t be i s o m o r p h i c e.g.
commutative r i n g s , l o c a l r i n g s , l e + t ' i o e t h e r i a n r i n o s ,
e t c ... have t h e l e f t I.B.N.
P r o p e r t y . We r e f e r t o P. Cohn, [231, f o r
more p r o p e r t i e s o f t h e s e r i n g s .
We use t h e I.B.N.
property i n the
following : 1.3.25.
L e t R be a s t r o n g l y graded r i n q such t h a t Re has t h e l e f t
Lemma.
and such t h a t f i n i t e l y g e n e r a t e d p r o j e c t i v e l e f t Re-modules
IBN property
a r e f r e e . Then
Proof.
F
< R >
i s exact.
in
S i n c e R o i s a l e f t p r o j e c t i v e R -module we f i n d R o e RL') e
Re-mod f o r some s e t I.From R,-1
bo
i n Re-mod.
Re
R u % Re i t f o l l o w s t h e n t h a t ( R O - l ) ( I ) =
E u t R o - l = RLJ) i n Re-mod f o r some s e t J . Hence t h e I B N p r o p e r t y f o r Re e n t a i l s t h a t c a r d ( I x J ) Consequently R o such t h a t Ru
5
:
1.
Re i n Re-mod. By Lemma 1.3.14.2.
l(Re)a
as Re-bimodules.
Let
e
t h e r e e x i s t s a u C Aut(Re)
: l(Re)au
+
R o be a bimodule
U
isomorphism. P u t t i n g uo = e ( 1 ) y i e l d s Ro = Reuo = uURe and f r o m = RR,1-,
= R
Ro-l
Ru =
we may d e r i v e :
Re and u o - l R R u = Re. Hence t h e r e e x i s t X, + e ' e CJ c Re such t h a t 1 = u 1 u o - l , and 1 = u - 1 p uo. I t f o l l o w s t h a t each uu i s UaRe.Reuo-l
=
an i n v e r t i b l e e l e m e n t and t h e r e f o r e 1.3.26.
E iR >
i s e x a c t a t C. as w e l l .
C o r o l l a r y . L e t R be a s t r o n a l y araded r i n g such t h a t Re i s a
semi-local r i n g ,
Rz R~ r &
,aR
I.
Re
A. Some General Techniques in the Theory of Graded Rings
34
Proof. Apply Lemma 1.3.15.
1.3.27.
Corollary.
and 1.3.20.
-
I f R i s a s t r o n g l y graded r i n g such t h a t Re i s a l e f t
p r i n c i p a l i d e a l domain then R
, aR1
Re [ .; < R >
and R i s a l e f t
graded p r i n c i p a l i d e a l r i n g . Moreover i f G = Z then R
CI
-1, c p 1 where
Ro[X,X
X i s a v a r i a b l e of degree one and q i s an automorphism o f Ro.
~
Proof. The f i r s t statement
i s c l e a r from t h e lemma. Recall t h a t a
graded r i n g i s a l e f t graded p r i n c i p a l i d e a l i s p r i n c i p a l . Now i f
I.3.8.,
L
r i n g i f every l e f t i d e a l
i s a graded l e f t i d e a l o f R then by C o r o l l a r y
R o I e = Iu for a l l
0
c
G. But Ie = Rea
,
hence I = Ra.
The l a s t statement i s e v i d e n t from C o r o l l a r y 1.3.24. We now proceed t o study i n some d e t a i l t h e graded i d e a l s o f s t r o n a l y graded r i n g s . I f R i s a s t r o n p l y graded r i n g o f type G , then Mod(R1,R)
c
w i l l denote t h e f a m i l y o f a l l two-sided R1-submodules o f R. I f A and u c G then p u t Ao = R o - l A Ru.
Mod(R1,R)
I t i s c l e a r t h a t A 5 C Mod(R1,R);
we
say t h a t Ao i s the o-conjugate o f A. 1.3.28.
F i x i n n u c G , t h e map
Lemma.
: Mod(R1,R)
cp
+
Pod (R,R),
A
+
A"
i s b i j e c t i v e and i t preserves i n c l u s i o n s , sums, i n t e r s e c t i o n s and products i n R o f elements o f Hod(R1,R). P r o o f . That q, i s b i j e c t i v e i s c l e a r , a c t u a l l y t h e map A i n v e r s i v e . For
U,T
c G we g e t (A')T
n o t hard t o see t h a t A, B C MOd(R1,R) c (AU)"-l =
AO
n
B"
n .
Q"
then A
(B")"-l,
--f
A'
-1
f o r a l l A c Mod(R1,R).
= Am
is its
It i s
preserves sums and i n c l u s i o n s . Now consider
n
B = RoRG-l(A
hence A
n
n
B = Ro(Au
I n a s i m i l a r way one checks t h a t
Q,
B) RuRRa-l c R(,A'
n
B')Ru-l
n B')
and thus (A
preserves products.
Ro-l c
n B)'
=
35
1. Graded Rings and Modules
We say t h a t A
C
Mod (R1,
R) i s G-invariant
i f A'
= A f o r a l l u F G.
It
i s straightforward t o check t h a t any ideal of R i s a G-invariant element
of Mod(R1,R). l e t I be a
1.3.29. Corollary. Let R be strongly graded of type G , and graded ideal of R . Then Ie i s an ideal of R1,
I = RI e = IeR. The correspondence I
+
which i s G-invariant and
I e defines a bijection between the
s e t of graded ideals of R and the s e t of F-invariant ideals
Proof. 1;
c
For any
T
F
Ie
RT c I and thus R,-1
u F G , then :
I e . Pick any
I = I u-1 and I: e e
G, RT-l
Ie
=
u 0-1
(Ie)
c ' :1
of Re.
Ie RT c Ie or c I e , hence
= Ie.
I = RIe
By Corollary I . 3 . 8 . ,
=
IeR. Conversely, l e t J be any G i n v a r i a n t
ideal of Re then RJ = JR i s a graded ideal of R i s e a s i l y seen. The statement about the correspondence I 1.3.30. Lemma.
+
Ie
i s a l s o e a s i l y checked.
Let R be a strongly graded ring o f type G .
! . If M i s a simple l e f t Re-module then f o r every u
C G,
Ru
M is
60
Re
a simple l e f t Re-module. (Ra @ M ) then : I = JO. 2. Putting I = AnnR ( M ) , J = Ann e Re R~ Proof.
Let N c R u
60
M be a l e f t Re-submodule, then since R u - i i s a
Re Projective R -bimodule i t follows t h a t R,-l e
60
BY Proposition 1.3.6. i t follows t h a t R u - i 60 N
Re Or Ro-l
@
c R u - l 60 Ru 60 El.
N
Re c
M thus
Re Re R u - l 60 N = P Re
N = 0. The l a t t e r case obviously e n t a i l s t h a t N = 0 while the
Re f i r s t case y i e l d s N
=
Ra 60 M.
Re
A. Some General Techniques in the Theory of Graded Rings
36
2 . Consider a F Ru,
i c I , b F Ru-l,
then a i b F 1'-
1
and on t h e o t h e r
hand ( a i b) ( R a 60 M) = a 60 i b R u M . Re Now b R n M abi
6
c AnnR
M and r
(M),
1
A n n R ( R u 60 M ) . Thus Iu Re
Using t h e f a c t t h a t M = Ro-l
-1
60
thus i b R M = 0 , consequently c J and I c J u .
(RO
Re
Jc
60
R
M ) we a l s o o b t a i n J
c
I 0-1, hence
c I . Combination of both i n c l u s i o n s y i e l d s I = J'.
1.3.31. Lemma.
Let R be a s t r o n a l y
graded r i n g of type G.
I f P i s a prime i d e a l o f Re then Po i s a prime i d e a l of Re f o r a l l u C G . I f I and J a r e i d e a l s of Re such t h a t I J c P' -1 = 1o-l Ja - l (P')5-1 = P , hence I 0 c P say. Proof.
Then I c P' By
then ( I J ) '
-1
=
follows.
Prim ( - ) we denote t h e s u b s e t of Spec ( - ) c o n s i s t i n g of the p r i m i t i v e
i d e a l s of t h e r i n g under c o n s i d e r a t i o n . If R i s a s t r o n g l y graded r i n g o f type G then the
1.3.32. C o r o l l a r y .
Jacobson r a d i c a l of Re, J ( R e ) , and the n i l r a d i c a l o f Re,
rad(Re), a r e
both G - i n v a r i a n t . Proof.
Since J(Re)
=
n I P , P c Prim Re},
and rad(Re) =
n
{ P , P C Spec(Re)l
we only have t o apply Lemma 1.3.30. and Lemma 1.3.31. The following theorem i s a g e n e r a l i z a t i o n of t h e c l a s s i c a l " C l i f f o r d theorem' cf. [88]
theorem 2.16 pag. 2%.
t h e proof of theorem
1.3.33. Theorem.
The proof we give here c l o s e l y resembles
2.16. in loc. c i t .
Let R be a s t r o n g l y graded r i n g o f type G , where
i s a f i n i t e group. I f
G
i s simple i n R-mod then p2 c o n t a i n s a simple Re-
37
1. Graded Rings and Modules
submodule W such t h a t
e
M
=
z
Ro W. I n p a r t i c u l a r C i s semisimple i n
uFG
Re-mod. W 2 W } = { u F G , Ro W = Id } and l e t MW be the Re W . Then H i s a subcroup sum of a l l Re-submodules X of M such t h a t X
put H =
{U
F G , R O 69
of G and MW i s a simple R(H)-module such t h a t
M then
I”r 2 R
MW.
69
R(H)
proof.
I f x # 0 in
Ro,
G, i s a f i n i t e l y Generated Re-module i t follows t h a t
L c
o F
=
Rx
= ( C
aC G
Ro)x
=
C
oCG
R,x.
Since each is a
f i n i t e l y generated Re-module, hence there i s a maximal proper Re-submodule K of
E.
P u t KO
=
o6G
RoK.
R-submodule of
Clearly R,Ko
M and
sequence in Re-mod :
then KO
0
+
c KO f o r a l l o F
+ F1 implies
M
--f
0 UCG
G. Therefore, KO i s an
KO = 0. Thus we find a n exact
M / RuK.
-
B u t each R u i s an i n v e r t i b l e Re-bimodule, therefore each RUK i s also a maximal proper Re-submodule of
M.
semisimple Re-module and then a l s o
I n other words,
0 U6G
!/Ro
i s a semisimple Re-module. From
t h i s we deduce the existence of a simple Re-submodule of
M
=
RW e n t a i l s t h a t M = C
aCG
K is a
y,
W say. Clearly
Ro W. Again, RG being an i n v e r t i b l e Re-bi-
module, i t follows t h a t each ROW i s a simple Re-module. The canonical multiplication map Ra 69 W
+
ROW i s a n isomorphism because ROW i s
Re simple i n Re-mod a n d Ra i s i n v e r t i b l e . The second statement o f the theorem follows now immediately.
A, Some General Techniques in the Theory of Graded Rings
38
1.4. Graded Division R i n g s A graded r i n g of type 6 i s s a i d t o be a x a d e d d i v i s i o n r i n p i f every
nonzero homogeneous element i s i n v e r t i b l e . C l e a r l y i f R i s a g r a d e d d i v i s i o n r i n a then Re i s a s k e w f i e l d .
Let R be a graded r i n n of type G , then the followinc:
1.4.1. Lemma.
statements a r e e q u i v a l e n t : 1. R
has no graded l e f t i d e a l s d i f f e r e n t from
0,
R.
2 . R has no graded r i g h t i d e a l s d i f f e r e n t from
0,
R.
3. R i s a graded d i v i s i o n r i n g .
Proof.
Straightforward.
1.4.2. Theorem. Let R be a graded d i v i s i o n r i n g . 1. I f G' = { g
C
G , Rg # 0 } then 6 ' i s a subgroup of G and R i s a domain
i f G' i s ordered. 2 . The r i n g R ( G ' ) = 3. R(")
=
Re
[ E
c
Q
R
OF G'
i s a s t r o n g l y GI-graded r i n q .
R > , aR 1 where
i s t h e following e x a c t sequence
of groups :
where R*,
=
Ro - { O } for a l l
u F
G , and
aR
i s t h e group morphism
Proof. 1. T h e f i r s t s t a t e m e n t i s obvious. I t i s a l s o c l e a r t h a t t h e r e a r e no
homogeneous z e r o d i v i s o r s and under t h e assumption of 1. we only have
39
1. Graded Rings and Modules
t o c o n s i d e r t h e s i t u a t i o n where ( 1 t a where 1 < u < a.
n
or bT
= 0
m
=
...
and 1
m : n ai T’ F J ( A 6n k ( T ) ) ; t h u s ai T’ f J ( A qh k ( k k and t h e r e f o r e ai F I . Pick
x c I, t h e n ( 1
+
xT)-’
t h e c o e f f i c i e n t s have t o
=
1 - xT
+
x2T2
-...
i n A 60 k ( ( T ) ) . However k
l y i n a f i n i t e d i m e n s i o n a l k-subspace o f A i . e
x i s a l g e b r a i c o v e r k . I f x were n o t n i l p o t e n t t h e n t h e p o l y n o m i a l e q u a t s a t i s f i e d by x w o u l d g i v e r i s e t o a nonzero i d e m p o t e n t e l e m e n t i n t h e Jacobson r a d i c a l , a c o n t r a d i c t i o n . I n view o f p r o p o s i t i o n 1 . 7 . 1 3 we may produce s t a t e m e n t s s i m i l a r t o I . 7 . F and 1.7.18.,
a two s i d e d
v e r s i o n o f 1 . 7 . 1 9 . and a analogon f o r 1.7.21.;
w i t h r e s p e c t t o t h e s y m p l e c t i c r a d i c a l J s ( R ) o f a graded r i n g . L e t us j u s t m e n t i o n t h e main r e s u l t and r e f e r t o [ 8 4 ]
f o r more d e t a i l .
I. Graded Rings and Modules
61
1.7.22. Proposi t i o n . Let R be a graded r i n g of type 2. T h e n : 1. J s ( R )
=
n tM,
M maxima
i d e a l of R} i s graded.
2. I f A i s any r i n g , X a v a r i a b l e commuting with A then : JS(A [ X I )= I [ X I
, JS(A
[X,
X-l])
= I"X,
X-l]
f o r i d e a l s I , I ' of A.
1.7.23. C o r o l l a r y .
Let R be a graded r i n g of type Z. I f M i s a maximal i d e a l o f R then M
9
i s t h e i n t e r s e c t i o n of t h e maximal i d e a l s c o n t a i n i n g Mg. I n p a r t i c u l a r any gr-maximal i d e a l i s an i n t e r s e c t i o n of maximal i d e a l s . 1.7.24. Example.
Let k be a f i e l d , R
=
k [XI
a l l maximal i d e a l s of R i s zero. Note t h a t M maximal i f M i s maximal; indeed M
=
(X-1)
t h e n the i n t e r s e c t i o n o f
9 i s n o t n e c e s s a r i l y gry i e l d s Mg = 0 c ( X ) .
A. Some General Techniques in the Theory of Graded Rings
62
1.8. Almost s t r o n g l y graded r i n g s 1.8.1. D e f i n i t i o n .
The r i n g R i s c a l l e d an almost
s t r o n g l y graded r i n g
over t h e group G i f t h e r e e x i s t s a decomposition sum ( n o t n e c e s s a r i l y d i r e c t sum) : (1) R =
I:
u6 G
Ro
such t h a t Ro RT
( a s a d d i t i v e groups) i n t o a d d i t i v e subgroups Ro, = Rm
for all
U,T
C
u C G,
G.
I f t h e decomposition sum (1) i s d i r e c t , then R
i s a s t r o n g l y graded
r i n g a5 i n 1.3. In 1 2 4 1 , Dade c a l l s an almost s t r o n g l y graded
r i n g : C l i f f o r d system.
1 . 8 . 2 . Remarks. 1) I f R =
Z Ro i s an almost s t r o n g l y graded r i n g , then R/I i s a l s o
0 6G
an almost s t r o n g l y graded r i n g , f o r a l l two-sided
i d e a l s I of R .
In p a r t i c u l a r i f R = 0 Ro i s a s t r o n g l y graded r i n g , then R/I i s an
d G
almost s t r o n g l y graded r i n g f o r a l l two-sided i d e a l s I of R . ( n o t necessarily 2) If R =
graded i d e a l s ! ) . 2 Ra i s an almost s t r o n g l y graded
d G
r i n g , then f o r a l l u
m
t h e n f o r some i :
ic 0.
z
Because
5
n
Ai)* 3
#
3 ( -
n
2 A ); j#i
i t follows that (
A:
N, a c o n t r a d i c t i o n . Thus r a n k
we have t h a t r a n k RON we have :
0
(M/RoN)
Re
+
M
+
5
8
oCF
Re
(M/N)
m for a l l u
M/RoN,
z
j#i
A . rl Air# J
5 in.
c
G. Because
and so r a n k
Re
M5m.n.
t h e n Re has
C o r o l l a r y . I f R has t h e f i n i t e G o l d i e dimension
f i n i t e G o l d i e dimension. Moreover, i f R = G o l d i e dimension ~
8
oCG
RU i s a s t r o n g l y graded r i n g , t h e n R has
i f and o n l y , i f
Re has
finite
finite
G o l d i e dimension.
Proof.
F o r t h e f i r s t s t a t e m e n t we a p l l y Lemma 1.8.7..
R =
RU i s a s t r o n g l y graded r i n g . S i n c e R o i s a p r o j e c t i v e f i n i t e l y
8
Now, suppose
that
UCG
generated Re-module f o r each dimension as an R -module. e dimension as an Re-module.
u C G,
t h e n Ro has f i n i t e
Because G i s f i n i t e ,
R has
Goldief i n i t e Goldie
Hence R has f i n i t e G o l d i e dimension as an
R-module. 1.8.9.
*
N
=
Lemma.
Let M
c R-mod and N 5 M be an Re-submodule o f
C such t h a t
0. Then :
i ) M i s a l e f t N o e t h e r i a n Re-module i f and o n l y i f M/N i s l e f t N o e t h e r i a n Re-module. i i ) K.dim
I4 = K.dim (P/N) i f e i t h e r s i d e e x i s t s . Re Re
iii) G. dimRe M = G.dimRe(M/N)
i f either side exists.
Note t h a t I4 embeds i n 8 M/RoN. uc G Lemma 1.8.5.
\
The r e s u l t now f o l l o w s f r o m
0.
66
A. Some General Techniques in the Theory of Graded Rings
1.8.10.
L e t M c R-mod. Then Y i s l e f t N o e t h e r i a n i f and o n l y
Theorem.
if
M i s l e f t N o e t h e r i a n . I n D a r t i c u l a r R i s l e f t N o e t h e r i a n i f and Re o n l y i f Re i s l e f t N o e t h e r i a n . Proof.
Suppose t h a t RM i s N o e t h e r i a n . By Lemma 1.8.9.
t o show t h a t M/N i s a l e f t N o e t h e r i a n R -module where N e
it i s sufficient
5P
i s a maximal
Re-submodule w i t h r e s p e c t t o N* = 0. L e t M1/N of M/N.
c M2/N
Then M1
. ..
c
j
be an ascending c h a i n o f non-zero R-submodules
N and hence 0
# MT
c
MI.
Since
M; i s R-submodule o f M
i t f o l l o w s by N o e t h e r i a n i n d u c t i o n t h a t M/N;
a l e f t N o e t h e r i a n Re-module.
Thus t h e c h a i n must t e r m i n a t e and hence M/N
i s a l e f t N o e t h e r i a n Re-
module. The converse i s c l e a r . 1.8.11.
L e t M c R-mod and supoose t h a t K.dim
Lemma.
K.dimRM = K. dim Proof. a
Re
M.
C. L e t K.dimRM = a. By i n d u c t i o n on Re 5 a. I t i s easy t o reduce t h e p r o b l e m t o t h e
C l e a r l y , K.dimR M 5 K.dim
we show t h a t K.dim
Y
Re case where M i s a - c r i t i c a l . L e t r e s p e c t t o N* = 0. I f X
so K.dim K.dim
Re
M e x i s t s . Then Re
Re
M/X*
e t h e n
II. Some General Techniques
f'=X
- y i e l d s t h a t t h e r e exists x c X such t h a t x
with
H) i s a graded R(H)-module with g r a d a t i o n given by ( M ( ' i s H ) ) h
=
Mho,
f o r a l l h F H. 1
11.3.12. Lemma. RP
Let P be a graded R(H)-submodule of M('iyH) then
n ~ ( ~ i =y P .~ ) If y F h(RP
Proof.
xk F h ( P ) , k = l . . . n .
.
n
z n Xk xk with X~ c h ( R ) , k=l = dea x k , k= 1,..., n . Then xk c ( M ( u i y H))T
Mtui'H))then y
Put
T~
=
I f we w r i t e deg y = T i n M ( u i y H )then y F M T u , . I t follows k i 1 from a l l t h i s t h a t : -cui = (deg X k ) . ~ k ~ fi o r a l l k = 1, ..., n . =
MT
-1 On the o t h e r hand deg(X ) = -c-ck f o r a l l k = 1, ...,n . k Hence Xk F R ( H ) f o r a l l k = l , . . . , n and t h e r e f o r e y RP
n M(0i.H)
C
P
i.e.
c P.
11.3.13. C o r o l l a r y . I f M i s a l e f t gr-Noetherian R-module then f o r every i c I , M ( O i Y H ) i s a l e f t gr.-Noetherian R(H)-module.
k
=
ll. Some General Techniques
91
11.3.14. C o r o l l a r z . IfH i s of f i n i t e i n d e x i n G t h e n t h e f o l l o w i n s s t a t e m e n t s are equivalent : 1. M i s a l e f t g r . - N o e t h e r i a n R-module
2. ~
(
proof.
~ i s il e f~t g r~- N o )e t h e r i a n i n R(H)-mod f o r e v e r y i c 1 . 1
-
2 i s j u s t 11.3.13.
2 = 1. S i n c e I i s f i n i t e and s i n c e MZ @.M('iyH) i 61
a l e f t g r . - N o e t h e r i a n R(H)-rnodule b u t t h e n
P
i t follows that M i s
i s c e r t a i n l y l e f t gr.-
N o e t h e r i a n i n R-mod.
11.3.15. C o r o l l a r y . R(H)
I f R i s a l e f t gr-Noetherian r i n g then the r i n g
i s l e f t gr.-Noetherian.
A. Some General Techniques in the Theory of Graded Rings
92
11.4. I n v e r t i b l e I d e a l s . The Rees R i n o and t h e G e n e r a l i z e d Rees Rina. L e t R be any r i n g , I an i d e a l o f R. The Rees r i n o t h e r i n g R(1) = R
+
+ ...
IX
t
In Xn
+ ...
associated t o I i s
c R [ X I which obviously i s
2 a graded s u b r i n g o f R [ X I i s o m o r p h i c t o R 8 I @ I @ . . . @ A r i n g T i s an o v e r r i n g o f R
In Q
...
i f R c T and lT = lR. An i d e a l I o f R i s
an i n v e r t i b l e i d e a l i f t h e r e i s an o v e r r i n g T o f R c o n t a i n i n g an R-bimodule J such t h a t J I = I J =
R.
11.4.1.
R
Example.
Let
be a l e f t N o e t h e r i a n p r i m e r i n g and l e t x E R
be a n o r m a l i z i n g e l e m e n t ( i . e . x R = R x ) . By G o l d i e ' s theorems t h e i d e a l I = Rx
xR i s a n i n v e r t i b l e
idea
:
o f R w i t h inverse x-lR = R x
-1
i n t h e r i n g o f q u o t i e n t s o f R. I f I i s an i n v e r t i b l e i d e a l o f R ( i n t h e o v e r r i n ? T) t h e n we can d e f i n e t h e g e n e r a l i z e d Rees r i n g o f t y p e Z, graded s u b r i n g o f T[ X,X-.'].
t o be # ( I ) = 8 In Xn w h i c h i s a ncZ
Note t h a t , w i t h n o t a t i o n s as b e f o r e , # ( I ) + = R ( 1 ) . 11.4.2.
Proposition.
If R
i s l e f t N o e t h e r i a n and t h e i d e a l I o f R i s
i n v e r t i b l e t h e n # ( I ) and R ( 1 ) a r e l e f t N o e t h e r i a n r i n a s . Proof.
C l e a r l y # ( I ) i s s t r o n g l y graded o f t y p e Z
# ( I ) i s l e f t N o e t h e r i a n and by P r o p o s i t i o n 11.3.4. L e t I be an i d e a l o f R. On an
(*): M =
M0
3
M
1
2
. . _3
Mn
2
M c
...
,
t h u s by Lemma I I . 3 . 7 . ,
R ( I ) i s l e f t Noetherian.
R-mod we c o n s i d e r t h e I - a d i c f i l t r a t i o n ; (More a b o u t f i l t r a t i o n s i s i n t h e
f i n a l appendix i n t h i s book). Note t h a t t h i s f i l t r a t i o n has t h e p r o p e r t y
IMn c Mn+l,
f o r a l l n . The module R(M) = Po @
... Q Mn
Q.. i s called
t h e Rees module a s s o c i a t e d t o P and t h e f i l t r a t i o n ( * ) . I t i s c l e a r t h a t R(M) i s a graded R(1)-module.
The f i l t r a t i o n (*) i s s a i d t o be I - f i t t i n g i f t h e r e i s a
D 6
N
such t h a t
II. Some General Techniques
M I,
93
f o r a l l n 2 p. As an example one may consider t h e f i l t r a t i o n
= Mn+l
...
...
n Mn = I M f o r a l l n 3 0. I f M = M0
J
f i l t r a t i o n then f o r any submodule
N o f M we may d e f i n e an I - f i t t i n p
MI
f i l t r a t i o n on M/N by p u t t i n g (M/N),
3
3Mn
3
i s an I - f i t t i n g
Mn+N/N.
=
Lemna. L e t R be a l e f t Noetherian r i n g and l e t I be an i d e a l o f
11.4.3.
R. Consider a l e f t Noetherian M 6 R-mod and equip i t w i t h the I - a d i c f i l t r a t i o n (*) M = Mo
2
MI
...
3
3
Mn
... .
3
The f o l l o w i n g statements a r e e q u i v a l e n t : 1. The f i l t r a t i o n (*) i s I - f i t t i n g .
2. R(M) i s an R(1)-module o f f i n i t e Proof. 1 = 2. I f I C n = Mn+l
type.
f o r a l l n 3 p f o r a certain p P
c l e a r l y a f i n i t e l y generated’R-module,
whence t h e statement f o l l o w s .
2 = 1. L e t R(M) be generated by elements xl,.
11.4.4.
i=l,
l e f t Noetherian. L e t
M c
Mo = M
2
Hl
. . ,xn
o f degree dl,.
} and check t h a t M+,l
=
. . ydn
IMP.
L e t R be a r i n g and I an i d e a l o f R such t h a t R ( I ) i s
Lemma.
2
...n
0 then
M as an R(1)-module. The l a t t e r i s
R(M) i s generated by Mo 8 V1 8 . . . @
resp. Put p = max { di,
?
Mn
3...3
R-mod be f i l t e r e d by the I - a d i c f i l t r a t i o n (*)
... .
If M i s f i n i t e l y generated and (*) i s I - f i t t i n g then f o r every R-sub-
module
N o f M, t h e induced f i l t r a t i o n
(*)N : N =
-
N nM
0
2
N
n
M~
3
...
3
:
N r l Cn
2
...
i s also I - f i t t i n g .
Proof. By the f o r e g o i n g lemna R(M) i s a f i n i t e l y generated R(1)-module hence a l e f t Noetherian R(1)-module.
Now R(N) i s an R(1)-submodule of
R(M) and t h e r e f o r e R(N) i s f i n i t e l y generated as an R(1)-module. Lemma 11.4.3 implies t h a t
(*)N i s I - f i t t i n g .
94
A. Some General Techniques in the Theory of Graded Rings
11.4.5. C o r o l l a r y .
(Strong Artin-Rees p r o p e r t y )
L e t R be a
ring,
I an i d e a l o f R, such t h a t R(1) i s l e f t Noetherian. L e t M be a f i n i t e l y generated R-module c o n t a i n i n g t h e R-submodules K and L , then t h e r e i s a p
c
JY
I"-P
such t h a t f o r a l l n 2 p :
n L)
(IPK
Proof. Put Mi
=
= I ~ Kn L .
i I M. From Lemma 11.4.3. we d e r i v e t h a t R(M) i s a l e f t ( I n K n L ) i s a f i n i t e l y generated @ n?O By Lemma 1 1 . 4 . 4 . t h e f i l t r a t i o n
Noetherian R(1)-module. Thus R(1)-submodule
R(M).
L, n 3 0 } i s I - f i t t i n g . The r e s u l t f o l l o w s .
{ In K fl
11.4.6.
of
Corollary.
L e t R be a l e f t Noetherian r i n g . I f I i s an
i n v e r t i b l e i d e a l o f R then I has the s t r o n g Artin-Rees p r o p e r t y .
11.4.7. Remark.
L e t R be a l e f t Noetherian r i n g and l e t I be an i d e a l
generated be c e n t r a l elements. We c l a i m t h a t R ( I ) i s a l e f t Noetherian r i n g . Indeed, suppose al,
...,
canonical homomorphism
: R [ X
Obviously
Q
cp
a
n
c Z(R) generate I , then we have a Xn]
+
R(1) d e f i n e d by o(Xi)= aiX.
i s s u r j e c t i v e , thus R ( I ) i s l e f t Noetherian and by C o r o l l a r y
11.4.5. i t f o l l o w s t h a t I has the s t r o n g Artin-Rees p r o p e r t y .
95
II. Some General Techniaues
11.5. Krull Dimension of Graded Rings. The Krull dimension of ordered s e t s has been defined by P. Gabriel and Rentschler, i n [ 3 3 1 , f o r f i n i t e ordinal numbers,
and G.Krause 59 1 . Let us
generalized the notion t o other ordinal numbers c f . recall some d e f i n i t i o n s and elementary f a c t s . Let ( E , {
5)
x t E, a
be an ordered s e t . For a , b E E we write [ a , b ] 5
x c b
} and we
p u t r(E) =
f o r the s e t
( a , b ) , a 5 b }.
{
By t r a n s f i n i t e recurrence we define on T ( E ) the following f i l t r a t i o n : = t ( a , b ) , a=b}
T-l(E)
r o ( E ) = { (a,b) t
supposing T,(E)
rP ( E )
T(E), [ a , b ] i s Artinian }
has been defined f o r a l l p
= { (a,b) t T ( E ) ,
W b 2 bl
n F N such t h a t
?
[ bi+l
ia,
... 5
bn ?
bi] t
r 0( E )
We obtain an ascending chain r-l(E)
c
ro(E)
There e x i s t s an ordinal 5 such t h a t
r
E)
I f there e x i s t s an ordinal u such t h a t have Krull dimension.
r,(E)
then
5
c
... 2
... c
= T(E)E+l
r(E)
=
a
for all i r,(E) =
r(E),
c
there i s an ?
n 1.
... .
... .
then E i s said t o
The smallest ordinal with the property t h a t
= r ( E ) will be c a l l e d the Krull dimension of
E a n d we denote
i t by K dim E . 11.5.1.
Lemma.
Let E , F be ordered s e t s and l e t :
E
+
F be a s t r i c t l y
increasing map. I f F has Krull dimension then E has Krull dimension and K dim E 5 Kdim F. ( c f . 1 43 1 ) .
1 1 . 5 . 2 . Lemma.
E x
I f E, F a r e ordered sets with Krull dimension then
F has Krull dimension and Kdim ExF = sup (Kdim E, Kdim F )
(Note t h a t E x F has the product ordering).
96
If
A. Some General Techniques in the Theory of Graded Rings
A
i s an a r b i t r a r y a b e l i a n category and M i s an o b j e c t o f
A
then we
M inA ordered by i n c l u s i o n .
consider the s e t EM o f a l l subobjects o f
If EM has K r u l l dimension then M i s s a i d t o have K r u l l dimension and
we denote
i t by KdimA M o r simply Kdim M i f no ambiguity can a r i s e . -
I n t h i s case we may reformulate t h e d e f i n i t i o n o f K r u l l dimension as f o l lows : I f M = 0; Kdim = -1; i f
a
i s an o r d i n a l and Kdim M
provided t h e r e i s no i n f i n i t e descending c h a i n : M subobjects Mi o f M i n A such t h a t f o r i = 1, An o b j e c t M o f A having Kdim M =
a
...,
4: a 3
then Kdim M =
Mo
3
M1
3
Kdim (Mi-l/Mi)
...
4:
a
of a.
i s s a i d t o be a - c r i t i c a l i f K d i m ( M / M ' ) i a
f o r every non-zero subobject M ' o f M. For example, M i s o - c r i t i c a l i f and o n l y i f M i s a simple o b j e c t o f Also, i t i s obvious t h a t any nonzero subobject
A.
o f an a - c r i t i c a l o b j e c t
i s again a - c r i t i c a l . 11.5.3.
Lemma.
I f M i s an o b j e c t o f
Kdim M = sup { Kdim(M/N),
A
Kdim N } and
and N i s a subobject o f M then e q u a l i t y holds p r o v i d e d e i t h e r
side exists. c f [43 1 .
Proof.
Using Lemna I I . 5 . 2 . ,
11.5.4.
Lemma.
Proof.
See P r o p o s i t i o n 1.3. [ 4 3 I .
11.5.5.
Lemna.
Every Noetherian o b j e c t o f
Suppose t h a t
A
d i r e c t sums. I f an o b j e c t M o f
A
has
K r u l l dimension.
i s an a b e l i a n category a l l o w i n g i n f i n i t e
A
has K r u l l dimension then M cannot
c o n t a i n an i n f i n i t e d i r e c t sum o f subobjects. I n p a r t i c u l a r M has f i n i t e Goldie dimension. 11.5.6.
Lemna. Suppose t h a t A i s an a b e l i a n category
d i r e c t sums and suppose the o b j e c t M o f
A
allowing i n f i n i t e
has K r u l l dimension.
97
II. Some General Techniques
p u t u = sup I 1 a
t.
subobject of M}. Then
N a n essential
KdimA (M/N); -
KdimM.
proof.
For I I . 5 . 5 . , 11.5.6. we r ef er t o
[
43
Let us now come back down t o earth and p u t
A
_ L
]
= R-mod.
The Krull dimension
of M i n R-mod i s c al l ed the l e f t Krull dimension of M and we denote i t The Krull dimension of R considered as a l e f t R-module i s by Kdim M. R called the l e f t dimension of R and we denote i t by Kdim R . If R i s a graded r i n g , putting
A
for t h a t p a r t i c u l ar M
C
R-gr). The purpose for M
Kdim ( M ) and Kdim (M) R R-gr 11.5.7. Lemma.
R-gr we define Kdim M
=
C
R-qr
07
C
t h i s paragraph i s t o compare
R-gr.
Let R be a graded ring of type
(or r i g h t ) limited M
as before ( i f i t e x i s t s
z and
consider a l e f t
R-gr, t h e n :
1. M has Krull dimension i f and only i f !v! has Krull dimension and in
t h i s case we have : Kdim M = Kdim M . R-gr R 7 . If M i s a-crit'ical then i s a-critical. Proof. 1. Follows from Corollary 11.2.4. 2. If
X#
Kdim R
(F/X)5
0 i s any submodule of M then Kdim ( M / f " ) < a . R-gr BY proposition 11.2.3. and Corollary 11.2.4. and Lemma 11.5.1. we have
11.5.8.
Kdim ( M / f " ) . R-gr
Consequently
,
is a-critical.
Proposition.
Let R be a p r o s i t i v el y graded ring of type Z . For M F gr the following Properties hold : 1. M has Krull dimension i f and only i f M has Krull dimension and i f t h i s
i s the case then Kdim M = Kdim M. R-gr R
A. Some General Techniques in the Theory of Graded Rings
98
2. I f M i s a - c r i t i c a l then Proof.
i s a-critical.
For u E S, the p o s i t i v e elements o f G, we d e f i n e uM,
=
8 .M, T>U
i s a graded submodule o f M and M,/,
I f i s c l e a r t h a t, ,M
is right
l i m i t e d . From t h e f o r e q o i n q lemma we r e t a i n : Kdim M R-qr u' and a l s o Kdim (M/M>u) R-gr
= Kdim
R
= Kdim M (,),
R -
(M/M > u). --
A p p l i c a t i o n o f Lemma 11.5.3. y i e l d s the statement. 2. Consider a nonzero submodule
Combining t h e f a c t t h a t
M
X o f E . For
# 0.
some u E G,X fl, ,M
i s a - c r i t i c a l w i t h t h e r e s u l t of t h e foregoing
fl Mu) < a. Now t h e r e e x i s t s a s t r i c t l y lemma we deduce t h a t Kdim X M (/,, R-gr i n c r e a s i n g map from t h e l a t t i c e o f submodules o f i n t o t h e set-product
y/X
o f the l a t t i c e s o f submodules o f JM , X fl, ,M -~ By Lemma I I . 5 . 2 . ,
and because Kdim (M/M,u R --
Kdim (M/Z) < a; hence
R -
11.5.9.
Lemma.
and o f MM / u,
-
, g i v e n by
) < a, i t f o l l o w s t h a t
i s a-critical.
L e t R be a l e f t Noetherian graded r i n g o f type Z , and
l e t M 6 R-gr be a u n i f o r m o b j e c t . Suppose t h a t e i t h e r M i s l i m i t e d ( l e f t o r r i g h t ) o r t h a t P i s p o s i t i v e l y graded. Then
i s uniform i n
R-mod. Proof.
Theorem 2. 7. o f [ 431 s t a t e s t h a t an o b j e c t from any a b e l i a n
category, having K r u l l dimension a, c o n t a i n s a nonzero subobject which i s a-critical.
Since R i s l e f t Noetherian, we may apply Lemma 11.5.4.
and t h e f o r e g o i n g remark t o deduce t h a t M c o n t a i n s a graded Rsubmodule
N which i s a - c r i t i c a l . U n i f o r m i t y o f M e n t a i l s t h a t M i s an e s s e n t i a l extension o f N i n F g r . then
i s an e s s e n t i a l extension o f
Lemma 11.5. 7. and P r o p o s i t i o n
11.5.8.
yield that
(see 1.2.8.)
i s anacritical
99
II. Some General Techniques
i s u n i f o r m too.
R-module, hence i t i s u n i f o r m and t h e r e f o r e 11.5.10.
Theorem. L e t R be
R-gr then
uniform o b j e c t o f
proof.
M -x c -
M, -u
For u E G l e t
such t h a t f o r some
n M,IJ)
Indeed i f x E h(Xwith x = y
and
a graded r i n g o f type Z . I f M E P-gr i s a
u1
Q;(R)
M
where ag(m)
=
6O
R
M
-+
pg ( z ’ q i
160 m and
e n t a i l s t h a t : K(Q:(R)
Q,(M) i
M)
60
R
i s c l e a r because Ker a c
=
“h
mi)
=
C ’ 9.m.. The assumptiom 3. 1 1
0 hence Ker aq
5(M);t h u s Ker
oq =
2
K ( M ) . That K ( M )
K ( M ) . Then
morphism of degree e Q:(M)/Img9 that
4.
pg i s
Ker a‘
pq i s a mono-
morphism. Because of P r o p o s i t i o n 11.10.9. pg i s Q!(R)-linear,
hence
Note t h a t p g i s a graded
submodule o f Q:(M).
Im pg i s a graded Q:(R)-
3
c G ! Since Im pg c o n t a i n s M/K ( M ) i t follows t h a t
i s 1:-torsion and a graded Q:(R)-module.
T h u s by 3. we o b t a i n
s u r j e c t i v e , hence an isomorphism i n R-gr.
1. For every L c
we have
“ K )
Q:(L)
=
Q:(R)
because R / L i s
K - t o r s i o n . From 4. i t follows then t h a t :
Q:W
=
=
P~(Q:(R)
11.10.11. D e f i n i t i o n . property T i f Q:
P
R
L ) = Q ~ ( R )j K ( L ) .
A r i q i d kernel f u n c t o r
Let
a s s o c i a t e d kernel f u n c t o r
K K-
be a r i g i d kernel f u n c t o r on R-qr with on R-mod, then t h e following a s s e r t i o n s
are equivalent : K
on R-or i s s a i d t o have
i s e x a c t and commutes w i t h d i r e c t sums.
11.10.12. C o r o l l a r y .
1.
K
has p r o p e r t y T.
2 . 5 has p r o p e r t y T.
I f t h i s i s the case then Q:(M) -
-
=
QK(M)
14s
II. Some General Techniques
~
Proof. 1. = 2 . From the foregoinq theorem we r e t a i n t h a t , f o r a l l L c
&)(K),
L contains a graded l e f t ideal of f i n i t e type L ' which i s s t i l l i n L ( K )
we have : Q:(M)
By Proposition 11.10.2. I f now H Thus Q:(R)
c
.C(K_)
then H
9
j K ( H g ) = Q:(R)
_
c
=
_
a,-(!)
L ( K ) by d e f i n i t i o n of
f o r a l l M < R-gr. K -.
and from the above i t i s then obvious t h a t
Q K ( R ) j K ( H ) = Q K ( R ) follows i . e . K has property T . -
-
-
2. = 1. I f K- has property T then i n p a r t i c u l a r 5 has f i n i t e type and
then Lemma 11.10.1. y i e l d s t h a t Khas f i n i t e type on R-qr. Proposition 11.10.2. then implies t h a t Q:(R) ~t h i s t h a t K has property T.
=
Q K ( R ) and one e a s i l y
deduces from
Some p a r t i c u l a r r e s u l t s concerning l o c a l i z a t i o n of qraded rinqs of type
Z will be given i n Section C . I . l . , a f t e r we have deduced the araded version of Goldie's theorem(s).
150
A. Some General Techniques in the Theory of Graded Rings
11.11. I n j e c t i v e Modules, Prime Kernel Functors and L o c a l i z a t i o n a t Prime I d e a l s , In t h i s s e c t i o n we use the n o t a t i o n s of S e c t i o n 11.9. and i n p a r t i c u l a r Theorem 11.9.7. i s very u s e f u l . For any M F R-qr, E q ( M ) denotes the i n j e c t i v e h u l l of M i n R-or; i t i s c l e a r t h a t Eg(M(o)) = Eq(M)(,) all u
F
G and consequently : Ass M
=
for
Ass ( E g ( M ) ( o ) ) f o r o C G (terminolosy
of 11.7). The followin9 Lemma i s a s l i q h t s t r e n g h t e n i n g of a s t a t e m e n t of S e c t i o n I I . g . ( o n e of t h e remarks 1 1 . 9 . 7 . ) . Let R be a qraded r i n g of type G .
11.11.1. Lemma.
c R-gr i s s h i f t i n v a r i a n t i . e .
If M
r i g i d . Every r i g i d kernel f u n c t o r
0
6 G
on R-gr i s of the form
then K~
is
K~
f o r some
c R-gr.
s h i f t invariant M Proof.
K
M Z M(o) f o r a l l
The f i r s t s t a t e m e n t i s e a s i l y checked. Conversely, suppose
i s a r i g i d kernel f u n c t o r on R-gr. That
K
=
f o r some N
K~
c l e a r by general t o r s i o n t h e o r y . Now f o r any
u
F E we
F
K
R-gr i s
o b t a i n , f o r every
L F R-gr : = fl
KN(,)(L)
=
n
i Ker f , f {
F
Horn
R-qr
=
K
N ( a ) =K~
@N(o)
}
(Ker f ) (u), f - c Hom ( L ( o - l ) , E q ( N ) ) R-qr
= (KN(L(u-l)))
Hence
( L , Eg(N(o)))
=
}
KN(L).
f o r a l l o F 6 . This implies t h a t
K~
=
A i K N ( r r ) 30 6
=
and we may t a k e M = 0 N( 0). 0
U
Recall from 11.9. ( s e e Theorem 1 1 . 9 . 7 . ) t h a t t o M 6 R-or t h e r e corresponds a r i g i d kernel f u n c t o r such t h a t E ( K ~ 1)1 L ( R ) (1 I t follows t h a t t h a t K ~ ( M ) = 0.
K ;
I(;
defined by
= E(K$
K:
= A { K ~ ( o ) ,o F
6 1 , which i s
and a l s o , f o r X F R-qr : K ~ ( X ) = K,(?). -
i s the larqest r i q i d
kernel f u n c t o r on R-qr such
151
II. Some General Techniques
An M c R - g r i s s a i d t o be a araded s u p p o r t i n g module functor
K
f o r the kernel
on R-gr i f K ( M ) = 0 b u t ti(M/N) = M/N f o r e v e r y araded submodule
N o f M ( d e l e t i n g "graded" everywhere y i e l d s t h e d e f i n i t i o n o f a s u p p o r t i n q
I).
module f o r a k e r n e l f u n c t o r 5 on R-mod, u t i l i z e d i n [40 functor
K
on R-gr i s s a i d t o be qraded p r i m e
a graded s u p p o r t i n g module. K~
-
i s a graded p r i m e
A kernel
i f i t i s coqenerated by
I t i s c l e a r t h a t i f M F R-qr i s such t h a t
k e r n e l f u n c t o r on R-mod w i t h s u p p o r t i n q module M
t h e n z M i s a qraded p r i m e k e r n e l f u n c t o r on R-or w i t h s u p p o r t i n a module El. i . e . M F R-?r does n o t make
Warning : remember Remark 11.9.10. By d e f i n i t i o n o f c o g e n e r a t i o n i n
(see 11.9.)
R-clr
tiM
-
araded!
it i s clear that
tiM
depends on Eg(M), n o t on M. T h e r e f o r e l e t us i n v e s t i q a t e Eg(M) j u s t a be an i n j e c t i v e h u l l o f
l i t t l e more. L e t E(4)
M
i n R-mod. Suppose M C R-ar
i s such t h a t t h e s i n g u l a r r a d i c a l Z(M) equals z e r o . , LT x c MTo P u t E d = { x C E(M)
l e f t ideal L
for
T
c
G, f o r some nraded e s s e n t i a l
.
of R }
E' = 0 E l c i s i n R-or and i t c o n t a i n s M a s a qraded e s s e n t i a l
Obviously
0
R-submodule. 11.11.2.
Lemma.
M c
R-qr be such t h a t Z(M) = 0, t h e n E ' i s t h e l a r q e s t
R-submodule o f E(M) - w i t h t h e p r o p e r t y t h a t E l i s graded and c o n t a i n i n a
M as a graded submodule. Proof.
Suppose we have M c N i n R-gr such t h a t
x E No t h e n
( f j:
Since ( M : X ) ~ x c 11.11.3.
M c N
_. -.
c
E(M) - . If
x ) i s a graded l e f t i d e a l o f R w h i c h i s a l s o e s s e n t i a l .
PI
n
Proposition.
NTO = MTo,
x
c E' follows.
I f M F R-gr i s such t h a t Z(M) = 0 t h e n Eq(M)
E'
i n R-gr. Proof.
I n an a r b i t r a r y
Grothendieck category, i n j e c t i v e h u l l s o f o b j e c t s
may be c h a r a c t e r i z e d as b e i n g maximal e s s e n t i a l e x t e n s i o n s o f t h o s e o b j e c t s .
152
A. Some General Techniques in the Theory of Graded Rings
Eg(M) i s a g r - e s s e n t i a l o f M,then by Lemma I . Z . B . ,
However, i n R-gr,
&is
an e s s e n t i a l e x t e n s i o n o f M i n R-mod i . e . we may assume
c E(M).
The f o r e g o i n g lenuna then y i e l d s Eg(M) c E ' and t h e m a x i m a l i t y o f E a ( M ) w i t h r e s p e c t t o being p r - e s s e n t i a l then y i e l d s E ' = Eq(M). 11.11.4.
A kernel functor
Remark.
i n R-gr i s coqenerated by M i f i t
K
i s cogenerated by Eg(M), i f and o n l y i f K i s coaenerated by E(M). We say t h a t a r i g i d kernel f u n c t o r functor i f
K
= A {
f u n c t o r on R-gr, i . e . module f o r
K
~
u F G }
K,(u), K
=
K;
K
on R-gr
, where
i s a rigid-prime kernel i s a graded prime k e r n e l
K~
f o r some M F R-qr which i s a araded s u p p o r t i n q
.
denote by ( E ) ~
L e t K be an a r b i t r a r y k e r n e l f u n c t o r on R-mod, then we the r i g i d k e r n e l f u n c t o r on R - o r a i v e n
by t h e araded f i l t e r
X ( (-K ) ~ )=
5 K b u t e q u a l i t y o n l y holds i f K_ i s I ( K ) rl L (R). Note t h a t ( K ) 9 9 a graded k e r n e l f u n c t o r on R-mod. By Theorem 11.9.7. i t f o l l o w s t h a t f o r =
r ( K ~ = ) K ~~ For . X F R-gr we i n t r o d u c e the f o l l o w i n q n o t a t i o n s :
M E R-gr,
Since
need n o t be a qraded kernel f u n c t o r on R-mod t h e f o l l o w i n a
K~
-
m o d i f i c a t i o n o f t h e techniques o f P r o p o s i t i o n 11.10.4. By g y,
a,(?)
we now mean t h e
-
=
UF
I x
F Q,
-
(J),
Lz x c ( X h M ( X ) & 11.11.5.
qraded module 0
Theorem.
c-
Yo,
i s necessary
where
t h e r e i s an L 6 I(K;) such t h a t for all
T
F G
1.
L e t R be a graded r i n g o f type G. Then w i t h n o t a t i o n s
as b e f o r e : 1. g QM(R) i s a graded r i n g c o n t a i n i n q R / K ~ ( R )as a qraded s u b r i n g -
.
II. Some General Techniques
2 . For every X c R-or, g
Q,(Z) -
153
i s a gradedgQ, (R)-module. Moreover : -
f o r a l l X t R-gr :
Qi(x)9 QM(&). -
___
=
Proof. 1. As i n Proposition 11.10.4. 2 . Consider the following diagram i n R-mod : 0
If x c
=X -Qi(x)
x/rM(x)
+
Qi(X)o then,
U Q M (- Z )
__
because coker
ax
i s Kp,-torsion, we have -
x F q Q,(X)".
O n the other hand, the construction of 9QM(X) i s such t h a t coker p x i s Kh-torsion i s an e s s e n t i a l i n R-qr. Proposition 11.10.7. and and the f a c t t h a t Q;(X) extension of X / K , ~ ( X ) i n R-mod, e n t a i l t h a t gQ,(X) c hence LT x
S Q (X) -M -
c ( X / K ~ ( X ) ) f~o~r every T
=
c
G, i . e .
Qi(X),
Qi(x).
__
Now consider a graded prime ideal P of a l e f t Noetherian graded rina R of type G. P u t M = R / P and write C.I.3.
5
i t will be shown t h a t E g ( R / P )
I n the Z-graded
= K
~
/
-
~K ; ,
=
K:/~.
I n Section
i s a P-cotertiary waded R-module.
case we will be able t o deduce a more complete r e s u l t ,
see Section C . I . l . ,
here i n the oeneral G-graded case we have :
11.11.6. Proposition.
Let R be a l e f t Noetherian ring graded of type G and l e t P F Spec ( R ) , then K~ and
1.
q
K ;
may be described by the followino f i l t e r s :
= { J l e f t ideal of = { J l e f t ideal
of R / P 1
R , ( J : a ) n G(P) # 0 f o r a l l a F R
}
of R , J maps t o an e s s e n t i a l l e f t ideal
154
2.
A. Some General Techniques in the Theory of Graded Rings
~(KF)
=
I J graded l e f t i d e a l o f
R such t h a t J maps t o an e s s e n t i a l
graded l e f t i d e a l o f R/P } Proof.
By Theorem 11.9.7.
and t h e f a c t t h a t
(I
k . N e x t c o n s i d e r t h e case where q(X),, t (R[ s ] [ 1 R [ s ] be such t h a t a h(X) = q(X) q(X) + r ( X ) . Then a h f t ) = 0 and
(R [ s ] [ X ] ) Let a
c
t h e n deqXr(X) < degXg(X) e n t a i l s r ( X ) = 0
g(t) = 0 y i e l d r ( t ) = 0 but
and ah(X) = q ( X ) q ( X ) . S i n c e h ( X ) i s homoqeneous t h i s y i e l d s ab,h(X) = t h u s a g a i n q r 4 ( t ) = 0 o r q,(t)
= gM(X) q,(X)
Then degX q,(X) Therefore a',qM(X)
2 deqX OM( X)
t h e r e is a n
a'
F R [ s ] such t h a t a'qbz(X)
degX qh(X) 2 degX q,(X),
1. Therefore t h e r e
4
PR-t R P t
c
P n . B u t then R-t P t c Po c Pn y i e l d s t Pn, hence R u t c P . However ( R - l ) c R-t then i m p l i e s R-l c P
but R R - l
=
R c o n t r a d i c t s P # R . Consequently i f n
e x i s t s a t c Z such t h a t P t
>
1, then Po
4
Pn.
1 1 . 2 . 6 . P r o p o s i t i o n . Let R be a s t r o n q l y qraded qr-Dedekind rino, then :
1. For each graded prime i d e a l P of R , P = R P o . 2. For every graded i d e a l of R i s generated by i t s p a r t o f deqree z e r o . 3 . I f Ro i s a p r i n c i p a l i d e a l r i n q then R i s a o r - p r i n c i p a l i d e a l r i n o
and R 2 Ro [ X , X - ' ] Proof.
where X i s an i n d e t e r m i n a t e .
1. and 2 . hold f o r any s t r o n g l y w a d e d rinq, c f . 1 . 1 . 3 .
3.
Immediate from 2. and the r e s u l t s of S e c t i o n A . I . 3 . 11.2.7. Theorem.
Let R be a s t r o n g l y qraded ar-Dedekind r i n g then there
e x i s t s a f r a c t i o n a l i d e a l I of Ro such t h a t R = H 0 ( I ) ( f o r t h e d e f i n i t i o n of t h e g e n e r a l i z e d Rees rinq, s e e A . I I . 4 . ) There i s a canonical group epimorphism
~i
: C1(Ro)
+
C l ( R ) , t h e kernel of
TI
i s e x a c t l y t h e suboroup
of C1(Ro) generated by t h e c l a s s of I . Poreover , TI i s an isomorphism
11. Arithmetically Graded Rings
183
i f and o n l y i f I i s a p r i n c i p a l i d e a l and i n t h i s case R
Also, every
fl0
.
Ro [ X , X - l ]
( I ) f o r some f r a c t i o n a l i d e a l I o f Ro i s a qr-Dedekind r i n q .
P r o o f . By A . I I . 4 . ,
Proposition A.II.4.2.,
the rinas x0(I) are a l l
N o e t h e r i a n r i n g s , so t o show t h a t N 0 ( I ) i s qr-Dedekind i t w i l l s u f f i c e to
establish t h a t i t i s i n t e a r a l l y closed i n i t s f i e l d o f fractions, w h i l s t
graded p r i m e i d e a l s a r e qr-maximal. Let
5
be t h e i n t e g r a l c l o s u r e o f S = f l 0 ( I ) i n i t s f i e l d o f f r a c t i o n s
c (5 -
Suppose y
S)n f o r some n C Z .
Then S-n y c
5
b u t S-n y
because SS-, = S f o r a l l n F Z. T h e r e f o r e t h e r e i s a z F
z s a t i s f i e s T~ Tn t (an-l)o
t
Tn-’
Dedekind r i n q , z
Tn- 1
an-l
+...+(a
c So
)
+...t
0 0
=
a.
w i t h ai
0 w i t h (ai)o
of
S,
- S)o. I f
F S then i t also s a t i s f i e s F
S o . S i n c e So = Ro i s a
follows, a contradiction.
us now show t h a t f o r any p r i m e i d e a l
(5
4
Q(S)
Therefore
5
= S. L e t
Po o f Ro, SPo i s a ar-maximal i d e a l
s.
Indeed S/SPo i s a s t r o n g l y graded r i n q o f t y p e Z w i t h (S/SPo) = Ro/Po 0
b e i n g a f i e l d . Thus S/SPo i s a q r - f i e l d and hence SPo i s ar-maximal i n S .
So f a r we have e s t a b l i s h e d t h a t e v e r y q e n e r a l i z e d Rees r i n q N 0 ( I ) i s a gr-Dedekind r i n g . C o n v e r s e l y i f R i s a qr-Dedekind r i n p w h i c h i s s t r o n a l y graded t h e n i t i s c l e a r t h a t , i n o r d e r t o i n v e r t a l l elements o f h(R) i t suffices
-1 f xdl
Kg
t o i n v e r t t h e elements o f Ro-tO}. Take x - ~F R-l-{O}.
y i e l d s t h a t K’I = KO [ x - ~ ,
XI:] where
Then
KO i s t h e f i e l d o f f r a c t i o n s
of Ro. L e t J be t h e l a r q e s t f r a c t i o n a l i d e a l o f Ro such t h a t J x - ~c R and I t h e l a r a e s t
J X - =~ R-l,
( f r a c t i o n a l ) i d e a l o f Ro such t h a t I
-1 I x - ~= Rl.
From R-l
XI: c R.
Then :
R1 = Ro i t f o l l o w s now t h a t IJ = J I = Ro
i . e . J = I-l.S i m i l a r l y , i f I2 i s t h e i d e a l o f Ro n i v e n by (R2 : xl)2
2
t h e n 121-1 c I f o l l o w s f r o m R-l R2 c R1, hence I2 I . On t h e o t h e r 2 2 2 hand, I x1 = (Ixl) c R2 y i e l d s I c 12. R e p e a t i n o t h i s arqument,
184
B. Commutative Graded Rings
we o b t a i n R 2 N0(I). Finally
,let
TI
: C1(Ro)
+
C l ( R ) be d e f i n e d as t h e c l a s s map o b t a i n e d
f r o m mapping an i d e a l o f Ro t o i t s e x t e n s i o n i n R. T h a t
If H i s
TI
i s surjective
Certainly the class o f I ,
f o l l o w s f r o m P r o p o s i t i o n 11.2.6.. contained i n Ker
TI
s i n c e No(I).X-'
=
I
say, i s
No(1).I.
an i d e a l o f Ro such t h a t RH i s p r i n c i p a l t h e n w r i t e RH = Rh f o r
some h c h(RH), o f degree t say. T a k i n q p a r t s o f degree n y i e l d s (RH) = n n
= HI X
= Rn-th'
where h ' = hX
Consequently : H I n = I"'h
t
(we i d e n t i f i e d R and No(I) h e r e ) .
and I? i s i n t h e subrlroup o f C1(Ro) g e n e r a t e d
by t h e c l a s s o f I . F i n a l l y i f I = R i i s p r i n c i p a l t h e n R 0
and C l ( R ) '2' C I R o ) .
Conversely i f
Ro [ i X , ( i X ) - ' ]
i s a l s o i n j e c t i v e t h e n R I = RX-'
TI
entails
that I i s principal. 11.2.8.
Proposition.
If R =
0
( I ) 2 N0(J) t h e n I and J
rewesent the
same e l e m e n t o f C1(Ro). Conversely, i f I and J a r e i n t h e same c l a s s t h e n
__ P r o o f . W i t h o u t l o s s o f g e n e r a l i t y we may assume t h a t I and J a r e Ro- i n t e g r a l i d e a l s . From R I = RX-' #,(J)
=
2
n
O(i.e.
up t o i n t e r c h a n g i n q X and X-',
e
Y
such t h a t X = XY P e q u a l s t h e r a n k [ (S/P)o[Y,Y-l] :(S/P)o[ X,X- 11 1 .
and Y - l i f n e c e s s a r y ) . So we f i n d a n a t u r a l number e with X
c
(S/P)o. Clearly t h i s
The norm o f P
0
e
P i n S o i s t h e number o f elements i n t h e r e s i d u e f i e l d
so/po = (S/P)o w h i c h i s a f i n i t e e x t e n s i o n o f t h e p r i m e f i e l d Ro/Po. i s c l e a r t h a t a d d i t i o n o f X,X-'
It
made S/P i n t o an i n f i n i t e r i n q , however
i t makes sense t o d e f i n e t h e graded norm o f P as f o l l o w s N(P) = No(Po)ep,N(P)n= = N(P)n and i f I = P1"1 ...Pr'r
t h e n we p u t N ( I ) = nr N(Pi)
'.
V.
i=l 11.2.22.
The graded
f u n c t i o n o f t h e (Fraded) i n t e q r a l c l o s u r e o f a. graded
Dedekind r i n g i n Q ( t ) . Definition.
t: ( s ) 9
=
z I
N(I)-',
( s a complex v a r i a b l e ) where t h e
sunwnation
190
B. Commutative Graded Rings
i s o v e r graded i n t e g r a l i d e a l s o f S. The d e f i n i t i o n o f t h e norm i m p l i e s t h a t
c0(s) i s
r: 9 ( s )
2.
Jg(S)
i s dominated by ~ ~ ( s where ) ,
t h e z e t a f u n c t i o n o f t h e p a r t o f deqree 0.
11.2.23. Proposition. 1.
r: 9 ( s )
F o r Re(s) > 1 we have :
converqes a b s o l u t e l y and a l m o s t u n i f o r m l y =
n
PcSpec ( S ) 9
(l-N(P)
-1 -1
1
Proof. 1.,2.
S l i g h t m o d i f i c a t i o n o f t h e p r o o f i n t h e un9raded case, c f . f o r
example [ 99 1 . Note t h a t t h e p r o d u c t a p p e a r i n g i n 3 i s f i n i t e . 3. Obvious f r o m 2. L e t us end t h i s s e c t on w i t h some r e s u l t s on t h e c l a s s group.
11.2.24. Proposition
I f R i s a gr-Dedekind r i n g t h e n t h e r e e x i s t s e F
N
such t h a t t h e f o l l o w i n g diagram o f group homomorphisms i s commutative.
c1 (R) Where
7
i s t h e c l a s s o f I i n C1(Ro) and I i s t h e f r a c t i o n a l i d e a l o f Ro
describing the structure o f R(e). Proof.
Take e and I as i n Theorem 11.2.12.
I d e n t i f y R(e) w i t h a s u b r i n a
o f R by f o r g e t t i n g t h e g r a d a t i o n . E x t e n s i o n o f i d e a l s f r o m Ro t o R ( e ) , "from"
t o R and f r o m Ro t o R d e f i n e s t h e arrows i n t h e diagram. One
e a s i l y checks c o m m u t a t i v i t y o f t h e diagram.
191
II. Arithmetically Graded Rings
I f RR1 # R t h e n we r e f e r t o t h e i d e a l RIR-l
o f Ro as b e i n g t h e
o f t h e qr-Dedekind r i n q R and i t w i l l be denoted by S ( R ) .
discriminator
1 1 . 2 . 1 5 . Lemma.
L e t R be a gr-Dedekind r i n g and
K
be an
k e r n e l f u n c t o r on t h e c a t e g o r y o f graded R-modules. Then,
dempotent f S denotes
Q j ( R ) , S i s a gr-Dedekind r i n g w i t h
{P,
Cl(S) = Cl(R)/ c
P 6 L(K)} >
where L ( K ) i s t h e f i l t e r o f
K ,
and P i s graded p r i m e .
S i n c e R i s a gr-Dedekind r i n g ,
Proof.
l o c a l i z a t i o n Q:
has p r o p e r t y T(A 11.9) and t h e
K
i s p e r f e c t . F o r e v e r y i d e a l L o f 5 , we t h e n have L = S.(LnR)
and t h e r e f o r e we o b t a i n an epirnorphism, Y = Cl(R)
+
Cl(5).
I f J i s an i d e a l o f R such t h a t S J = Sa f o r some homogeneous a c S, t h e n t h e r e e x i s t s an i d e a l L 6 L ( K ) such t h a t La c J . The g r a d e d Dedekind p r o p e r t y y i e l d s La = J.H where H i s such t h a t SH = S i . e . H F
L(ic).
H may
"1 " be w r i t t e n as H = P ... Pnn and each Pi i = 1,..., n belongs t o L ( K ) . So J = H-'
La and
-
J =H-'i.
prime belonqing t o
L(h)}>,
Now
thus
i 5
and
R
belong t o c {
belongs t o
P, P
i s a oraded
i t t o o , hence Ker y e q u a l s
t h i s subgroup. 11.2.26.
L e t R be a gr-Dedekind r i n n and S a m u l t i p l i c a t i v e
Corollary.
c l o s e d s u b s e t o f h*(R). and P
n
S #
Then : C l ( Q ? ( R ) ) = C l ( R ) / < I P ,
P praded p r i m e
@I>.
11.2.27.
C o r o l l a r y . L e t R be a gr-Dedekind r i n g and I a praded i d e a l o f R.
Then :
Cl(Q:(R))
= Cl(R)/
where P1 ,..., Ps a r e t h e graded
p r i m e i d e a l s c o n t a i n i n g I. P r o o f . I n t h i s case L ( K ~ c) o n t a i n s o n l y a f i n i t e number o f graded rsrime i d e a l s c o n t a i n i n g I.
192
B. Commutative Graded Rings
11.2.28. C o r o l l a r y . W e have a commutative )d Cl(R)
where P1,.
.. ,Ps
>-
diagram.
C1
=
C l ( ? ) / < p l,...,ps>
a r e the graded prime i d e a l s c o n t a i n i n ? R,;
t h e prime i d e a l s of Ro containinq F ( R ) i . e . p i = Pi
n
p l , . . . ,ps a r e
Ro, and
6(R)
is
( R ) definin? the generalized t h e f r a c t i o n a l i d e a l of Qa 6(R) 0 Rees r i n g s t r u c t u r e of Q:R (R). 1
t h e c l a s s of
Proof.
Apply c o r o l l a r y 11.2.27. with I
=
RR1 and note t h a t Qg
aR)
a s graded r i n g s because of Lemma II.2:ll.
(R)
Q
RR1
(R)
11.2.29. P r o p o s i t i o n . Let R be a qr-Dedekind r i n g such t h a t there i s an N
C
N such t h a t t h e g e n e r a l i z e d Rees r i n g
C1(R(N))
+
R ( N ) has the property t h a t
Cl(R) i s i n j e c t i v e . Then
Cl(Qg ( R ) ) RR1
C1(Ro)/,
where I i s the i d e a l of Ro determining t h e s t r u c t u r e of t h e a e n e r a l i z e d Rees-ring R ( N ) and p i , i
=
1, ..., S , a r e the p r i m i t i v e f a c t o r s of the
discriminator in Ro. Corollary I I . 2 . 2 8 . , we w r i t e I , ( R ) = Q : ( R ) ( I ' )
-.In
f o r some i d e a l
Commutativity of t h e diagram y i e l d s t h a t I ' i s i n the kernel
I ' of Ro.
of t h e composition C1(Ro)
--f
C1(R(N))
v1 "SqY i f we w r i t e I ' = p1 ... ps ...qt
-f
Cl(R).
, then
(where e . , j = 1, ..., s a r e t h e r a m i f i c a t i o n J
r e s p e c t i v e l y ) and S I '
=
(SQ,)
C l ( Q & R ) ( R ) ) . Hence,
+
RI'
=
elVl P1 ...Psesv;
i n d i c e s of p l ,
. . . (SQt)Wt where
Q, . . . Q9 t
..., P,
S denotes Q:(,)
(R).
I/. Arithmetically Graded Rings w
Consequently, (SQl)wl...(SQt)
i s a p r o d u c t o f t h e P1,
p r i n c i p a l R - i d e a l s . S i n c e C1(R(N))
--f
193
...,P s up t o
Cl(R) i s i n j e c t i v e i t follows
0
t h a t the class whence
: I' 6
o f q y '...q t t i s
.
pl,
...,p,.
-
-
I;
Therefore :
11.3. References, Comments, E x e r c i s e s .
11.3.1.
References.
- H. Bass, A l g e b r a i c K - t h e o r y , New York, Benjamin 1968. - I . Beck, I n j e c t i v e Modules o v e r a K r u l l Domain, J . o f A l g e b r a 17,1971, 116- 131.
- N. Bourbaki, AlgPbre Commutative I , 11, P a r i s , Hermann 1961. - L . Claborn, Note g e n e r a l i z i n q a r e s u l t o f Samuel's , P a c i f i c J . Math. 15, 1965, 805-808.
- L. Claborn, R . Fossum, G e n e r a l i z a t i o n o f t h e n o t i o n o f c l a s s qroup, Ill. J . Math. 12, 1968, 228-253. - L. Claborn, R. Fossum, C l a s s groups o f n - N o e t h e r i a n Rinos, J . o f A l g e b r a 10, 1968, 263-285.
-
P. E a k i n , W . H e i n z e r , Some open q u e s t i o n s on m i n i m a l p r i m e s o f K r u l l domains, Canad. J . Math. 20, 1968, 1261-1264.
- P. Eakin, W . H e i n z e r , Non f i n i t e n e s s i n f i n i t e d i m e n s i o n a l K r u l l domains J . o f A l g e b r a 14, 1970, 333-340.
- R. G i l m e r , M u l t i p l i c a t i v e
I d e a l Theory
- A. G r o t h e n d i e c k , L o c a l Cohomoloqy, L.N.M. Y 1, B e r l i n , S p r i n q e r V e r l a q 1967.
- W. K r u l l , I d e a l t h e o r y , E r g e b n i s s e d e r Math. 46, B e r l i n , S p r i n p e r V e r l a g 1968.
B. Commutative Graded Rings
194
- W . K r u l l B e i t r a g e z u r A r i t h m e t i k k o m m u t a t i v e r I n t e a r i t a t s b e r e i c h e , Math. Z . 41, 1936, 545-577. - Y . M o r i , On t h e I n t e q r a l C l o s u r e o f an i n t e r o a l domain, B u l l . Kyoto U n i v . Ser. B.,
7, 1955, 19-30.
- C. N x s t i s e s c u , F. Van Oystaeyen, Graded and F i l t e r e d Rinos and modules, L.N.M.
758, B e r l i n , S p r i n a e r V e r l a q 1980.
and Modules
- M. Nagata, A g e n e r a l t h e o r y o f a l g e b r a i c qeometry o v e r Dedekind domains I ( a l s o 11, 1 1 1 ) , Arner. J . Math. 78, 1956, 78-116. - M. Nagata, L o c a l Rings, I n t e r s c i e n c e T r a c t s i n Pure and A p p l i e d Math. 13, New York, 1962.
- P. Samuel, Anneaux graducs f a c t o r i e l s e t modules r e f l e x i f s , Math. France, 92, 1964, 237-249.
- P . Samuel L e c t u r e s on U.F.D., Bombay 1964.
-
B u l l . SOC
T a t a I n s t . f o r Fundamental Research no 30
J . P . Van Deuren, F. Van Oystaeyen, A r i t h m e t i c a l l y Graded Rings, I, R i n g Theory 1980, L.N.M.
825, B e r l i n 1981, S p r i n g e r V e r l a a .
- F. Van Oystaeyen, G e n e r a l i z e d Rees Rings and A r i t h m e t i c a l l y Graded Rinqs J . o f Algebra, t o appear i n 1982. - 0. Z a r i s k i , P . Samuel, Commutative A l q e b r a 1960.
,
I, 11, Van N o s t r a n d 1958,
195
11. Arithmetically Graded Rings
11.3.2.
Exercises.
a field.
1. L e t R be p o s i t i v e l y qraded N o e t h e r i a n and suppose t h a t Ro i s L e t M c R-gr be QR-R,
f i n i t e l y g e n e r a t e d . Then M i s f r e e i f and o n l y i f
(M) i s p r o j e c t i v e .
2. L e t R be a p o s i t i v e l y qraded N o e t h e r i a n r i n g and l e t M F R-gr be f i n i t e l y generated. Then M i s f r e e i f and o n l y i f Ro R -module and Torl(Ro,M)
=
0
60
R
M i s a free
0.
3. L e t R be a p o s i t i v e l y graded K r u l l domain such t h a t Ro i s a f i e l d k . Consider a f i e l d e x t e n s i o n k ' o f k . I f R ' = R 60 k ' i s a K r u l l domain then R ' i s a
k
f a i t h f u l l y f l a t R-module and C l ( R )
--f
Cl(R') i s injective.
4. ( I . Beck) L e t A be t h e i n t e ' a r a l c l o s u r e o f t h e N o e t h e r i a n i n t e g r a l domain B and l e t { a l,...,ar}
be a s u b s e t o f A; t h e n B : ( B : C B a.) c j J Deduce f r o m t h i s f a c t t h a t a d i v i s o r i a l ( p r i m e ) i d e a l
A:(A:zAa.). J o f A intersects B i n a d i v i s o r i a l (prime)ideal.
5. Consider K r u l l domains A and B , A c B . We say t h a t P 0 E
(Pas
d ' e c l a t e m e n t = no b l o w i n q up) i s s a t i s f i e d i f P F (Spec B ) l ht(P
n
A) c 1 . C o n d i t i o n
principal
ideals o f A
P.D.E
map
i s e q u i v a l e n t t o demandinq t h a t
t o p r i n c i n a l i d e a l s o f B under t h e map
( d i v ) i i n d u c e d by t h e i n c l u s i o n i : A
+
B.
Show t h a t PDE h o l d s i n any o f t h e f o l l o w i n a cases : a. B i s a f l a t A-module b. B i s an i n t e g r a l e x t e n s i o n o f A C.
B i s a subintersection o f A
Deduce t h e graded a n a l o
entails
o f t h e above s t a t e m e n t s .
6. Commutative Graded Rings
196
6. Examples. ( C f . [
29 ] and a l s o P. Samuel's T a t a l e c t u r e s . L e t k be a
f i e l d o f c h a r a c t e r i s t i c d i f f e r e n t from
z.
( a ) I f F i s a non-degenerate q u a d r a t i c f o r m i n k [X1,X2,X31/(F) C l ( k [X1,X2,X3] i s factorial. =
/(F)
i s c y c l i c o f order 2
o r e l s e k [Xl,X2,X31 /(F)
I f k i s a l g e b r a i c a l l y closed then
C l ( k [ X1,X2,X3]
/(F))=
2/22.
( b ) I f F i s a non-degenerate q u a d r a t i c f o r m i n k [X1,X2,X3,X4] t h e n C l ( k [ X1,X2,X3,X4]
/(F))
i s e i t h e r i n f i n i t e c y c l i c o r zero. I t i s
i n f i n i t e c y c l e f o r example i n t h e case where k i s a l q e b r a i c a l l y closed. ( c ) If F i s a non-degenerate q u a d r a t i c f o r m i n k [X,,...,X,] n 2 5, t h e n k [XI
,..., Xnl /(F)
with
i s a factorial ring.
2 2 2 2 /(Xl + X2 + X3 + 2X4) i s a f a c t o r i a l graded 2 2 2 2 . r i n g . However t h e f o r m X1 + X2+2X3 + 2X4 y i e l d s a graded r i n a 2 2 2 2 . Q [ X,,. . . ,X4 ] /X, + X2 + 2X3 + 2X4 w i t h c l a s s group Z. E x e r c i s e . Q [X1,X2,X3,X41
7. L e t D be a commutative graded domain. Check whether t h e f o l l o w i n g conditions are equivalent. a. D i s a gr-Dedekind r i n g b. A l l graded i d e a l s o f D a r e d i v i s o r i a l . 8. L e t C be a K r u l l domain and l e t I be a
d i v i s o r i a l i d e a l o f C. C o n s i d e r
the r i n g S Cl(C)
+
( I ) d e f i n e d i n 11.1.16. Check whether t h e k e r n e l o f C C l ( S l C ( I ) ) i s g e n e r a t e d by t h e c l a s s o f I.
9. L e t C be a K r u l l domain and l e t M be a
d i v i s o r i a l C-module. I s t h e
i n j e c t i v e h u l l of M i n C-mod a a a i n d i v i s o r i a l . What a b o u t maximal r i n g s o f q u o t i e n t s ? What a b o u t l o c a l i z a t i o n s a t t o r s i o n t h e o r i e s . Call a torsion
theory d i v i s o r i a l i f the associated f i l t e r o f i d e a l s
197
II. Arithmetically Graded Rings
has a f i l t e r basis c o n s i s t i n a o f d i v i s o r i a l i d e a l s . Is t h e l o c a l i z a t i o n o f a d i v i s o r i a l module a t a d i v i s o r i a l t o r s i o n theory d i v i s o r i a l over the l o c a l i z e d r i n g ?
R be a commutative qraded semiprime r i n q such t h a t
10. L e t
t h e o n l y idempotent elements o f of
0 and 1 a r e
R. Show t h a t t h e i n v e r t i b l e elements
R a r e homogeneous.
11.3.3.
Comnents
1. I f G i s a t o r s i o n f r e e a b e l i a n group s a t i s f y i n g t h e ascending c h a i n c o n d i t i o n on c y c l i c subgroups then t h e theory o f G-graded K r u l l domains p a r a l l e l s t h e Z-graded t h e o r y . The zealous reader may d e r i v e most o f the results i n
S e c t i o n 11.3.
f o r t h e G-graded case, u s i n g t h e t o o l s
o f chapter A.
2. The g e n e r a l i z e d Rees r i n g s i n t r o d u c e d h e r e may be f u r t h e r g e n e r a l i z e d t o t h e non-commutative case, e.g.
the P.I.
case, y i e l d i n g an i n t e r -
e s t i n g new c l a s s o f r i n g s . This c l a s s o f r i n g s c o n s i s t s o f subrings o f c e r t a i n crossed products R [ G,Q,c] group morphism, c : G x G a "nice" P.I.
L.
: G
+
Aut(R) i s a
U(Z(R)) a 2-cocycle and when R i s u s u a l l y
r i n g l i k e a tame order, an HNP r i n g , a maximal o r d e r
o v e r a K r u l l domain, by
-t
where
...
For a l l t h i s we r e f e r t o a r e c e n t paper
Le Bruyn and F. Van Oystaeyen, [ 64 1 .
198
Ill: Local Conditions for Noetherian Graded Rings
I n t h i s c h a p t e r , a l l r i n a s c o n s i d e r e d w i l l be commutative and N o e t h e r i a n u n l e s s e x p l i c i t e l y mentioned o t h e r w i s e . A l l graded r i n o s c o n s i d e r e d w i l l be o f t y p e Z . 111.1.
I n j e c t i v e dimension o f qraded r i n g s called a
L e t us i n t r o d u c e t h e f o l l o w i n g terminology : a r i n o R i s ~
local ring
i f i t has e x a c t ? y one maximal araded i d e a l ( t h e s e i d e a l s
a r e s a i d t o be gr-maximal) i n t h e s e t o f p r o p e r araded If R =
0
ic Z
2-
Ri
i d e a l s o f R.
i s g r - l o c a l w i t h M t h e maximal graded i d e a l t h e n Ro i s
a l o c a l r i n g and M
n
Ro i s t h e maximal - i d e a l o f Ro.
L e t P be a p r i m e i d e a l o f R and l e t S be t h e m u l t i p l i c a t i v e l y c l o s e d s e t S = h(R)-P. I f M 6 R-gr we denote t h e clraded module o f t h e f r a c t i o n s Qz(M) by Q i - p ( M ) . I f
Q
=
(P),
then : Qi-p(M) =
Qi-p(R) i s a r - l o c a l w i t h t h e maximal qraded
QaR-Q ( C ) .
Clearly
ideal (P)gQi-p(P). L e t
kg(R) denote t h e graded d i v i s i o n r i n g Q ~ - p ( R ) / ( P ) g Q ~ - p ( R ) .
111.1.1. Lemma.
L e t P be a p r i m e i d e a l o f R, t h e n :
1. F o r any M c R-gr, Q o n l y i f Q:(M)
=
R-P
(M) = 0 i f and o n l y i f Q R - ( p ) (M) - = 0 i f and 9
0.
2. P u t supp M = { P c Spec R, QR-p(M) = 0
1,
t h e n P c supp M i f
c supp M. 9 3. I f P i s a graded p r i m e i d e a l t h e n : Eg (ka(P)) Qi-p(R)
and o n l y
i f (P)
4. L e t P be a graded p r i m e i d e a l and l e t Ai
=
{x
Ei(R/P)
c Eq(R/P), Pix
then t h e f o l l o w i n g p r o p e r t i e s h o l d : a. Ai
i s a graded I?-submodule o f Eg(R/P) and E q ( R / P )
=
U Ai.
il.1
=
0
}
199
111. Local Conditions for Noetherian Graded Rings
b. Ai+l/Ai
i s a graded kg(P)-rnodule
c. A1
kg(P).
5. Any i n j e c t i v e o b j e c t i n R-qr i s a d i r e c t sum o f some E g ( R / P ) ( n ) , f o r some graded p r i m e i d e a l s P o f R and some n F 2 . 6. Denote by gr-inj.dimRI.c t h e i n j e c t i v e dimension o f M i n t h e c a t e g o r y R-gr. I f S
c
g r . i n j . dim
h(R) i s any m u l t i p l i c a t i v e l y c l o s e d s e t M 6 R-gr, t h e n S - l M C. q r . i n j . DimRM
S-'R
P r o o f . l . , 2., 3. a r e easy consequences o f t h e f o r e p o i n g chapters;4., 5.,
6.,
f o l l o w as i n t h e ungraded case.
L e t M F R-gr. and c o n s i d e r m i n i m a l i n j e c t i v e r e s o l u t i o n s i n R - q r and R-mod r e s p . :
0
+
F!
+
Eog
+
Elq
+
._.
0
+
Y
+
Eo
+
El
+
.i.
F o r a graded p r i m e i d e a l P o f R, l e t L{(P,M), number o f c o p i e s o f E i ( R / P ) i n E,: 111.1.2.
Propositions.
r e s p un(P,M),
be t h e
r e s p . o f ER(R/P) i n En.
L e t M F R-gr, P
be a graded p r i m e i d e a l o f R. functor
F o r n o t a t i o n a l convenience we w r i t e Q g f o r t h e l o c a l i z a t i o n on R-gr a s s o c i a t e d t o h(R)-P, t h e n :
i s a f r e e graded k'(P)-module
1. The group Qg Exti(R/P,M)
2. clq(P,M)
4. I f S
c
= wi(P,M)
h(R) i s any m u l t i p l i c a t i v e l y c l o s e d s e t such t h a t P
t h e n $(P,M) Proof.
o f r a n k LL:(P,M).
n S
= pn(S-lP,S- 1M) f o r a l l n 2 0.
We have :
Exti(R/P,M)
=
QgExti(R/P,M)
EXTi(R/P,M) =
=
HOMi(R/P,Qy) = HomR(R/P, QS),
QgHomR(R/P,Q;)
=
Horn
Q4(R) and t h e l a t t e r i s a f r e e kg(P)-module.
( k a ( P ) , Q:)
hence
=
d
8. Commutative Graded Rings
200
g
I f P l i s a graded p r i m e i d e a l o f R such t h a t P1 # P t h e n
Q tbm(R/P,El(P1))=O,
hence QgHomR(R/P,Qq) i s a f r e e kg[P)-module o f r a n k equal t o pY(P.M). S i n c e Exti(l?/P,M) i s a f r e e k ( P ) m o d u l e o f r a n k ki(P,M)
w h i l s t QExt(R/P,M)
a t P ( Q t h e l o c a l i z a t i o n i n R-mod
i s t h e l o c a l i z a t i o n o f QgExt(R/P,M)
a s s o c i a t e d t o P ) . 2 ) f o l l o w s i m m e d i a t e l y and 3) f o l l o w s f r o m 2 ) . F o r t h e s t a t e m e n t 4 ) we a p p l y t h e s t a t e m e n t 1). 111.1.3.
L e t M c R-gr,
Proposition.
P a graded p r i m e i d e a l o f R
a C h(R) a nonzero d i v i s o r on M and R, t h e n f o r a l l n pn(P/aR, 9
M/aM) =
(P;M)
E:
L e t O+M+
Proof.
d A
d-1
E:
be a m i n i m a l i n j e c t i v e r e s o l u t i o n o f
, E24 CI
0 :
-
M on R-gr and N
I f K 6 R-gr i s a r b i t r a r y , t h e n HornR (R/aR, K )
3
=
and
d
0
'
(E0')
(K:a) = { x
f
1
ax =O}
E2g)
+...,
K
From t h e commutative diaqram :
O+
M ? E o a A
N
iqa
1.a
1.a
M---t
Eoq-)N
---to where Q ~ ( X =) ax
O+
we have (N:a)
=
0
3
M/aM so Hom(R/aR,N)
h
C/aM.
From t h e e x a c t sequence
0-J
R
ip
a,R +R/aR
we deduce t h a t
-0
E x t i ( R / a R , M) = 0 f o r a l l i > 1.
T h i s i m p l i e s t h a t t h e sequence : (1):
0-
HomR(R/aR,N)+
i s e x a c t . Because HornR (R/aR,
HornR (R/aR, Elq)+HomR Eig)
h
(Eiq:a)
(R/aR,
i s an i n j e c t i v e module
o v e r t h e r i n g R/aR we have t h a t (1) i s a m i n i m a l i n j e c t i v e r e s o l u t i o n o f t h e R/aR-module M/aM = HornR (R/aR,N).
201
111. Local Conditions for Noetherian Graded Rings
Now, i f P i s a graded p r i m e i d e a l such t h a t a v e r i f i e d t h a t HornR (R/aR,
ki
(P/aR, M
111.1.4.
M) =
P, t h e n i t
i s easily
envelope o f t h e
i s the injective
E(R/P))
R/aR-module R/P = (R/aR)/(P/aR).
c
T h e r e f o r e , f r o m ( 1 ) we o b t a i n t h a t
(P,M). L e t P be a p r i m e i d e a l o f R such t h a t P #
Proposition.
t h e n f o r a l l n t N a n d M t R-gr we o b t a i n : P ~ ( ( P ) ~ , M )= P r o o f . By P r o p o s i t i o n 111.1.2.4. w h i l e untl(P,M)
= pn+l
Qg(R) and Qg(M) r e s p . , gr-maximal i d e a l
we have : u,,((P)~,M)
(Q4(P),Qg(M)).
(P),
(P,M).
= (Qq((P)g),Qg(M))
So we may r e p l a c e R and M by
we may suppose t h a t R i s g r - l o c a l and
i.e.
of R . I n t h a t case P i s a maximal i d e a l o f R. S i n c e
i s a qraded d i v i s i o n r i n q , t h e r e e x i s t s a t R such t h a t P = ( P ) +Ra. 9 9 From t h e e x a c t sequence :
R/(P)
( * ) O-Rl(P)g
a
4
R/(P)q-R/P
where ma denotes m u l t i p l i c a t i o n by a, we o b t a i n :
+ Extn(R/(P)g,M) 4 -+ Extn+l(R/(P)g,M)--t Note t h a t Extn(R/(P)g,M)
Extntl(R/P,M)
+
...
... -,Ext"(R/(P)
9
m , M A
+ Extn+l(R/(P)g,M)
"'a
. .. = EXTn(R/(P)9, M) i s a qraded R/(P)q-module,
hence E x t n ( R / ( P ) ,M i s a f r e e araded R / ( P ) q module. Consequently 9 Extn(R/(P) ,M) --$ Extn(R(P) ,M)is rnonomorphic. ma: 9 9 The l o n g e x a c t sequence o b t a i n e d above y i e l d s a s h o r t e x a c t sequence: m Extn(R(PjQ,M) + Ext"'(R/P,M) 30 0 4 Extn(R/(P)g,M)
2
I n t h i s sequence V = Extn(R(P)q,M) whereas V ' = Ext"'(R/P,M)
i s f r e e graded o f r a n k p,,((P),,M),
i s a v e c t o r space o f
d i m e n s i o n P,,+~(P,M)
I n R/(P) -mod we have t h e e x a c t sequence : 9 v -V' 3 0 kn((P)g'M) we deduce f r o m ( * ) and (*) t h a t Since V 2 (R/(P)g)
o v e r t h e f i e l d R/P. m (**) o + v
a,
,
B. Commutative Graded Rings
202
V'
(R/P)
111.1.5.
Pn ( (P)
.M)
.
On t h e ' o t h e r hand V ' = ( R / P )
Corollary. I n the s i t u a t i o n o f III.1.4.,
Pn+ 1(P 9 M)
P,((P)~,M)
,
therefore
=
0 i f and
o n l y i f F ~ + ~ ( P , M )= 0. 111.1.6.
C o r o l l a r y . L e t P be a q r a d e d p r i m e i d e a l o f R.
The m i n i m a l i n j e c t i v e r e s o l u t i o n o f Eg(R/P) i n R-mod i s t h e f o l l o w i n q :
+ E(R/P)-Q
O+
E(R/P')--+O P' where t h e d i r e c t sum i s o v e r a l l p r i m e i d e a l s P ' # P o f R such t h a t (P')
E~(R/P)
9
= P.
-
P r o o f . The r e s o l u t i o n has t h e f o r m : 0-
~ gR/P) (
where Q, =
-0
E( R/P)-Q~
c
P'tSpec R
E(R/P ' )
111(~ , E ~ ( R / P )
and C o r o l l a r y 111.1.5. we g e t Q,
c i n j . dim.R
__ Proof.
E ( R / P ' ) where ( P ' )
9
= P.
o
c n
t
1
D i r e c t l y f r o m t h e p r o p o s i t i o n 1.4. Remark. I f M i s a maximal i d e a l o f R w h i c h i s graded t h e n
111.1.8. Ei(R/'M)
P'
C o r o l l a r y . I f M t R-qr w i t h g r . i n j . dim.R M = n, t h e n
111.1.7. n
= 8
. A p p l y i n g P r o p o s i t i o n 111.1.4.
=
ER(R/M), e . 9 . i f
R
= k [XI
,..., X n l
and
M=(X1 ,..., X n ) .
(k i s a field). 111.1.9.
Lemma. L e t R be a graded r i n g and suppose P
c
Q are d i s t i n c t
graded p r i m e s w i t h no o t h e r graded p r i m e i d e a l between t h e n . I f M i s a graded
f i n i t e l y g e n e r a t e d module t h e n
Proof.
By P r o p o s i t i o n 111.1.2.4.
a c h(R) such t h a t a t Q, a
P.
a;(P,M)
# 0 i m p l i e s g;+l(Q,M)
# 0.
we may assume R = Q g (R) . Choose R-Q We denote A = R/P and B = A/aA = R/(a,P)
203
111. Local Conditions for Noetherian Graded Rings
C l e a r l y , B 1s a n o b j e c t of f i n i t e l e n g t h i n R - g r and a i s a nonzero The e x a c t sequence : O
d i v i s o r on A .
an e x a c t sequence : ExtR(A,M) n
cpa
d
A -6
A L
+Exti
(A,M)
-+
-0 Ext;+l(B,M).
yields By
h y p o t h e s i s R i s g r - l o c a l w i t h Q as t h e or-maximal i d e a l and a t Q, t h e n
# 0. Now, b y i n d u c t i o n on 1(B) ( 1 d e n o t i n g l e n r j t h ) , and Extntl(B,M) R ( . .M) we o b t a i n t h a t u s i n g t h e r i g h t e x a c t n e s s o f f u n c t o r s Ex’:t (R/Q,
Ex$”
111.1.10.
M) # 0, and hence P:+~(Q,F”)
Corollary.
# 0.
L e t R be a commutative N o e t h e r i a n r i n g w i t h l e f t
l i m i t e d g r a d i n g . I f M t R-gr i s a f i n i t e l y t h e n : g r . i n j . dim. R M = i n j . dimR Proof.
Put n = or. i n j . dim.R M c
g e n e r a t e d qraded module
#.
-.
I f i n j . dimR
# n, t h e n i n j . d i m R
= n+l.
Now pn+l (P,M) f 0 f o r some p r i m e i d e a l P o f R, so by t h e f o r e g o i n g results P i s
maximal. C l e a r l y P i s n o t qraded and by P r o p o s i t i o n 111.1.4.
t h a t p n ( ( P ) .M) # 0 hence P:((P)~,M) 9 i s graded maximal, i n d e e d i f n o t , t h e n (P)
# 0. We c l a i m t h a t
i t follows
(P),
graded t h e n
G:+~(P~,M) # 0 by Lemma I I I . 1 . 9 . ,
c P1, b u t i f Plis g c o n t r a d i c t i o n . Since R
has l e f t l i m i t e d a r a d a t i o n i t f o l l o w s t h a t (P) i . e . P = (P) 111.1.11.
9’
i s a maximal i d e a l ,
c o n t r a d i c t i o n . Consequently, i n j . dim.RM = n .
Corollary.
L e t R be a commutative N o e t h e r i a n r i n g w i t h l e f t
l i m i t e d g r a d i n g . L e t M c R-gr. b e a f i n i t e l y g e n e r a t e d qraded module with gr.inj.dim.
M
=
n, n
c
-.
If V M + E q * E F + 0
... + E i *
0
i s a m i n i m a l i n j e c t i v e r e s o l u t i o n of M i n R-ar t h e n E i i s i n j e c t i v e i n R-mod.
# 0. By Lemma
Proof.
L e t P be a araded p r i m e i d e a l such t h a t F:(P,F”)
111.1.9.
we o b t a i n t h a t P i s a qr-maximal i d e a l . Because R has l e f t
l i m i t e d g r a d i n g P i s a maximal i d e a l . By Remark 111.1.8. t h a t E i ( R / P ) i s an i n j e c t i v e an R-module.
R-module.
i t follows
T h e r e f o r e E,“ i s i n j e c t i v e as
204
8. Commutative Graded Rings
I n e x t e n d i n g f r o m R t o R [ X I we need t h e f o l l o w i n g lemma where R has t r i v i a l g r a d a t i o n and R IX ] i s graded as u s u a l . 111.1.12. Lemma.
I f P i s a graded p r i m e i d e a l o f R [ X I t h e n e i t h e r
P = pR [ X I o r P = pR [ X I Proof. S i n c e P
+
(X),
where p = P
1/
pR [ X 1 and R [ X
3
n
R.
pR [ X I = (R/P)[ X I we may assume p = 0,
i . e . R i s a domain. L e t K be t h e f i e l d o f f r a c t i o n s o f R . S i n c e P fl R = 0, PK
I X I i s a graded p r i m e i d e a l o f K [ X I i . e . PK [ X I = 0 o r P K
[ X
1
=
X.
o
111.1.13. P r o p o s i t i o n . L e t R be a N o e t h e r i a n commutative r i n g and t a k e Spec R P u t E = ER(R/P). C o n s i d e r t h e r i n g R [ X I
P C
Denote M = E [ X I
t o t h e degree i n X.
=
E
rn R [ X I R
graded c o r r e s p o n d i n g
.
The m i n i m a l i n j e c t i v e r e s o l u t i o n o f M i n t h e c a t e o o r y R [ X I - q r i s : O-M-Ei(R[X]
(R[Xl/PR[XI+
/PR[X])-+Eg
(X))+O
and t h e m i n i m a l i n j e c t i v e r e s o l u t i o n o f M i n R [XI-mod i s :
t h e d i r e c t sum b e i n g t a k e n P'
n
over t h e prime i d e a l s P' o f
R [ X I such t h a t
R = P.
Proof.
Consider t h e minimal i n j e c t i v e r e s o l u t i o n o f
0-M
-1:
+
Proposition III.1.3.,
'i+1 (PR [ X I
+
(R
= Eq
[ X]/PR [ X I ) . Taking i n t o account
we o b t a i n :
(X),M)
[ X I + (X), E [ X I /
= wi(PR
S i n c e p . ( P . E ) = 1 i f i = 0 and pi(P,E) 1
q(PR [ X I
=
1
+
(X),M)
= 1 and pi(PR
P ' F Supp. M and t h e r e f o r e P '
n
+
[ X I
L e t P ' be a p r i m e i d e a l o f R [ X ]
By Lemma 111.1.12.
R [ X I - gr :
....
If +I;-
S i n c e A s s M = { PR [ X I }, 1:
M in
0 if i
0, we have
= 0 f o r a l l i > 1.
(X),M)
such t h a t +(P',M) R
3
P. I f P '
i t follows that P'
=
X E ( X ) ) = pi(P3E)
n
z
0. Then n e c e s s a r i l y
R # P, p u t P '
n
Q R [ X I or P' = Q R [ X I
R = Q.
+ (X).
205
111. Local Conditions for Noetherian Graded Rings
I n case P '
Q R [X]
=
p i ( P ' , M ) = P.1 ( Q R
we have :
[ X l , E [XI ) = d i m k ( p ) ( E x t A I X l( R [Xl/QR [ X I , E [ X I )
However,
Exti
( R [ X I /QR [ X I , E [ X I )
=
E x t i (R/Q,E)
a n d as Q # P we obtain E x t i ( R / Q , E )
Thus
p.
(P',M) = 0 for a l l i
In case P '
Q R [XI
=
0 R -
[
+
2
-
=
R 1x1
@
R
0 for a l l i
2
.
0.
0.
( X ) , we have an exact sequence mX
X ]/Q R [ X I
R [ X l/QR [ X I d
R [ X ]/PI -0
Yielding a long exact sequence :
... j E x t i ( R
mX
[ X ] / Q R [ X I , M) -Exti
( R [ X I , Q R [ X I , M)
-
( R [X]/P',M) 4 E i + l ( R [ X l / Q R [XI, M) &'Exti+'(R
+ Exti"
[ X]/QR
( a l l E x t a r e E x t R [XI ) ' From the above we deduce : Ext
1
( R [ X]/P',M) = 0 f o r a l l i 2 1.
Finally t h i s amounts t o 1: = Eg ( R [Xl/PR [XI + ( X ) ) a n d Igk = 0 f o r all k
P
1.
For the second statement of the proposition, l e t the minimal i n j e c t i v e in R [ XI-mod be qiven by
resolution f o r
II-J
M -Io+
0-
...
12-
I t i s c l e a r t h a t I. = E R
:
( R [X]/PR [XI). From Proposition 111.1.2.
( R [ X]/P') where P ' R [XI = PR[X] o r varies i n the s e t of prime ideals of R [ X I such t h a t
and 111.1.4. i t follows t h a t I 1 contains
P'
=
PR [ X I
Indeed i f P '
+
n
( X ) . Therefore ( P ' )
R
=
9 P and P ' # ( P ' )
# PR [XI then h t ( P ' ) h t ( P ' ) 2 2 + h t ( P ) . On the PI
n R
=
P, h t ( P ' )
5
g
2
=
9
@
PI
E
PR [XI i f and only i f P ' n R = P . then h t ( P ' )
1 + h t ( P R [XI)
=
=
1 + h t ( P ' ) q , thus i f
1 + h t ( P ) . Thus
other h a n d , since R i s Noetherian and
1 + h t ( P ) , contradiction. Since the converse i s
obvious, t h i s proves the statement.
[XI, M)+
6. Commutative Graded Rings
206
3
L e t P l b e a p r i m e i d e a l o f R [ X I w h i c h i s n o t graded and such t h a t P1 il R f P Then (Pl)g
PR [ X I
would i m p l y
P1
n
# PR [ X I + (X) because(Pl)q
and (Pl)q
R = (P )
l g
From P r o p o s i t i o n 111.1.4.
n
PR [ X I + (X)
R = P.
we r e t a i n t h a t ui((P
) ,M) = pitl(Pl,M). 1 q ,M) = 0 f o r a l l i z 0, hence
already established pi((P.) ' g = 0 f o r a l l i 2 1. T h i s y i e l d s t h a t I1 =
However we have p . (P1,M)
=
8
P'
ER [Xl(R
[Xl/P')
as s t a t e d . I f I 2 were nonzero t h e n t h e r e e x i s t s a p r i m e
where P ' i s
i d e a l P2 o f R [ X I
such t h a t p2(P2,M) # 0.
I f P 2 i s graded t h e n from P r o p o s i t i o n 111.1.2. we deduce $(P2,M)
and t h u s I!
# 0
# 0, c o n t r a d i c t i o n .
I f P2 i s n o t graded, t h e n we deduce f r o m P r o p o s i t i o n 111.1.4. t h a t
p2(P2,M) = PR [ X I +
=
u ~ ( ( P ~ ) ~ , fM 0. ) I n t h i s case i t i s necessary t h a t (P2) ( X ) , hence P2 fl R = ( P 2 )
fl R = P. S i n c e P2
q
=
# (P2)g, h t ( P 2 ) =
h t ( ( P ) ) = 2 + ht(PR [ X I ) = 2 + h t ( P ) . B u t f r o m P2 n R = 2 9 = P, h t ( P 2 ) 5 1 t h t ( P ) f o l l o w s hence we r e a c h a c o n t r a d i c t i o n . = 1
+
T h e r e f o r e p2(P2,M)
=
0 and I p = 0 f o l l o w s .
111.1.14.
C o r o l l a r y . L e t R be a commutative N o e t h e r i a n
i n R-mod,
then :
g r . i n j . dim.
R [XI
E [XI
+
1 = i n j . dim.R
ring,
E [ X I .
E injective
207
111. Local Conditions for Noetherian Graded Rings
111.2. R e g u l a r , G o r e n s t e i n and Cohen-Macaulay R i n g s . r i n g and M a nonzero f i n i t e l y a e n e r a t e d
L e t R be a commutative N o e t h e r i a n
R-module. We p u t : V(M) = supp.(M) = V(AnnRF1) = { P C
M,
Recall t h a t t h e K r u l l dimension o f
Spec R, AnnRM
=
P
1.
denoted by K.dimRM i s d e f i n e d t o
be t h e supremum of t h e l e n g t h s o f c h a i n s o f p r i m e i d e a l s o f V(M) i f t h i s supremum e x i s t s
,
and
-
i f n o t ( t h e r e a d e r may v e r i f y t h a t K.dimRM
c o i n c i d e s w i t h t h e K r u l l dimension o f M d e f i n e d i n s e c t i o n 1 1 . 5 . ) . We have htMP = K.dim QR-p(R) (QR-P(F)). I f I i s an i d e a l o f R such t h a t IM # M t h e n t h e l e a s t r f o r which
# 0 w i l l be c a l l e d t h e grade o f I on P, denoted by : grade (1,M);
Extk(R/I,M) grade
(1,M)
i s e x a c t l y t h e common l e n p t h o f a l l maximal sequences
c o n t a i n e d i n I (see Lemma 111.3.3. o f I f R i s a l o c a l r i n g w i t h maxima!
9
3). s a i d t o be a
i d e a l Q , then M i s
Cohen-Macaulay module o r a C.M.-module
i f grade (si,M)
=
K.dimRM. A n o t
n e c e s s a r i l y l o c a l r i n g i s s a i d t o be a Cohen-Macaulay r i n g o r C.M-ring i f QR-p(R) i s a Cohen-Macaulay QR-p(R)-module and i n t h a t case P i s a C.M-module i f QR-p(M) i s a C.M.-QR-P(R) module. I t i s easy t o v e r i f y u s i n g P r o p o s i t i o n 111.1.2.
t h a t M i s a C.M.-module
each maximal i d e a l Q c V(M), ki(S1,M)
i f and o n l y i f f o r
= 0 whenever i < h t M
(a).We
M i s a Gorenstein-module i f f o r e v e r y maximal i d e a l 9 c V(M), if and o n l y i f i # h t M ( Q ) . P r o p o s i t i o n 1 1 1 . 1 . 2 . e n t a i l s t h a t
say t h a t
wi(Q,M)
= 0
M is a
Gorenstein-module i f and o n l y i f QR-p(M) i s a G o r e n s t e i n QR-p(R)-module for a l l P
c
V(M).
A G o r e n s t e i n r i n q R i s a r i n g w h i c h i s a Gorenstein-module when c o n s i d e r e d as a module o v e r i t s e l f . F o r a d e t a i l e d s t u d y o f we r e f e r t o Bass, graded case.
[ 5 ] . F i r s t we g i v e some l o c a l
these classes o f r i n g
-
g l o b a l theorems i n t h e
B. Commutative Graded Rings
208 111.2.1.
Theorem. L e t R be a commutative N o e t h e r i a n graded r i n g and l e t
M c R-gr be f i n i t e l y g e n e r a t e d . The f o l l o w i n q s t a t e m e n t s a r e e q u i v a l e n t . 1. M i s a Cohen - Macaulay module. 2. F o r any graded p r i m e i d e a l P o f V(M), QR-p(M) i s a Cohen-Pacaulay QR-p( R)-module. ~
P r o o f . 1 = 2 i s t r i v i a l . To p r o v e t h e converse l e t us assume f i r s t t h a t QR-p(M) i s a C.M.
QR-p(R)-module f o r e v e r y graded P
c
V(M). L e t P ' c V ( P ) .
If ( P ' ) g = P ' t h e n we a r e done. So suppose t h a t P ' # we have t h a t h t H P ' = h t M ( P ' ) a
+
I n t h i s case
1. Choose i < h t M ( P L ) .
hence p . ( ( P ' ) ) = 0. By P r o p o s i t i o n 111.1.4. 1-1 g,M pi(P',M)
= 0 and t h e r e f o r e M i s a C.M.-module.
111.2.2.
Theorem.
Then i - 1 c h t M ( ( P ' ) q ) ,
i t follows then t h a t
L e t R be a graded commutative N o e t h e r i a n r i n p and l e t
M C R - g r be f i n i t e l y generated. The f o l l o w i n a s t a t e m e n t s a r e e q u i v a l e n t : 1. M i s a Gorenstein-module 2. F o r each graded P c V(M), QR-p(H) i s a G o r e n s t e i n QR-p(R) module. P r o o f . 1 = 2. Easy. F o r t h e converse, suppose t h a t QR-p(M)
i s a Gorenstein
QR-p(R)-module f o r each qraded P 6 V(M). L e t P ' be a r b i t r a r y i n V(M). I f
P'
(P')
t h e n t h e a s s e r t i o n i s c l e a r . Supposinq t h a t P ' # ( P ' )
and 9 i # h t M ( P ' ) , we o b t a i n i - 1 # h t M ( ( P ' ) ) . S i n c e Q R - ( p , ) ( P ) i s a G o r e n s t e i n g 9 Q R - ( P l ) (R)-module i t f o l l o w s t h a t : 9 =
9
~ i _ l ( ( p ' ) ~ M= )p i - 1 ( Q R - ( p l ) ( ( P ' ) a ) , Q R - ( p l ) ( H ) ) = 0 9 9 From P r o p o s i t i o n 111.1.4. we deduce t h a t pi(P',M) = 0. I n a s i m i l a r way i t may be deduced f r o m v i ( P ' , M )
G o r e n s t e i n module
.
= 0 that i
+ htM(P')
hence
M
is a
0
A l o c a l r i n g w i t h maximal i d e a l Q i s s a i d t o be r e g u l a r i f q l . dim R < o r e q u i v a l e n t l y , ifK.dim R = dim
R/Q
2
(Q/Q ). I f 9 i s not a local rinq,
209
Ill. Local Conditions for Noetherian Graded Rings
t h e n R i s c a l l e d a r e g u l a r r i n g i f f o r e v e r y maximal i d e a l Q,QRTS i s regular.
111.2.3. C o r o l l a r y . L e t R be a commutative N o e t h e r i a n r i n a and M a f i n i t e l y g e n e r a t e d R-module.
I f M i s a Cohen-Macaulay
-
t h e n M [ X I i s a Cohen __ Proof. L e t P
c
n
R t h e n p c V(M).
P = p R [ X] or P = p pi(P,M)
and
M [XI c P
R
[X]
+
if P = pR [ X I
By Lemma 111.1.11. we have
( X ) . I f P = p R [ X I we have vi t
(P,M [ X I ) =
(X) we have by P r o p o s i t i o n 111.1.3. t h a t
'L~+~(P,M [ X I ).
v~(P.M) Moreover h t P = pR [ X I
111.2.2.
Macaulay ( r e s p . F o r e n s t e i n ) R-module.
V(M [ X I ) be a graded p r i m e i d e a l ; t h e n AnnR
I f we w r i t e P = P
=
( r e s p . G o r e n s t e i n ) module
M [ X I
+
(X).
P = htMP i f P = p [ X ] and h t
M [XI
P = 1 t htMP i f
Now t h e c o r o l l a r y f o l l o w i n q f r o m Theorem 1 1 1 . 2 . 1 . and
finishes t h e proof.
:
1 1 1 . 2 . 4 . Theorem. L e t R be a graded commutative N o e t h e r i a n r i n a t h e n t h e f o l l o w i n g statements are equivalent :
1. R i s a r e g u l a r r i n g . 2 . QR-p(R) i s r e g u l a r f o r e v e r y graded p r i m e i d e a l P o f R. Proof. __
2
=
The i m p l i c a t i o n 1 = 2 i s o b v i o u s .
1. If P ' i s an a r b i r a r y p r i m e i d e a l o f R t h e n t h e r i n p Q R - ( p , ) (R) i s q
r e g u l a r i . e . g l . d i m . Q R - ( p , ) (R) < -. S i n c e Q R - p , ( R ) i s t h e l o c a l i z a t i o n 9 of S = QR-(pb) (R) a t t h e p r i m e i d e a l g e n e r a t e d by P i i n i t , i t f o l l o w s 9 t h a t we may assume t h a t we have chosen R t o be g r . l o c a 1 w i t h gr.maximal By h y p o t h e s i s g l . dim S = n c -. C o n s i d e r a f i n i t e l y g' g e n e r a t e d R-module M. By P r o p o s i t i o n 111.1.2. we have t h a t ideal (P')
i n j . dimS Q R - ( p , ) (M) 5 n and hence g r . i n j . dim R M 5 n. T h e r e f o r e 4 3 r . g l . d i m R c m and C o r o l l a r y 111.1.10. y i e l d s q l . d i m . R c m , hence g l . - d i m QR-p,(R)
degree i n Q -52'
( i f there i s no such x then R
P' =
R'/ ( X I ,
K.dim R '
=
=
then R
Ro
=
Rofollows). P u t
Q / ( x ) . Now R' i s a qraded r e q u l a r r i n a and
Ro
= =
(R')o [Y1
t h a t R = Ro [ U1,..
,..., Yk]
-
with deq Y i > 0 , i
0 i t follows t h a t ( R ' ) o
d then d
t
i n R f o r t h e images o f Y 1 . . . Y k
.
.,Uk+l]
homomorphism v! : Ro [ X1
k
=
=
i s an isomorphism.
=
1,..., k .
RO'
n-1. Choose r e p r e s e n t a t i v e s U1,...,Uk i n R ' , and
put:Uk+l
=
x . One e a s i l y checks
Therefore we have an epimorphic graded r i n q
,..., X k t l ]
+
Ro [ U 1
,...,Uk+l]
t h e f a c t t h a t K.dim R = k + d t 1 = K.dim Ro [ X1,...,Xk+ll 'v
=
The induction hypothesis y i e l d s : ( R ' ) o i s a r e q u l a r
l o c a l r i n g and R'
So i f K.dim Ro
0
0 , choose a nonzero homoaeneous element x of p o s i t i v e
$2'
n-1.
Since ( x ) f7
=
=
R . However, yields that
o
111.2.6. Remarks.
1. I f R i s a p o s i t i v e l y graded r e g u l a r
r i n a , then Ro i s r e g u l a r . Indeed,
i f w i s a maximal i d e a l i n Ro then Q = W O R+ i s a maximal graded i d e a l . Since t h e l o c a l i z a t i o n QR-Q(R) i s r e g u l a r and a l s o p o s i t i v e l y sraded (note t h a t i t coincides w i t h the
qraded r i n g of q u o t i e n t s Q:-Q(R))
i t follows from t h e foreqoing t h a t ( Q R - Q ( R ) ) o (QR-Q(R))o
=
QR JRo). 0
i s r e g u l a r . However
I t f o l l o w s t h a t Ro i s r e g u l a r t o o .
111. Local Conditions for Noetherian Graded Rings
2 . See a l s o J . M a t i j e v i c [
7 1 . If
R
21 1
0 Riis C-orenstein, hence Cohenizo Macaulay, t h e n i t i s n o t n e c e s s a r i l y t r u e t h a t Ro i s G o r e n s t e i n o r C . V . .
2
/ (X , X Y ) , where R i s any f i e l d . P u t Q1
Indeed, p u t T = k [ X , Y ]
Q2 = ( X ) ,
=
t h e n t h e image of Q ,
n
o f degree 1 and p u t S = T [ W ] , Q ;
Q, i n = Q,S,
T
=
(Y),
i s z e r o . L e t W , V be v a r i a b l e s
QE = Q 2 S +WS and I = Qf fl Q E .
2
P u t U = S / I , R = U [ V ] / ( x V + yV,V ) where x, y denote t h e imaaes o f X, Y r e s p . T h i s c o n s t r u c t i o n y i e l d s a qraded r i n a R w i t h K.dim R = 1 and Ro = T. However
T
is
n o t a C.C-rina.
I f 8 denotes t h e u n i q u e maximal graded i d e a l Gorens t e i n.
o f R then Qq-Q(R)
is
212
5. Commutative Graded Rings
111.3. Graded Rings and M-sequences. R i s a commutative graded r i n a throuclhout t h i s s e c t i o n . Such a r i n a R
i s s a i d t o be c o m p l e t e l y p r o j e c t i v e o r i s s a i d t o have p r o p e r t y C.P. i f f o r any graded i d e a l I and f o r any f i n i t e s e t o f araded p r i m e i d e a l s
PI,
. . . ,Pn
w i t h h ( 1 ) c P1 U . . . U
a t l e a s t one Pi.
Pn i t f o l l o w s t h a t I i s c o n t a i n e d i n
It i s n o t hard t o v e r i f y t h a t , i f R i s completely
p r o j e c t i v e t h e n so i s S-’R
f o r every m u l t i o l i c a t i v e l y closed subset
S o f h ( R ) . I t i s a l s o s t r a i g h t f o r w a r d t o check t h a t e p i m o r p h i c images
o f a c o m p l e t e l y p r o j e c t i v e r i n g have p r o p e r t y C.P. 111.3.1.
too.
L e t R be a commutative r i n g such t h a t a l l graded p r i m e
Lemma.
i d e a l s o f R a r e i n P r o j R, t h e n R i s c o m p l e t e l y p r o j e c t i v e .
4
P r o o f . Assume I
1 5 i 5 n and assume t h e pi a r e n o t comparable.
Pi,
I f n = 1 then h(1)
Q
PI f o l l o w s . We p r o c e e d by i n d u c t i o n on n. The
t h e r e i s an ai
,
c
h(I
n
Pi)
n
Pi)
Q
P 1 5 i 5 n hence j’ such t h a t ai f P . f o r each j # i. L e t J
induction hypothesis y i e l d s h ( I
U j+i
r! a . . C l e a r l y xi c I and xi c P . f o r j # i b u t xi 4, Pi. S i n c e j#i J J no graded p r i m e i d e a l o f R c o n t a i n s R, i t i s easy t o see t h a t we may
x. =
choose deg xi > 0 f o r each i, 1 5 i 5 n. P u t d 1. = dea x . and d = d l . . . . .dn, y,
=
P’di
x.
Then deg yi = d and i t i s o b v i o u s t h a t y = yl+ ...+yn i s
a homogeneous element such t h a t y c I and y f PIU 111.3.2.
... U
Pn.
Examples.
1. I f R i s a p o s i t i v e l y graded r i n g such t h a t Ro i s a f i e l d t h e n R i s completely p r o j e c t i v e . 2 . L e t R be a g r - l o c a l r i n g w i t h qr-maximal
i d e a l Q such t h a t R / Q i s a
non t r i v i a l graded g r - d i v i s i o n r i n g . Then R has p r o p e r t y C.P. Indeed, i f P
.
i s a graded p r i m e i d e a l t h e n P c Q. S i n c e R / Q i s non-
t r i v i a l l y graded, t h e r e i s a homoqeneous t o f p o s i t i v e deqree, t f Q , hence t t P.
213
Ill. Local Conditions for Noetherian Graded Rings 2. L e t O v be a d i s c r e t e v a l u a t i o n r i n g w i t h maximal i d e a l av g e n e r a t e d
i n R we have t x = 0.
by t say. P u t R = O v [ X ] / ( t X ) i . e . (t,x)
i s a graded i d e a l and e v e r y homogeneous e l e m e n t o f ( t , x )
z e r o - d i v i s o r ; however t
+
x i s n o t a z e r o - d i v i s o r . I f P1,...,Pn
t h e prime i d e a l s associated t o R , then these h ( 1 ) c PIU
...U
Pn b u t I
Q
Pi
Then is a are
a r e araded and c l e a r l y
f o r each i, 1 5 i 5
n.
So O,[
X I
does n o t
.
have p r o p e r t y C.P.
L e t R be a commutative N o e t h e r i a n r i n g and M a f i n i t e l y g e n e r a t e d Rmodule. A sequence al,
. . . ,n,
and f o r each i = 1,
,..., ai-l)M.
M/(al
...,an ai
i n R i s s a i d t o be an M-sequence i f (al,
. . . ,an)M#M
i s n o t an a n n i h i l a t o r f o r t h e module
I f I i s a n i d e a l o f R such t h a t I M # M , t h e n we s h a l l
denote by grade (1,M)
t h e common l e n g t h o f a l l maximal Wsequences
contained i n I .
111.3.3. Lemma.
L e t R be a commutative N o e t h e r i a n r i n q and l e t M be a
f i n i t e l y g e n e r a t e d R-module. I M # M. Then grade (1,M)
Proof.
Suppose t h a t I i s an i d e a l o f R such t h a t
i s t h e s m a l l e s t number r such t h a t Extk(R/I,M)#O.
F i r s t we show by i n d u c t i o n on r t h a t i f al,
sequence c o n t a i n e d i n I t h e n E x t i ( R / I , E x t L ( R/ I .M)
=
Hom( R/ I ,M/ ( a l , . . . ,ar) M)
-
a*,
...,a,
i s an M-
M) = 0 for 0 5 i 5 r, and
.
Indeed, from t h e e x a c t sequence
v
0-
C
a1
I M
M/alM
-0
We d e r i v e t h e e x a c t sequence : E x t r - l ( R / I ,M)
'a 4 Ext'-l(R/I
--jr Extr(R/I,M)
'a1 M
By t h e t h e i n d u c t i o n
pal
=
0. Hence E x t r - '
, W ) +xtr-'(R/1
...
Extr(R/I,M)-
h y p o t h e s i s : E x t r - l (R/I,M) ( R / I , M/alM)
,M/alM)
h
E x t r (R/I,M).
=
0 and t h e homomorphism
214
6. Commutative Graded Rings
-
BY i n d u c t i o n : Extr-' Extr(R/I,M) M" = i x
( R / I , M/alM)
5
Hom (R/I,P/(a l...ar)F.l)
Hom(R/I,M/(a l...ar)M).
I f Hom ( R / I , N ' )
I x = 0 } i s not zero. Therefore
C MI,
En' such t h a t I c
t h e r e e x i s t P F Ass M" c Ass
. . . ,an
sequence 1 al,
and t h e r e f o r e
# 0 t h e n t h e module
Ass M" # gj
i.e.
P, and t h e r e f o r e t h e
} i s an M-sequence o f maximal l e n q t h .
C o r o l l a r y . L e t R be a commutative N o e t h e r i a n r i n a w h i c h i s
111.3.4.
graded and c o m p l e t e l y p r o j e c t i v e . L e t F.1 F R-qr be f i n i t e l y g e n e r a t e d and I a graded i d e a l o f R such t h a t I M # M . I f grade (1,M) = m, t h e n t h e r e e x i s t s i n I an M-sequence o f l e n g t h m, c o n s i s t i n o o f homogeneous elements. Moreover any M-sequence formed by homooeneous elements o f I has t h e same l e n g t h grade (1,M).
Proof.
For m = 0 the assertion i s clear. Let m
>
1 and l e t P1'....pS
be graded p r i m e i d e a l s a s s o c i a t e d t o M. S i n c e qrade (1,M)
I4
.U
PIU..
Ps and because o f Lemma 111.3.1. h ( I )
t h e r e e x i s t s fl c h ( 1 ) such t h a t fl n o n - a n n i h i l d t o r o f M. L e t grade (I,M)
Q
P1 U . . . U
Q
0,
P1 U . . . U P,.
P s , t h e r e f o r e fl i s a
M1 be t h e qraded R-module M/flM,
= m - 1 and we a p p l y
Hence
then
t h e i n d u c t i o n h y p o t h e s i s . The second
s t a t e m e n t may a l s o be p r o v e d u s i n g a s i m i l a r i n d u c t i o n aroument. 111.3.5.
C o r o l l a r y . L e t R be a commutative N o e t h e r i a n araded r i n a h a v i n q
p r o p e r t y C.P. L e t P be a p r i m e i d e a l o f R h a v i n g h e i g h t n. Then t h e r e e x i s t homogeneous elements al, Ra
1
t.. .t
Proof.
...,an
i n R such t h a t P i s m i n i m a l o v e r
Ra .
F o r n = 0 t h e r e i s n o t h i n a t o prove, so we may assume n > 0
and proceed by i n d u c t i o n on n . L e t { Q,,
...,Qk
} be t h e s e t o f m i n i m a l
i = 1, . . . ,k
p r i m e i d e a l s o f R and as R i s graded, each one o f t h e Qi
,
i s graded. S i n c e h t ( P ) 3 1, P i s n o t c o n t a i n e d i n any Qi
and t h e r e f o r e k
Lemma 111.3.1. e n t a i l s h(P)
4
QIU
... U
Q,.
P i c k al
F h(P) - U
i=1
Qi.
215
111. Local Conditions for Noetherian Graded Rings
I n R/Ral,P/Ralis
a a r a d e d p r i m e i d e a l w i t h ht(P/Ral)
h y p o t h e s i s y i e l d s t h a t P/Ral i = 2,
ai,
ai
ai f o r
..., n
n-1. The i n d u c t i o n
5
i s m i n i m a l o v e r ( a *+...,an),
a r e homogeneous e l e m e n t s o f R/Ral.
the
where
Choosinq r e p r e s e n t a t i v e s
i n R i t i s o b v i o u s t h a t P i s m i n i m a l o v e r ( a l,...,an).
C o n s i d e r a g r - l o c a l r i n a R w i t h maximal w a d e d i d e a l Q and suppose t h a t R i s c o m p l e t e l y p r o j e c t i v e . L e t M c R-gr have f i n i t e t y p e . The r i n g ?i = R/annRP i s a g r . - l o c a l and
r i n q w i t h maximal q r a d e d i d e a l
= B/annRM
R has p r o p e r t y C.P. W r i t e n f o r t h e K r u l l d i m e n s i o n o f M, t h e n
n = K.dim
R
= h t ( 6 ) . By C o r o l l a r y I I I . 3 . 5 . , t h e r e e x i s t homoqeneous
,..., an c 6
elements
al
i s minimal over
such t h a t
t . . .+anM =
Our h y p o t h e s i s y i e l d s t h a t M/alM
M/alM+..
_-
. .+ Ra i s a g r - A r t i n i a n r i n q , hence M/alMt..
R/Ral+.
(R/Ra,+. =
6
On t h e o;her
..+!%,)-module.
R/annRM
+
R a p ...tRan, t h u s i t f o l l o w s t h a t M/alMt
g r . A r t i n i a n (R/annRM
t
Ral+.
..+Ran)-module;
.+anM.
Furthermore
.+an
i s a qr.Artinian
that
hand we have
- -
+ . . . t R an i n R .
&'Ra,t.
...+anM
. .+Ran
=
is a
consequently, t h e l a t t e r
module a c t u a l l y i s g r . A r t i n i a n i n R-gr. A s y s t e m o f homogeneous p a r a m e t e r s i s a s e t o f e l e m e n t s al,
...,an
A r t i n i a n i n R-gr (hence o f
f o r t h e module M w i t h K.dim M = n
c h ( Q ) such t h a t M/alM+
...+anM
is
f i n i t e lenqth i n R-qr!).
1 1 1 . 3 . 6 . Remarks. 1. As i n t h e ungraded case one may e s t a b l i s h t h a t e v e r y M-system may be i n c l u d e d i n a system o f homoqeneous p a r a m e t e r s f o r #. 2. I f R i s p o s i t i v e l y g r a d e d and such t h a t Ro i s a f i e l d t h e n t h e r e e x i s t s a system o f homogeneous p a r a m e t e r s f o r R+.
However i f Ro i s
n o t a f i e l d b u t a l o c a l r i n g w i t h maximal i d e a l w t h e n t h e g r a d e d i d e a l
B
=
'A
+ Rt may n o t have a system o f homogeneous p a r a m e t e r s , as t h e
f o l l o w i n g example shows.
B. Commutative Graded Rings
216
111.3.7. ideal o
Example. L e t V V
be a d i s c r e t e v a l u a t i o n r i n g w i t h maximal
and u n i f o r m parameter t . L e t R = V [ X I / ( t X ) ;
then R i s
N o e t h e r i a n and p o s i t i v e l y graded. Moreover R i s a g r . l o c a 1 r i n g w i t h K r u l l d i m e n s i o n 1. One e a s i l y v e r i f i e s t h a t t h e r e c a n n o t e x i s t a homogeneous al
c (%+ R + ) / ( t X ) such t h a t {al}
i s a system o f homoqeneous
parameters. I n t h e l i g h t o f t h e f o r e g o i n g o b s e r v a t i o n s w e now suppose t h a t t h e qraded N o e t h e r i a n r i n g R has t h e f o r m Ro 8 R1 8
...,
where Ro i s a f i e l d . Then
R i s g r . l o c a 1 w i t h g r . maximal i d e a l R+.
111.3.8. C
Proposition. L e t M
c
R-gr have f i n i t e t y p e and l e t { a l,....aR}
h(R+). A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r al,
...,an
t o be a
system o f homogeneous parameters f o r M i s t h a t M has f i n i t e t y p e as an o b j e c t o f Ro [al, Proof.
I f al,.
..., a n ]
. . ,an
- g r w h i l e n i s m i n i m a l as such.
i s a system o f parameters f o r M t h e n
i s A r t i n i a n i n R-gr. S i n c e On t h e o t h e r hand s i n c e graded, t h e g r a d i n g o f
k M
k
M
=
V/alM
+. . .+anM
has f i n i t e t y p e i t s g r a d e d i s l e f t l i m i t e d .
i s A r t i n i a n i n R-gr and s i n c e R i s p o s i t i v e l y
i s a l s o r i g h t l i m i t e d . Consequently
M
has t o
be a f i n i t e d i m e n s i o n a l R - v e c t o r space. Nakayama's lemma t h e n y i e l d s 0
that M i s finitely
g e n e r a t e d as a n Ro [ al,
..., a n ] -
module. Conversely,
i f we s t a r t f r o m t h e assumption t h a t M i s a f i n i t e l y g e n e r a t e d Ro [al, module, t h e n M/alM
+.. .+ anM
. . . ,a n ] -
i s a f i n i t e d i m e n s i o n a l R o - v e c t o r space
t h u s A r t i n i a n i n R-gr. N e x t l e t us check m i n i m a l i t y o f n as such. I n d e e d i f { b l,...,bm}
i s a s e t o f homogeneous elements such t h a t M i s a f i n i t e l y
g e n e r a t e d Ro [ bl,
..., b],
-module t h e n M/blM
of f i n i t e l e n g t h t o o . B u t t h e n K.dim M mcn.
5
i s A r t i n i a n i n R-gr,
i.e.
n excludes the p o s s i b i l i t y
217
111. Local Conditions for Noetherian Graded Rings
IV.3.9.
P r o p o s i t i o n . L e t M c R-gr have f i n i t e t y p e . I f al , . . . , an i s a
...,an a r e
system o f homogeneous parameters f o r M t h e n t h e elements al, a l g e b r a i c a l l y i n d e p e n d e n t o v e r R,. ~
...,an
P r o o f . L e t t i n g al,
... , X n ] w i t h g r a d a t i o n 1, ..., n . S p e c i a l i z i n a Xi
r i n g S = Ro [ X1, =
i
deg ai,
=
homomorphism o f degree 0, We have t h a t K.dim Ro [ al,.. then
o f parameters, c o n s i d e r t h e
be such a system
'0 : Ro [
. ,an]
X1,. =
d e f i n e d by p u t t i n g deg Xi +
=
ai d e f i n e s a s u r j e c t i v e r i n g
..,Xn]
+
. . ,an 1
Ro [ al,.
.
K.dim M = n. However i f Ker '0 # 0
t h e K r u l l dimension s h o u l d have dropped, so '0 i s an isomorphism.
111.3.10.
Proposition.
Let M
c
R - g r be f i n i t e l y qenerated. The f o l l o w i n g
statements a r e equivalent. 1. M i s a Cohen-Macaulay module. 2. I f al,
...,an
i s a system o f homogeneous parameters f o r M t h e n M i s
a f r e e ( g r a d e d f r e e ) Ro [ a l,...,an]
-module.
3. E v e r y system of homogeneous p a r a m e t e r s f o r M i s an M-sequence. Proof. 1 = 2. P u t S = Ro [ a l
,...,a n ] .
By P r o p o s i t i o n 111.3.9..
S i s a regular
r i n g w i t h K.dim S = n. S i n c e M i s a Cohen-Macaulay module, grade (R+,V) = K.dim M = n and
bl,...,bn =
M/blM
by
C o r o l l a r y 111.3.3.
=
t h e r e e x i s t s an M-sequence
= h(R) i = 1, . . . , n. F o r i = 1, . . . , n ~ u Mt (i) +...tbiM and p u t M = M. C o n s i d e r i n a t h e f a c t t h a t S i s a
with b
c
(0)
r e g u l a r r i n g , we may d e r i v e f r o m t h e e x a c t n e s s o f m. ltl O+ M M(i) M ( i t l ) +'> (i) (where mitl denotes mu1 i p l i c a t i o n by bi+l) t h a t D.dimSM(i+l)
'
*
= l+p.dim M
s
= M/blM+ ...+ bn M i s an S-module o f f i n i t e l e n g t h . (n) i s a f i n t e l y d i m e n s i o n a l R o - v e c t o r space. Because S i s
On t h e o t h e r hand, M Therefore M
(n) g r - l o c a l we have t h a t p dimSM(,,)=
n = p.dim M
s
(0)'
Now p.dim M
s
(n)
=
n
(i)
218
8. Commutative Graded Rings
y i e l d s p.dimSM(o) = p.dimst4 = 0, c o n s e q u e n t l y M i s a f r e e S-module. 2 = 3, as w e l l as 3 = 1 a r e r a t h e r easy.
o
Supplementay B i b l i o g r a p h y . 1. S. Goto and K. Watanabe, On graded r i n g s I, J. Math. SOC. Japan, 30 (1978) 172-213. 2 . S. Goto and K. Watanabe, On graded r i n g s I1 (Zn-graded r i n g s ) Tokyo J. Math., 1, n r . 2 ( 1 9 7 8 ) . 3. R. Fossum The S t r u c t u r e o f
ndecomposable I n j e c t i v e modules ( p r e p r i n t . )
Exercises 1. L e t R be a graded r i n g o f t y p e Z . Prove t h a t i f i d e a l PI QR-p(R)
f o r any araded p r i m e
i s N o e t h e r i a n t h e n f o r any p r i m e i d e a l Q,QR-Q(R)
i s Noetharian. 2. Suppose t h a t R i s graded N o e t h e r i a n r i n g and
2
i s a araded i d e a l i n
r R. F o r each n C Z we d e f i n e ( R g r ( a ) ) n = l i m ( R / a ) n and we denote by r Rgr(a) = 0 ( R g r ( a ) ) n .Rsr(a) i s a graded r i n g ; t h i s r i n q i s c a l l e d nFZ t h e graded a - a d i c c o m p l e t i o n o f R. Prove t h a t : 1) The r i n g
Rgr(a) i s
Noetherian
2) Rgr(a) i s f l a t R-module. 3 ) I f we denote by i : R
i( a ) Rgr
+
Rgr
(3the
c a n o n i c a l homomorphism t h e n
i s c o n t a i n e d i n t h e Graded Jacobson r a d i c a l o f
4) Ifa i s gr-maximal i d e a l t h e n
Rgr(a) i s
Rgr(a)
g r - l o c a l with i(a)Rgr
t h e gr-maximal i d e a l . 5 ) If2 i s c o n t a i n e d i n t h e graded Jacobson r a d i c a l of R t h e n t h e c a n o n i c a l homomorphism i : R
--f
Rgr(d)
i s t h e 2-adic completion o f R then R a
,gr@)
- c o m p l e t i o n of Rqr@)
i s i n j e c t i v e . Moreover i f c
Rgr@)
c
6 and
i s the
(a) as
219
111. Local Conditions for Noetherian Graded Rings
3. Let R be a graded NoetherIan r i n g and P a graded prime i d e a l . Prove
i s isomorphic t o t h e graded completion
t h a t t h e r i n g FNDR(E:(R/P))
Qi-p(R)
of t h e r i n g 4. Let R be a
a t i t s graded maximal i d e a l .
local graded r i n g with M the gr-maximal i d e a l . Suppose
t h a t R i s graded M-completed ( i . e . R = R g r ( M ) ) .
E
=
E;(R/M).
Prove t h a t :
1) I f X i s gr-Noetherian R module then H O M R ( X , E )
if Y
Write :
i s g r - A r t i n i a n and
i s a g r - A r t i n i a n R-module then H O M R ( Y , E ) i s gr-Noetherian
R-module.
2 ) The f u n c t o r H O M R ( . , E )
i s a d u a l i t y between t h e category
of g r -
Noethbrian modules and t h e category o f g r - A r t i n i a n modules. 5 . Let R = 8 R i be a graded r i n g of type Z and i CZ
T i s an indeterminate
with deg T = 1. Then we can form t h e graded ( T ) - a d i c completion o f
R [ T I ; we denote by gr R [ [ T I ] t h e ( T ) - a d i c completion o f R[ T I . Prove t h a t : 1) For every n C Z , ( g r R [ [ T I ] ) , = {
2 ) I f Ri
=
-?
i=n
a i T”’
,a c
Ri)
0 f o r every i < 0 , then
gr R “ T I 1
=
.
R [TI
3 ) I f R i s Noetherian then gr R [ [ T I ]
i s Noetherian and gr R [ [ T I 1
i s a f l a t R-module. 6 . Let R be a graded Noetherian r i n g . T h e following s t a t e m e n t s a r e
equivalent : 1) The r i n g R i s Gorenstein 2 ) inj.dim Q R - p ( R ) 3 ) i n j . dim
M -0
L2, . . . a r e qraded f r e e modules and K e r di
where Lo, L1,
c
Li
f o r a l l i 3 0. 2 ) TorL(R/m,M) R
s
Li/m Li
3 ) I f f o r a l l i < 0 Ri
=
0 and Mi
=
0 then prove t h a t t h e p - t h
R component o f t h e graded v e c t o r space Tori (Rtm,M) 4 ) p.dim M = n R
8. L e t R
Ln # 0 and Li
Ro 8 R1 8 R2 8
=
1 ) Prove
...
=
=
0 for all i > n
be a graded N o e t h e r i a n r i n g .
that there exists
with ki al,
-
i s z e r o f o r p c i.
homogeneous elements al,...,aj
C R
deg ai such t h a t R i s g e n e r a t e d as an Ro-algebra by
a2,, . . ,as.
2 ) Suppose t h a t R o i s an A r t i n i a n r i n g . I f M = Mo @MI
8
... i s
a
graded f i n i t e l y g e n e r a t e d R-module t h e n f o r a l l i 2 0, Mi i s an Po-module o f f i n i t e l e n g t h . 3 ) I f we denote by l ( M n ) t h e l e n g t h o f Mn i n Ro-mod,
we can c o n s i d e r
t h e power s e r i e s P(M,T)
=
,z0
l ( M n ) Tn
c
ZlITll.
Prove t h a t P(M,T) i s a r a t i o n a l f u n c t i o n i n T s k. f ( T ) / 11 (1-T ’ ) , where f ( T ) c Z [ T I . i=1 4)
I f m r e o v e r k. = 1, i = 1, 1
...,s,
o f t h e form
p r o v e t h a t for a l l s u f f i c i e n t l y
l a r g e n, I(Mn) i s a p o l y n o m i a l i n n. 9. L e t R be a graded N o e t h e r i a n r i n q of t y p e Zn(n 2 1 ) . I f M i s
a
f i n i t e l y g e n e r a t e d graded R-module t h e n t h e f o l l o w i n g s t a t e m e n t s are equivalent :
221
111. Local Conditions for Noetherian Graded Rings
1) M i s Cohen-Macaulay ( r es p . Gorenstein) module. 2 ) For any graded prime ideal P of V ( M ) , QR-p(M)
i s Cohen-Macaulay
(resp. Gorenstei n) QR-p( R) -module 10. Let R = Ro
Q
R1
...
Q
m = f i e l d . We p u t -
be a graded Noetherian ring where Ro = k i s a
R i . Let M 0 Mi be a graded R-module. We define i CZ M* = HOMR (M,k) and cal l i t the graded R-dual of M. Prove t h a t : Q
iz1
1) M* i s a graded R-module with
2 ) I1 =
F1**
{
i f and only i f dimkMi
= n-1
?
n - 1 and h t ( P ) 5 h t ( P )+l. 9
L e t R be a N o e t h e r i a n commutative rim. I f P i s a graded
1.1.1O.Corollary.
ppime i d e a l w i t h h t ( P ) = n, t h e n t h e r e e x i s t s a c h a i n of graded p r i m e i d e a l s = Po Proof.
2
P1
2 ... 9
P,
= p.
The s t a t e m e n t i s t r u e f o r n = 1. I f h t ( P ) = n > l t h e n t h e r e i s a
c h a i n o f d i s t i n c t p r i m e i d e a l s : Qo
2
Q, >... Qn
= P.
If Qn-l i s graded t h e n a p p l i c a t i o n of t h e i n d u c t i o n h y p o t h e s i s f i n i s h e s the proof.
Suppose t h a t Qn-l i s n o t graded and t h e n t h e f o r e g o i n g
c o r o l l a r y y i e l d s t h a t ht((Qn-l)g)
n-2. The i n d u c t i o n h y p o t h e s i s a s s e r t s
=
t h a t we may f i n d a c h a i n o f d i s t i n c t graded p r i m e i d e a l s Po c P1 c
...
cPn-2 = ( Q
)
n-1 g
.
Consider t h e graded r i n g R/(Qn-l)g
and denote by P,On-,
t h e image of P, Qnm1
homogeneous
a
element
of
P,
hence
a
E
in
on-1.
R.
=
R
Choose a nonzero
L e t Pn-l
be a m i n i m a l p r i m e
228
C. Structure Theory for Graded Rings of Type Z
ideal containing ht(P) =
Ci.. Then Pn_l
i s graded and
# P because o t h e r w i s e
Pn-l
1 f o l l o w s f r o m t h e p r i n c i p a l i d e a l theorem. P u t t i n g Pn-l
5
Pn-l/(Qn-l)g
we f i n d a c h a i n o f d i s t i n c t p r i m e i d e a l s :
.. .
Po c P1 c
=
c Pn-l
1.1.11.Corollary.
c Pn = P .
0
L e t R be a N o e t h e r i a n commutative praded r i n g and l e t
I be an i d e a l o f R, t h e n : ht(1) 5 ht(I-), Proof.
ht(1)
5
ht(1-).
S i n c e h t ( 1 ) = h t ( r a d ( 1 ) ) and ( r a d ( I ) ) - c
rad(1-)
we may reduce t h e p r o o f t o t h e case where I i s a semiprime i d e a l , I = P1
n... n
Py.Py
...
ht(I-)
=
P s , Pi p r i m e i d e a l s o f R. Then P1...Ps
c Iyields
P y c I-. L e t Q be a p r i m e i d e a l c o n t a i n i n g I-, such t h a t
h t ( Q ) . F o r some i, P y c
Q. B u t I
c Pi
hence we may assume t h a t
I = P i s a p r i m e i d e a l i n p r o v i n a t h e s t a t e m e n t . I f P i s qraded t h e n I f P i s n o t graded t h e n we have : h t ( P ) = h t ( P ) t 1. C l e a r l y q P c P- and Po # P-. I t r e s u l t s f r o m t h i s t h a t h t ( P ) < ht(P-) and 9 9 t h e r e f o r e h t ( P ) 5 ht(P-). I n a s i m i l a r way, h t ( 1 ) 5 ht(1-) may be P = P-.
es t a b 1 ished. I f one i s i n t e r e s t e d i n h a v i n g a n i c e r i n g o f q u o t i e n t s i . e . n o t b e i n g p a r t i c u l a r about i t n o t being a
c l a s s i c a l r i n g o f f r a c t i o n s , t h e n one
may a v o i d e x t r a hypotheses as i n P r o p o s i t i o n 1.1.4.
and d e r i v e a n o t h e r
graded v e r s i o n o f G o l d i e ' s theorem. W r i t e E g ( R ) f o r t h e i n j e c t i v e h u l l o f R i n R-gr. 1.1.12.Proposition.
L e t R be a graded p r i m e G o l d i e r i n g t h e n Eg(R) i s
a g r - s i m p l e g r - A r t i n i a n r i n g such t h a t R
+
t h e c a n o n i c a l r i n g morphism
Eg(R) i s a l e f t f l a t r i n g epimorphism.
229
1. NonCommutative Graded Rings
Proof.
_c_
the i n j e c t i v e hull E ( R ) of R in R-mod i s a
By Theorem 1.1.3.
simple Artinian ring a n d one e a s i l y v e r i f i e s (using Goldie's theorems) t h a t t h i s makes E 9 ( R ) a Goldie ring. I n j e c t i v i t y of E q ( R ) in R-gr e n t a i l s i s a graded regular ring ( i n Von Neumann's sense). The proof
that Eg(R)
t h a t E g ( R ) i s gr-simple gr-Artinian i s now s i m i l a r t o the ungraded case. However, E g ( R )
i s the l o c a l i z a t i o n of R a t the graded torsion theory
corresponding t o the f i l t e r generated by the graded e s s e n t i a l l e f t ideals of R ( s e e a l s o A.II.9. e t c . ) . The f a c t t h a t E 9 ( R ) i s gr-simple Artinian then e n t a i l s t h a t t h i s torsion theory i s graded perfect ( i . e . corresponds t o a graded kernel functor s a t i s f y i n g property T, see A.II.lO.).
The l a s t statement of the proposition then follows by general
1oca 1 i za t i on theory .
1.1.13.Corollary. Let h ( R ) r be the m u l t i p l i c a t i v e l y closed s e t of regular elements i n h(R). P u t E ( R ) = Q , Eg(R)
=
Qg and l e t Qh be the graded ring
of f r a c t i o n s obtained by inverting the elements of h(R)'. ring homomorphisms : R
+
Qh
-t
Qg
+
We have canonical
Q . Each l o c a l i z a t i o n involved
is a perfect l o c a l i z a t i o n (property T ) . We have t h a t Q, = Q9 i f a n d only i f graded e s s e n t i a l l e f t i d e a l s of R contain homogeneous regular elements. Proof.
R s a t i s f i e s the l e f t Ore condition with respect t o the s e t of
regular elements of R because R i s a Goldie ring. Because of Proposition
...,
A.II.lO.
i t follows t h a t R a l s o s a t i s f i e s the l e f t Ore conditions
with respect t o h ( R ) r .
All a s s e r t i o n s of the corollary follow from t h i s
and the foregoing r e s u l t s .
1.1.14.
Theorem.
Let R be a o o s i t i v e l y graded l e f t Noetherian ring and
l e t P be a graded prime ideal of R. Then
K ;
=
K
~
(
where ~ )
K
h ( p ) i s the
r i g i d kernel functor on R-gr associated t o the m u l t i p l i c a t i v e system h(G(P))
=
h ( R ) 11G ( P ) . I f R s a t i s f i e s the l e f t Ore conditions
with respect
230
C. Structure Theory for Graded Rings of Type Z
t o G(P) t h e n i t a l s o s a t i s f i e s Proof.
these conditions w i t h respect t o h(G(P)).
From C h a p t e r A, P r o p o s i t i o n 11.11.6.
, we r e t a i n t h a t - C ( K ; )
c o n s i s t s o f t h e g r a d e d l e f t i d e a l s J o f R such t h a t t h e image o f J i n
By Theorem 1.1.6. i t f o l l o w s t h a t
R / P i s an e s s e n t i a l l e f t i d e a l of R / P .
J n h(G/P)) # 6 ; the f i r s t statement f o l l o w s . Consider s c h(G(P)), r C h ( R ) and l o o k a t (Rs : r ) . and p u t L =
icy
(Rs : r i )
W r i t e r = rlt
c ( R s : r ) . Then
L i s i n - C ( K ~ because ) R s a n d (Rs:ri) Since
rl
L
i s graded, L F - C ( K ; ) .
C R such t h a t
slr
=
L
...t r n w i t h
deg rl< ...c d e g rn,
i s a g r a d e d l e f t i d e a l , and
a r e i n - C ( K ~ ) f o r a l l 1 5 i 5 n.
T h e r e f o r e t h e r e e x i s t s1
c
h ( G ( P ) ) and
rls.
Since R i s l e f t Noetherian t h i s e s t a b l i s h e s t h a t R Ore c o n d i t i o n s w i t h r e s p e c t t o h ( G ( P ) ) .
s a t i s f i e s the l e f t
231
1. Non-Commutative Graded Rings
1.2. Graded Rings S a t i s f y i n g P o l y n o m i a l I d e n t i t i e s . F o r t h e t h e o r y of r i n g s s a t i s f y i n g p o l y n o m i a l i d e n t i t i e s o u r b a s i c r e f e r e n c e s a r e t h e f o l l o w i n g books : N. Jacobson [ 53
1 % C.
Procesi
1 ; L. Rowen [ 96 1 . L e t us j u s t m e n t i o n h e r e some r e s u l t s t h a t
[ 90
a p p l y most d i r e c t l y t o t h i s s e c t i o n . L e t C be a commutative r i n g and l e t R be a C-alqebra, L e t C { X s } be t h e f r e e C-algebra i n u n c o u n t a b l y many v a r i a b l e s . I f f ( X s ) 6 C { X s }
then
o n l y f i n i t e l y many v a r i a b l e s appear i n f ( X S ) and by "abus de l a n g a g e " we say t h a t f ( X s ) i s a p o l y n o m i a l . We say t h a t f ( X s ) i s a _polynomial i d e n t i t y f o r R i f any s u b s t i t u t i o n o f rl,
...,
rn C R f o r t h e v a r i a b l e s
o f f ( X s ) makes t h e p o l y n o m i a l z e r o . The C-algebra R i s s a i d t o be a
P . I . algebra
i f i t s a t i s f i e s a polynomial i d e n t i t y f ( X s ) which i s n o t
c o n t a i n e d i n I { X s } f o r some i d e a l I o f C. The s t a n d a r d p o l y n o m i a l of deqree n i s g i v e n by :
sn
(XI
,..., Xn)
=
c
E ( 0 )
X,(l)-Xo(n)
I
OCS,
where Sn i s t h e symmetric group on n elements and of
U.
O b v i o u s l y S2(x1,
~ ( 5 i )s
the signature
x2) = x x -x x y i e l d s a p o l y n o m i a l i d e n t i t y 1 2 2 1
s a t i s f i e d by a l l commutative r i n g s . However, n o t e v e r y P . I .
algebra
s a t i s f i e s a standard i d e n t i t y , b u t :
1.2.1. Theorem. (S. A m i t s u r )
I f R i s a P.I.
s a t i s f i e s t h e p o l y n o m i a l i d e n t i t y Sn(X l,...,Xn)n, N o t e t h a t R s a t i s f i e s t h u s an The s t r u c t u r e o f p r i m e
algebra over C then R f o r s u i t a b l e n and m.
i d e n t i t y w i t h c o e f f i c i e n t s +1 o r -1.
P. I - a l g e b r a s may now be d e s c r i b e d a c c u r a t e l y
i n the following :
1.2.2.
Theorem. ( E . P o s n e r ) . I f R i s a p r i m e P . I . a l g e b r a t h e n :
C. Structure Theory for Graded Rings of Type Z
232
1. R has a c l a s s i c a l r i n g o f f r a c t i o n s , Q(R) say. 2. Q ( R ) i s a c e n t r a l s i m p l e a l g e b r a 3. Q ( R ) i s a c e n t r a l e x t e n s i o n o f R, i . e . Q ( R ) = R Z Q ( R ) ( R ) .
4. Q(R) s a t i s f i e s a l l p o l y n o m i a l i d e n t i t i e s s a t i s f i e d by R. I n t h e above theorem one i n t u i t i v e l y hopes t h a t t h e
c e n t e r o f Q(R)
i s t h e f i e l d o f f r a c t i o n s o f t h e c e n t e r o f R. T h i s i s a c t u a l l y t h e case and i t f o l l o w s f r o m a r e s u l t o f E. Formanek on c e n t r a l p o l y n o m i a l s . F i r s t n o t e t h a t P o s n e r ' s theorem e n a b l e s us t o embed p r i m e P . I . a l g e b r a s ir. m a t r i x
r i n g s o v e r commutative f i e l d s , j u s t by s p l i t t i n g Q(R) ( s e e
Section A . I . l . ) .
This f a c t h i g h l i g h t s t h e importance o f t h e study o f
p o l y n o m i a l i d e n t i t i e s s a t i s f i e d by n x n m a t r i c e s o v e r a commutative, p r e f e r a b l e i n f i n i t e , f i e l d . A p o l y n o m i a l a ( X s ) i n C{Xs} i s s a i d t o be a c e n t r a l polynomial f o r n x n matrices w i t h c o e f f i c i e n t s i n C i f every e v a l u a t i o n o f g(Xs) i n some Mn(A), where A i s any commutative C-algebra y i e l d s c e n t r a l elements. B o t h E. Formanek and explicit
I.
Razmyslov c o n s t r u c t e d
c e n t r a l p o l y n o m i a l s (Razmyslov's p o l y n o m i a l i s even m u l t i l i n e a r )
A s a consequence one may deduce : 1.2.3.
Proposition.
L e t R be a s e m i s i m p l e P . I . a l g e b r a .
Then nonzero i d e a l s of R i n t e r s e c t t h e c e n t e r n o n - t r i v i a l l y . 1 . 2 . 4 . C o r o l l a r y . If R i s a p r i m e P . I . 1.2.5.
a l g e b r a then, Z ( Q ( R ) ) = Q ( Z ( R ) ) .
Convention. If C i s n o t i m p o r t a n t o r more p r e c i s e l y , i f we c o n s i d e r
Z - a l g e b r a s we w i l l u s u a l l y speak a b o u t
P.I.
r i n g s (not : algebras).
The Formanek c e n t e r F(R) o f a r i n g R s a t i s f y i n g t h e i d e n t i t i e s o f
n x n
m a t r i c e s i s t h e s u b r i n g o f Z(R) o b t a i n e d by e v a l u a t i n a a l l c e n t r a l p o l y n o m i a l s f o r R w i t h o u t c o n s t a n t t e r m . T h i s F may be viewed as a r i g h t semi e x a c t f u n c t o r and R i s an Azumaya a l q e b r a o f c o n s t a n t
rank
233
1. Non-Commutative Graded Rings
n2 o v e r
Z(R) i f and o n l y i f R F(R) = R h o l d s .
The p . i .
degree o f a p r i m e P . I . a l g e b r a R w i l l be
fl where
N = dim
z ( Q (R ) ) Q (R,
If P i s a p r i m e i d e a l o f R t h e n we d e f i n e p . i . deg P = p . i . deg R / P . 1 . 2 . 6 . Theorem (M. A r t i n ) . I f R s a t i s f i e s t h e i d e n t i t i e s o f n x n m a t r i c e s t h e n R i s an Azumaya a l g e b r a of c o n s t a n t r a n k n2 o v e r i t s c e n t e r i f and o n l y i f f o r e v e r y p r i m e i d e a l P o f R, p . i . deq P = n .
... o f I , ... .
For extensions, applications e t c . 3
above we r e f e r t o [
I , [ 90
t h e v e r y i n t e r e s t i n g theorem
L e t us now r e t u r n t o t h e s i t u a t i o n where R i s g r a d e d o f t y p e Z and c o n t i n u e w i t h n o t a t i o n s as i n 1.1.13.. 1.2.7.
I f R i s a p r i m e graded P . I . r i n g t h e n Qh = Qg i . e
Proposition.
Qg i s a g r - s i m p l e g r - A r t i n i a n ' r i n g o f f r a c t i o n s o f R .
Put C = Z(R).
Proof,
I f t h e r e e x i s t s a c F h(C) w i t h deg c > o t h e n
1. a p p l i e s , s i n c e c i s r e o u l a r . I f t h e r e i s a c c h(C) 2 o t h e n t h e graded r i n g o f f r a c t i o n s o f R a t I l , c , c , . . . I ,
Proposition I.1.4., w i t h deg c
o
1, i t f o l l o w s t h a t
has a g r - s e m i s i m p l e g r - A r t i n i a n r i n g o f f r a c t i o n s w h i c h o b v i o u s l y
has t o c o i n c i d e w i t h Qh = Q g . So we a r e l e f t w i t h t h e p r o b l e m C = Co. BY t h e graded v e r s i o n o f Wedderburn's theorem ( s e e A . I . 5 . ) , Qg 2 Mn(A)(cJ
Posner's
f o r some graded d i v i s i o n r i n g , n c N and d
c
Zn.
By
theorem, Q i s f i n i t e d i m e n s i o n a l o v e r t h e f i e l d o f f r a c t i o n s
of C and t h u s i t f o l l o w s t h a t f o r e v e r y x c h(D) we have an e q u a t i o n c 0 xm
+
... +
Cm-l
x + cm = 0 w i t h ci
C, i = 0, ...,rn.
B u t C = Co i m p l i e s
C. Structure Theory for Graded Rings of Type Z
234
deg x
=
0 hence D
=
DO, and t h e l a t t e r e n t a i l s t h a t P i s a s k e w f i e l d ,
thus Q g = Q f o l l o w s . I n case D i s a n o n - t r i v i a l l y graded d i v i s i o n a l g e b r a and a l s o a P . I . r i n g then t h e r e s u l t s of A.I.4. e n t a i l t h a t D = Do [ X , X - l , q ~ ] where skewfield and
'p
i s an automorphism of
Do some power of which i s an inner
automorphism ( c f . a l s o 6 . Cauchon [ 17 ] ) . P u t m k = Z(Do)"
the f i x e d f i e l d of
k [T,T-l]
where T = X Xe,
9
in Z(Do).
some e
C
Do i s a
N, X
2
=
[Do
: Z(Do) ]
The c e n t e r Z ( D ) e q u a l s Since D i s an Ore domain
F .D :
i t has a skewfield of f r a c t i o n s Q ( D ) = DO(X,,?)
having k ( T ) f o r i t s c e n t e r .
Furthermore one e a s i l y v e r i f i e s t h a t : [ Do(X,*o):k(T)] = [ D : k [T,T-'Il
1.2.8.
=
Theorem. (Graded v e r s i o n of P o s n e r ' s theorem).>
A graded prime r i n g R i s a P . I . r i n g i f and only i f R i s a graded o r d e r
of M n ( K ) ( ~ ) f o r some g r - f i e l d K , and some Proof.
4
C
Zn, n
C l e a r l y , each (graded) o r d e r o f M n ( K ) ( < )
C
N.
i s a P . I . r i n g . Conversely,
i f R i s a prime graded P . I . r i n g then by P r o p o s i t i o n 1 . 2 . 7 . i t follows t h a t Q h = Qg i s a gr-simple g r - A r t i n i a n r i n g o f f r a c t i o n s of R . By t h e graded v e r s i o n of Wedderburns's theorem ( c f . A . I . 5 . ) ,
Qg = M n , ( D ) ( d ' )
f o r some n ' F N, d ' c Z n ' . I f D = Do then t h e theorem follows from Qg = Q ( R ) s i n c e the l a t t e r can be s p l i t by some t r i v i a l l y graded commutative f i e l d . Now assume D # Do i s a n o n - t r i v i a l l y graded d i v i s i o n a l g e b r a . The remarks preceding the theorem imply t h a t
Consider a maximal commutative graded s u b r i n g of D , L s a y . I t i s n o t hard t o v e r i f y t h a t L i s a g r - f i e l d and t h a t i s a l s o maximal a s a commutative subring of 0.
235
1. NonCornrnutative Graded Rings
W r i t e Q(L) = k ( T )
@
k [ T,T-'
1
L ; t h i s i s a commutative s u b r i n g o f Q ( D ) .
Ify c Q(D) commutes w i t h Q ( L ) t h e n f o r some s 6 k [ T,T-'],sy
c L because
L i s a maximal commutative s u b r i n g o f D. Consequently, Q ( L ) i s a
maximal commutative s u b f i e l d of Q f D ) , hence a s p l i t t i n g f i e l d . Since
Now, D
D
@
k [ J,J-l]
@
k [ T,T-l]
L i s a p r i m e r i n g , we g e t :
L i s a gr-c.s.
a. w i t h c e n t e r L hence i t i s i s o m o r p h i c
t o some M k ( A ) ( d ) f o r some graded d i v i s i o n a l a e b r a A, k c
N,
d
k c Z .
C e n t r a l l o c a l i z a t i o n a t t h e p r i m e i d e a l 0 y i e l d s = Mk(Q(A)) 2' Mn(Q(L)), s o i t f o l l o w s i m m e d i a t e l y t h a t k = n and Q(A) = Q ( L ) hence A i s commutative o r A = L. F i n a l l y we have e s t a b l i s h e d t h a t Mn(L) 2
D
60
k [ T,T-l]
L
becomes an isomorphism o f graded r i n g s i f t h e g r a d a t i o n chosen on Mn(L) i s d e s c r i b e d by d c Zn.
1.2.9.
Remark. P o s n e r ' s theorem i m p l i e s t h a t e v e r y (graded) P . I .
ring
can be embedded as an o r d e r i n a f u l l m a t r i x r i n g Mn(K) o v e r a commutative f i e l d K. There e x i s t s an i n f i n i t e number o f n o n - i s o m o r p h i c Z - g r a d a t i o n s on M,(K)
e x t e n d i n g t h e t r i v i a l g r a d a t i o n o f K. Unless R i s i t s e l f t r i v i a l l y
graded none o f t h e s e g r a d a t i o n s w i l l e x t e n d t h e q r a d a t i o n o f R. The above graded v e r s i o n o f P o s n e r ' s theorem may t h u s be t h o u g h t o f as a s t a t e m e n t a b o u t t h e e x t e n s i o n o f t h e a r a d a t i o n of R t o some " n i c e " s u b r i n g o f M,(K). 1.2.10. = P.i.
Proof. n 6
N
P r o p o s i t i o n . I f R i s a graded P . I . deq R/P
4
r i n g t h e n p . i . deq R/P =
f a r e v e r y P c Spec R.
T h a t p . i . deg R / P 5 p . i . deq R / P
i s o b v i o u s . Now i f f o r some 9 t h e s t a n d a r d i d e n t i t y Sen vanishes on R/P, t h e n t h e i d e a l g e n e r a t e d
C. Strucrure Theory for Graded Rings of Type Z
236
be t h e e v a l u a t i o n s o f S2n i n h(R) i s a graded i d e a l o f R because Szn i s m u l t i l i n e a r , thus i f t h i s i d e a l i s i n P then i t i s i n P a l s o vanishes on R/P p.i.
deg R/P
1.2.11.
9
5
p.i.
g
a'
T h e r e f o r e Szn
and t h e r e f o r e : deg R/P.
C o r o l l a r y . L e t R be a P . I .
r i n g satisfying the identities o f
n x n m a t r i c e s and l e t R be graded o f t y p e Z. The r a d i c a l r a d ( I n ) o f ( c f . [ 96 1 , f o r more d e t a i l s , In = i d e a l generated t h e c e n t r a l k e r n e l In by t h e e v a l u a t i o n s i n R o f a m u l t i l i n e a r c e n t r a l p o l y n o m i a l f o r n x n m a t r i c e s ) i s a graded i d e a l o f R. Proof.
Prime i d e a l s c o n t a i n i n g Ina r e e x a c t l y t h o s e h a v i n a p . i .
degree < n ( c f . [ 90
I).
Hence P
3
In i f and o n l y i f P g
3
In, follows
from the foregoing p r o p o s i t i o n . 1.2.12.
Corollary.
(Graded v e r s i o n o f
r i n g R i s an Azumaya a l g e b r a o f
M.
c o n s t a n t r a n k o v e r Z(R),
e v e r y graded p r i m e i d e a l has p . i .
i f and o n l y i f
degree equal t o p . i . deg R, and i f
and o n l y i f f o r e v e r y maximal i d e a l M o f p.i.
A r t i n ' s Theorem).A graded P . I .
R t h e p . i . degree o f
M
9
equals
deg R.
P r o o f . S t r a i g h t f o r w a r d f r o m 1.2.10.
and 1.2.6.
.
Note t h a t i n g e n e r a l graded maximal i d e a l s , maximal graded i d e a l s and t h e graded p a r t s Mg o f maximal i d e a l s M a r e
very
poorly
r e l a t e d . L e t us
g i v e a t r i v i a l example t o show t h a t C o r o l l a r y 1.2.12 does a l l o w t o economize a l o t
when c h e c k i n g p . i . d e g r e e ' s o f p r i m e i d e a l s i n o r d e r t o
v e r i f y w h e t h e r a graded P . I . r a n k o r n o t . Take R = D [ 0
X,,p]
r i n g R i s an Azumaya a l q e b r a o f c o n s t a n t ,fp
an automorphism o f t h e s k e w f i e l d Do;
t h e n R i s Azumaya if and o n l y i f p . i .
deg(X) = p . i . deg R and i t i s
known t h a t t h i s happens o n l y i f f o r some 1 c U ( D o ) ,
X X i s c e n t r a l i n R.
1. NonCommutative Graded Rings
1.2.13.
237
Proposition. If R i s a graded P . I . ring such t h a t i t s center i s
a g r - f i e l d then R i s an Azumaya algebra of constant rank. Proof. Let f be a m u l t i l i n e a r central polynomial f o r R . Clearly, f
cannot vanish f o r every homogeneous s u b s t i t u t i o n i . e . there
e x i s t u1 ,..., u k
F h ( R ) such t h a t 0 # f ( u l ,..., u k ) 6
h ( C ) where C = Z ( R ) .
Since homogeneous elements o f C are i n v e r t i b l e by hypothesis, i t follows
t h a t f ( u l , ..., u k ) of rank n 1.2.14.
2
.
Corollary. Let G a f i n i t e abe i a n group of automorphisms of the
f i n i t e dimensional skewfield Do. The terated twisted polynomial ring -1 Do [ Xo,Xo , IS C G I i s an Azymaya algebra. Proof.
A r a t h e r straightforward
modification of the techniques used
in foregoing r e s u l t s . Let us conclude t h i s section
with a r e s u l t concernina s p l i t t i n q f i e l d s
and s p l i t t i n g g r - f i e l d s , f i r s t an example : 1.2.15. Example. Consider D = C[ X,X-’,q] where ‘p i s complex conjugation. 2 -2 2 -2 Then Z ( D ) = 9 X , X ] and [ D:Z(D)] = 4 . Clearly L = E [ X , X ]and K=R[ X,X-’1 M2(L)(d) f o r c e r t a i n are s p l i t t i n g f i e l d s for D in the sense t h a t D rn L 2 Z(D) d C Z determined by the n a t u r a l gradation of the tensor product ( s i m i l a r
2
f o r K ) . Also, t(X ) andR(X) a r e s p l i t t i n g f i e l d s of C(X,cp) and both a r e 2
graded r e a l i z a b l e f i e l d s ( i n the sense of B . I . 3 . ) over R(X ) . On the other handR(Xti) i s a maximal commutative subfields of t ( X , ( p ) which i s n o t graded r e a l i z a b l e . 1.2.16. Corollary.
Let D be a graded division P . I . algebra then Q(D)
has graded r e a l i z a b l e s p l i t t i n g f i e l d s . Proof.
Write D = D o [ X , X - l , q ] ,
Q(0) = D o ( X , , ~ ) . I t i s e a s i l y checked
t h a t a maximal graded commutative subring of D , L say, y i e l d s a r e a l i s a b l e S p l i t t i n g f i e l d Q ( L ) of Q ( D ) .
238
C. Structure Theory for Graded Rings of Type Z
1.3. F u l l y Bounded Graded Rings.
Throughout t h i s s e c t i o n R i s a graded r i n g o f t y o e Z which i s suDposed t o be l e f t N o e t h e r i a n . I n t h i s and t h e f o l l o w i n g s e c t i o n we i n t r o d u c e c l a s s e s o f graded r i n g s g e n e r a l i z i n g i n d i f f e r e n t d i r e c t i o n s t h e c l a s s o f graded P . I .
. We use r e s u l t s and
r i n g s , s t u d i e d i n S e c t i o n 1.2.
r e s u l t s and n o t a t i o n s from A . I I . 7 . ,
A.II.9.,
A.II.ll
.,...
L e t Zg(R) be t h e s e t o f e q u i v a l e n c e c l a s s e s uo t o s h i f t s and isomorDhisms
of t h e indecomposable i n j e c t i v e s i n R-gr. L e t +g : Zg(R)
E
t h e map d e f i n e d by a s s o c i a t i n g t o a c l a s s Ass(E) 1.3.1.
,
where E r e p r e s e n t s
Spec ( R ) be g
t h e graded p r i m e i d e a l
5.
Lemma.
I f P t Spec ( R ) t h e n Eg(R/P) 9 t h e map ag i s s u r j e c t i v e . Proof.
+
i s P-cotertiary.
I n particular,
Decompose Eg(R/P) as a d i r e c t sum o f indecomposable i n j e c t i v e s i n
R-gr, say Eg(R/P) = El @...@
En.
From P r o o o s i t i o n 1.1.8.
Eg(R/P) i s g r - s i m o l e l e f t g r - A r t i n i a n ,
hence E4(R/P) =
we r a t a i n t h a t
L1 Q...QLr
where
t h e Li a r e m i n i m a l graded l e f t i d e a l s o f Eg(R/P) such t h a t f o r each i, j t 11, ...,r ) t h e r e i s an nii,j
c
Z such t h a t Li
5
Krull-Remak-Schmidt theorem we o b t a i n : r = n and Ei
L . ( n . . ) . By t h e J 1J = Ln(i), f o r some
p e r m u t a t i o n n o f { l ,..., n } . T h e r e f o r e we have : Ass(Ei)
= Ass(L
u ( i ) ) = ASS(Lv(j)(nv(i),v(j) T h i s y i e l d s t h a t Eg(R/P) i s P - c o t e r t i a r y .
) ) = Ass(L
u(j)
= Ass(Ej)
R e c a l l t h a t a p r i m e r i n g i s bounded i f e v e r y e s s e n t i a l l e f t i d e a l c o n t a i n s a nonzero i d e a l . A r i n g i s f u l l y bounded i f each o f i t s p r i m e f a c t o r r i n g s i s bounded. A p r i m e graded r i n g R i s s a i d t o be graded bounded i f each gr-essential
l e f t i d e a l c o n t a i n s a nonzero graded i d e a l . A graded r i n g
R i s g r a d e d f u l l y bounded i f e v e r y
R/P,
P F Spec (R), i s graded bounded. g
239
1. Non-Commutative Graded Rings
1.3.2.
Theorem.
L e t R be a l e f t N o e t h e r i a n g r a d e d r i n q .
The f o l l o w i n g p r o p e r t i e s a r e e q u i v a l e n t : 1. ...
>
n . Any j e J may be written
and j x = 0 y i elds t h a t j T x = 0. 1 Ol = 0. Since xo f 0 by assumption, i t will suffic e 1 1 t o show t h a t J" i s es s en t i al as a l e f t ideal of R , (J- i s defined i n
as j
j
Tl <m Consequently J" xu
T~
>...>
T~
Section A.II.2.) Since J i s an es s en t i al l e f t ideal of R, we have tha t f o r any y c h ( R ) there e x i s t s a n r
r
=
r
+. . .t r
Y1
Yt
with y1 >-..
. > yt,
highest degree f o r which r
C
R such t h a t 0 f ry c J . Writing
we obtain r y yi Consequently
E
3- where yi i s the
y # 0. i n t e r s e c t s every "i . graded l e f t ideal of R in a nontrivial way, i . e . J" i s gr-e sse ntia l.
Now Lemma A.I.2.8. y i el d s t h a t J- i s a n es s en t i al l e f t ideal of R , so J- xol = 0 leads t o a contradiction and so R i s l e f t nonsingular A symmetrical argumentation may be used t o derive t h a t R i s a l s o r i g h t
nonsi ngul a r
.
1.5.10. Proposition.
Let G be an ordered group and l e t R be a d r - r e g u l a r
ring of type G . Then the conditions of Proposition 1.5.8. being equivalent t o R i s Gr-abelian regular, ar e also equivalent t o : 6 . R has no
nonzero nilpotent elements.
7. Every nonzero graded l e f t ideal of R contains a nonzero central idempotent. Proof.
That 6 . i s equivalent t o 3 in Proposition 1.5.8. i s obvious when
G i s ordered. Furthermore, condition 5 in Proposition I .5.8. obviously
implies 7. The implication, 7
-
1 follows from Proposition 1.5.9. by the
arguments used in proving Theorem 3.2. in
[
41 1 .
C. Structure Theory for Graded Rings of Type Z
262
C o r o l l a r y . L e t R be G r - r e g u l a r o f t y p e C- where G i s a n o r d e r e d
1.5.11.
group. I f R i s g r - a b e l i a n r e g u l a r t h e n : 1. A l l i d e m p o t e n t elements o f R a r e c e n t r a l . 2 . Every m i n i m a l graded p r i m e i d e a l i s c o m p l e t e l y p r i m e . Proof. 1. I f i i s i d e m p o t e n t i n R t h e n ( 1 - i ) R i c o n s i s t s o f n i l p o t e n t elements so, i f R i s G r - a b e l i a n t h e n ( 1 - i ) R i = 0 by P r o p o s i t i o n 1.5.10. and t h e n
i i s c e n t r a l because R i s semiprime and because o f Lemma 1.5.7. 2. I f P i s a minimal
.
graded p r i m e i d e a l o f R t h e n R/P i s a g r - d i v i s i o n
r i n g because o f P r o p o s i t i o n I . 5 . 8 . ,
2.
.
S i n c e C: i s ordered, g r - d i v i s i o n
r i n g s o f t y p e G c a n n o t have z e r o - d i v i s o r s . The s t r u c t u r e o f G r - a b e l i a n r i n g s may be c o m p l e t e l y d e s c r i b e d i n t h e s t r o n g l y graded case f o r a r b i t r a r y groups. We need : Lemma. L e t A be an a b e l i a n r e g u l a r r i n g . Then e v e r y i n v e r t i b l e
1.5.12.
A-bimodule P i s f r e e as a l e f t ( o r r i g h t ) A-module. Suppose P, Q a r e A-bimodules such t h a t :
Proof.
P ~ ~ Q ~ A ~ Q Q o P A A We t h e n know t h a t P and Q a r e f i n i t e l y g e n e r a t e d p r o j e c t i v e l e f t ( r i g h t ) A-modules. The s t r u c t u r e t h e o r y f o r f i n i t e l y g e n e r a t e d p r o j e c t i v e modules over a r e g u l a r r i n g , c f . [ 4 1 P = RX
1
8
y i e l d s t h a t we may w r i t e
... 8 RX n .
D e f i n e a morphism f : A l e f t A-morphism). Ker f = { a
1,
c A,
+
P by f ( a ) = axl t ax2
+
... +
ax,.
(f i s a
We have t h a t :
On annA(xi). Since A i s a b e l i a n regular, i=l each l e f t i d e a l i s an i d e a l , hence Ra c aR. So i f a x . = 0 t h e n a Rx = 0;
f ( a ) = 0} =
t h e r e f o r e any a 6 K e r f w o u l d a n n i h i l a t e P. B u t P i s
f a i t h f u l since i t
1. Non-Commutative Graded Rings
263
i s i n v e r t i b l e , hence f i s i n j e c t i v e . Since Q i s p r o j e c t i v e a s a r i g h t A-module we have the e x a c t sequence : 0
+
l?f
Q63A
A
Q m P - A A
The map 1 8 f i s l e f t A-linear and so we o b t a i n
0
--f
AQ
2
AA.
NOW
Q i s f i n i t e l y generated a s a l e f t A-module, hence t h e r e exists an
idempotent i i n t h e c e n t e r of A such t h a t AQ
A i . From ( l - i ) A i = 0 i t
follows t h a t ( 1 - i ) Q = 0 but Q i s f a i t h f u l a s a l e f t A-module s i n c e i t i s invertible,
hence 1-i = 0 and A Q 2 A A f o l l o w s . S i m i l a r l y AP
AA
f o l 1ows. 1.5.12. Theorem. Let R be a s t r o n g l y graded r i n g of type G . The following s t a t e m e n t s a r e e q u i v a l e n t : 1. R i s Gr-abelian r e g u l a r . 2 . R i s t h e c r o s s e d product Re [
E
.aR] where Re
i s an a b e l i a n r e g u l a r
r i n g ( f o r t h e s t r u c t u r e o f t h e crossed products s e e S e c t i o n A . I . 3 . ) . Proof. 2 = 1. Follows immediately from Corollary 1 . 5 . 3 . and then from D e f i n i t i o n
1.5.6. 1
2 From t h e lemma above and t h e r e s u l t s o f S e c t i o n A.1.3. i n p a r t i c u l a r
Lemma A. I .3.20. 1.5.13. C o r o l l a r y .
Let R be a s t r o n g l y graded r i n g of type Z then the
following s t a t e m e n t s a r e e q u i v a l e n t : 1. R i s Gr-abelian r e g u l a r . 2. R
=
Ro [ X,X-l,fo] where
'0
i s an automorphism o f t h e a b e l i a n r e g u l a r
r i n g Ro. 1 . 5 . 1 4 . Example. I f R i s graded o f type Z but not s t r o n g l y graded then
t h e r e e x i s t Gr-abelian r e g u l a r r i n g s , which need n o t be o f the form
C. Structure Theory for Graded Rings of Type Z
264
Ro
.
[ X,X-',?]
L e t k be a f i e l d and f o r each n F N consider k [ X
where t h e g r a d a t i o n on k [ X (,), Consider B = of all b
=
(bl,
1 Xin)]
X-' ] (n)' (n) i s d e f i n e d by p u t t i n g deg X t n ) = n.
n k[X X - l ] and l e t A be t h e s u b r i n g o f B c o n s i s t i n g ~ F N (n)' (n) bp,
...,
bn, .. . ) i n B which become e v e n t u a l l y constant
s e r i e s . I t i s c l e a r t h a t A i s Gr-abelian r e g u l a r and A,
i s the regular
subring o f
c o n s i s t i n g o f t h e e v e n t u a l l y constant s e r i e s . C l e a r l y ,
A # A.
f o r any p o s s i b l e X one m i g h t t h i n k o f .
[X,X-l]
L e t R be a graded r i n g o f type G; we d e f i n e :
c
R i R ( l - i ) R , where i runs through t h e s e t o f a l l idempotent i elements o f R ;
N(R) =
Ng(R) = E R j R ( l - j ) R , where j runs through t h e s e t o f idempotent
j
homogeneous elements o f R; S(R) = t h e s u b r i n g ( w i t h o u t i d e n t i t y perhaps) generated by t h e n i l p o t e n t elemenst o f R; Sg(R) = t h e graded s u b r i n g o f R generated by t h e homogeneous n i l p o t e n t elements o f R. 1.5.15.
Theorem. L e t R be graded o f type G where G i s an ordered group
and suppose t h a t R i s a Gr-regular r i n g
.
Then Ng(R) = N(R) = S(R)
=
Sg(R)
and R/Ng(R) i s g r - a b e l i a n r e g u l a r . Proof.
I f f F R/N9(R) i s an idempotent homogeneous element then t h e r e
e x i s t s an idempotent homogeneous j F R which l i f t s f. Since j R ( i - j ) c Ng(R) i t f o l l o w s t h a t 0 = f ( R / N g ( R ) ) ( l - f ) and thus f i s c e n t r a l i . e . R/Ng(R)
i s Gr-abelian r e g u l a r . By P r o p o s i t i o n 1.5.10.,6.,
we o b t a i n
Sg(R) c S(R) c Ng(R) c N(R). I f E i s an idempotent, E F N(R) image i n R/Ng(R) i s c e n t r a l because o f C o r o l l a r y 1.5.10. ER( I-E) c Ng(R), hence N(R) c Ng(R) and N(R) = Ng(R) .
, then i t s
. Therefore
I. @on-CommutativeGraded Rings
Let
US
now proceed t o show t h a t N g ( R )
c
sg(R). Let
265
j be an idemPotent
homogeneous elements of R, then : R j R ( 1 - j ) R = j R j R ( l - j ) R + ( 1 - j ) R j R(1-J) R c j R(l-j)R
+ (1-j)R J R.
sg(R)
Moreover : j R ( 1 - j ) R = j R ( l - j ) ( l - j ) R j + j R(1-j) R(1-j) Similarly
:(l-j)R j R c
Therefore
zR j
j R(l-j)R
s~(R). = Ng(R) c sg(R).
I n the sequel R will be graded of type Z . 1.5.16. Proposition. I f R i s Gr-regular of type 2 then J(R) Proof.
=
0.
In Section A . I . 7 . we established t h a t the Jacobson radical of
a Z-graded ring i s a graded i d e a l . As a graded ideal of a Gr-regular
or e l s e J(R)
ring J(R) then has t o contain a n idempotent element
= 0.
Since the f i r s t i s impossible the l a t t e r i s a f a c t . 1 . 5 . 1 7 . Proposition.
Let R be a Gr-regular ring of type Z such t h a t
f o r each prime ideal p of Ro there e x i s t s a graded prime ideal P of
R such t h a t Po = p . Then the following properties hold : 1. For a l l n 6 Z , Rn R-, 2 . For a l l n
F
Z , RRn
=
=
RR-,
R-n Rn i s an ideal of R .
Proof. 1. I n a Gr-regular ring each l e f t ( r i g h t ) graded ideal i s generated
as a l e f t ( r i g h t ) ideal of R by i t s p a r t of degree zero. Thus RR1 = RIo where I.
R1 =
=
R-l
R1 i s a n ideal of Ro. From RR1 = RR-lR1
RIR-l
R1 = Jo R1 where Jo
RIIo R-l
a n d RIIo = JoR1, R-l
=
=
R1 R-l.
Hence : Jo
Jo = I, R-l.
Since R,
=
i t follows t h a t
RIR-l.
RIR-l
=
i s regular, I 0
and J o a r e semiprime i d e a l s . I f p i s a prime ideal of R, containing I.
C. Structure Theory for Graded Rings of Type Z
266
t h e n t h e hypothesis on R i m p l i e s t h a t t h e r e i s a graded prime i d e a l P o f R such t h a t Po = p i . e . P contains R I. R1 IoR-l
R. Consequently p = Po contains
Jo and thus we o b t a i n J, c I o . S i m i l a r l y , by symnetry, we
=
f i n d I, c
Jo and I.
= Jo. I n f o r m a l l y t h e same way i t can be shown
t h a t Rn R-n = R-n Rn f o r a l l n t
Z.
Consider RR1 Rn f o r some p o s i t i v e n c Z.
2.
Rn = R1 I.
C l e a r l y : I.
R 1: Rn-l
=
R I o Rn-l
=
Rm c
RR1 R n - l .
=
Rn-l
and thus KR1 Rn c RR1 1, Rn-l
R e p e t i t i o n o f t h i s argumentation
f o r a71 m c 0 i n Z. B u t s i n c e RRl
RR-l
= RR-lR1
= RR1 R-l
i t f o l l o w s t h a t RR1 i s an i d e a l o f R. Now note t h a t :
= RR-l
Rn R1 = (RRl)n+l
= (RIR)n+l
holds f o r a l l n 6 Z.
= R1 Rn
To prove t h a t RRn i s an i d e a l we proceed
by i n d u c t i o n
on n (we w r i t e
t h e p r o o f f o r n > 0, the case n < 0 i s s i m i l a r ) . Suppose we know t h a t RRm i s an i d e a l f o r each m, 0 < m < n. I f f o r some m, then RRm i s as r i g h t i d e a l generated by i t s RRm = R-,Rm
R1 Rt RR
n
R
R = R,
R-,,
R
=
R,
=
t
Rt R1 f o r a l l 1 t Z. = RR-,
Rn Rt = RRn R-,
R f o l l o w s . Consider RRn Rt
R t c RRn Rt-n.Repetition where to = t mod n
hence RRnRt c RRn as b e f o r e . I f t
0 and choose a r e p r e s e n t a t i v e Y i n h(R) f o r X. O b v i o u s l y Y
c
h(G(P))+ and f o r any s
c
enough. I f r 6 h(R), s
c
h(G(P)),
c
Ym s
h(G(P))+ f o r some m l a r g e
h ( G ( P ) ) + a r e such t h a t r s = 0 t h e n t h e l e f t
Ore c o n d i t i o n s w i t h r e s p e c t t o h(G(P)) e n t a i l s t h a t s ' r = 0 s'
c
h(G(P)) and t h u s Y m s ' r = 0 f o r some Yms'
f o r given r
c
h(R), s
c
h ( G ( P ) ) + we f i n d r '
c
c
h(G(P))+.
f o r some
Furthermore,
h ( R ) , s ' c h ( G ( P ) ) such
t h a t s ' r = r ' s , hence Y m s ' r = ( Y m r ' ) s . Consequently, i f R s a t i s f i e s t h e l e f t Ore c o n d i t i o n s w i t h r e s p e c t t o h(G(P)) t h e n R+ s a t i s f i e s t h e s e c o n d i t i o n s w i t h r e s p e c t t o h(G(P))+. Conversely, i f r where m,l
c
h(R), s
c
h(G(P)) a r e such t h a t r s = 0 t h e n YmrsY1 = 0
a r e chosen such t h a t mdeg Y
+
deg r > 0,
1 deg Y
The l e f t Ore c o n d i t i o n s w i t h r e s p e c t t o h(G(P))+ y i e l d slym, some s '
c
p u t t i n g s" = s l y m we f o u n d s "
h(G(P))+,
s " r = 0. A l s o , f o r g i v e n s t h a t Y m r c R+, some r '
c
the proof.
Yls
h(R+),
c
c
h(G(P)),
r
c
c h(G(P))
+
deg s > 0. =
0 for
such t h a t
h(R) we f i n d a g a i n m, 1 such
1
h ( G ( P ) ) + . T h e r e f o r e we o b t a i n : s'Ym = r ' Y s, f o r
s ' c h(G(P))+
and s i n c e s l y m
c
h(G(P)) t h i s f i n i s h e s
1. Non-Commutative Graded Rings
1.5.22.
269
C o r o l l a r y . If P i s a graded prime i d e a l o f t h e Gr-regular r i n g R
such t h a t P
3
Ng(R) then R‘
s a t i s f i e s t h e l e f t Ore c o n d i t i o n w i t h r e s p e c t
t o h(G(P+)). Proof.
R s a t i s f i e s t h e l e f t Ore c o n d i t i o n w i t h r e s p e c t t o h(G(P)) i f
and o n l y i f R / K ~ ( R )s a t i s f i e s t h e l e f t Ore c o n d i t i o n w i t h respect t o t h e image o f P, where
K
h(G(P)) (see A . I I . 9 . ) .
P
i s the r i g i d
kernel f u n c t o r associated t o
B u t K ~ ( R )= P; indeed, i f p c h(P) then p = p y p
y i e l d s ( 1 - p y ) p = 0 w i t h I - p y c G(P) ! Furthermore R/K
P
(R) i s of t h e
f o r some s k e w f i e l d A . E v i d e n t l y R/P s a t i s f i e s
form A o r A [ X,X-’,,p]
t h e Ore c o n d i t i o n s w i t h r e s p e c t t o zero. The statement i s t h e n a d i r e c t consequence o f P r o p o s i t i o n 1.5.21. 1.5.23.
Remark.
I n a way f o r m a l l y s i m i l a r t o t h e above we can prove
t h a t a Gr-regular r i n g R o f type Z s a t i s f i e s t h e l e f t and r i g h t Ore c o n d i t i o n s w i t h r e s p e c t t o G(P), and h(G(P)), f o r any graded prime i d e a l P o f R c o n t a i n i n g Ng(R). 1.5.24.
An example : t h e Weyl-algebra.
L e t k be any f i e l d and l e t S = k { x , y } be t h e s p e c i a l i z a t i o n o f t h e f r e e algebra i n two i n d e t e r m i n a t e s d e f i n e d by t h e r e l a t i o n X Y - Y X = 1. Put deg X = 1, deg Y = -1. Then S i s a graded r i n g such t h a t So = k [ X Y ]
.
The t o t a l graded r i n g o f f r a c t i o n s o f S i s then given by Qg(S) = k(YX) [ X , X - ’ , Q ] by ,p(XY) = 1 + XY.
where
I f char k
,Q
i s t h e automorphism o f k(XY) d e f i n e d
# 0. then
‘P
has f i n i t e o r d e r and then Qg(S)
i s a f i n i t e dimensional g r - d i v i s i o n algebra over i t s c e n t e r k(XY)”[ T,T-’l where T = Xn,
some n 6
N.
I f char k = 0 then S i s t h e f i r s t Weyl-algebra
i . e . a simple Ore domain which i s n o t a r e g u l a r r i n g . Therefore we see t h a t Qg(S) i s a simple domain which i s G r - r e g u l a r and even G r - a b e l i a n r e g u l a r b u t c e r t a i n l y n o t a r e g u l a r r i n g . I t i s well-known
that
C. Structure Theory for Graded Rings of Type Z
270 S
k [XI
[y, 6 J
where 6 denotes t h e " d e r i v a t i v e w i t h r e s p e c t t o X " .
I t may be s u r p r i z i n g t o see how t h i s d i f f e r e n t i a l p o l y n o m i a l r i n g has
a graded r i n g o f f r a c t i o n s w h i c h i s i s a c r o s s e d p r o d u c t . The g r a d a t i o n on S shows t h a t t h e p a r t o f degree z e r o o f a s i m p l e and graded domain need n o t be simple ! One can c a l c u l a t e f u r t h e r m o r e t h a t t h e s u b r i n g s k [ XnYm]
w i t h (n,m) = 1 a r e maximal graded commutative s u b r i n g s o f S
and t h e s e a r e a l s o maximal as commutative s u b r i n g s o f S . F i n a l l y l e t us m e n t i o n some s t a t e m e n t s a b o u t t h e i n c o m p a t i b i l i t y of g r a d a t i o n s and commn Von Neumann r e g u l a r i t y . A non
# 0
t r i v i a l l y graded
i f Rn c o n s i s t o f n i l p o t e n t elements f o r
r i n g i s s a i d t o be n i l g r a d e d all n
-
; I f R i s n o t n i l g r a d e d t h e n we say t h a t t h e g r a d a t i o n o f R
i s reduced, o r s i m p l y t h a t R i s reduced.
I f R i s a reduced graded r i n g t h e n R i s r e g u l a r
1.5.25. P r o p o s i t i o n
o n l y i f R i s t r i v i a l l y graded.
Proof.
Suppose 0 # xn
c
Rn i s n o t n i l p o t e n t , say n > 0. I f R i s r e g u l a r
t h e n t h e r e e x i s t s y E R such t h a t (l+x), W r i t e y = yi
+. . .+yi
1
m
where i
P . .
1
y ( l + x n ) = l+xn.
.>i,.Then yi
degree i n t h e l e f t hand s i d e hence yi
m
= 1,
i
m
m =
i s t h e p a r t o f minimal
0 and il>.. .>i = 0 m
follows. Straightforward calculations y i e l d : 2xn+yi
+...+y. +x y . +...+yi
1
lm
Since a l l y.
'k
'1
xn+ 1
...
= l+xn
have p o s i t i v e degree i t f o l l o w s e a s i l y t h a t :
=O i f ik < n,yn = -xn, yn+, = 0 f o r 0 k ( l + x n ) ( 1-xn+xn-xn+. 2 3 . .) ( l + x n ) = l + x n .
yi
S i n c e t h e homogeneous d e c o m p o s i t i o n
0 f o r a l l a p p e a r i n g terms, Then
u s i n g t h e g r a d a t i o n o f S one e a s i l y c a l c u l a t e s , as i n t h e f o r e g o i n g
z~~ -- x:
p r o o f , t h a t zn = -xn, t o be n i l p o t e n t i . e .
R
=
a.s.0.
l e a d i n g t o t h e f a c t t h a t xn has
Ro = So and R i s r e g u l a r .
L e t us i n c l u d e some o b s e r v a t i o n s a b o u t graded r i n g s o f t y p e Z w i t h Gr-regular center. I f graded ( l e f t ) i d e a l s a r e i d e m p o t e n t t h e n R i s s a i d t o be G r - f u l l y I l e f t ) idempotent.
A G r - f u l l y ( l e f t ) i d e m p o t e n t r i n g has a G r - r e g u l a r
c e n t e r . A graded r i n g R such t h a t each graded p r i n c i p a l i d e a l i s g e n e r a t e d by a homogeneous c e n t r a l i d e m p o t e n t i s s a i d t o be G r - b i r e g u l a r So if R i s G r - b i r e g u l a r t h e n Ro i s b i r e g u l a r , R i s G r - f u l l y l e f t idem-
p o t e n t and Z(R) i s G r - r e g u l a r . The f o l l o w i n g r e s u l t i s an easy a d a p t a t i o n of P r o p o s i t i o n 1.2. and 1.3.
1.5.27.
Theorem.
i n [ 85
I.
L e t R be a graded r i n g w i t h c e n t e r C, t h e n t h e
following conditions are equivalent :
1. C i s G r - r e g u l a r and f o r each the set o f 2. R i s & - f u l l y
M c s?
r-maximal i d e a l s o f R .
9
(R),
M
=
R(P n C) where 8 (R) i s 9
l e f t i d e m p o t e n t and a Z a r i s k i e x t e n s i o n o f C ( c f . 1 . 4 .
f o r Zariski extensions).
C. Structure Theory for Graded Rings of Type Z
272
3. R i s Gr-fully idempotent and a Zariski extension of C .
4. R i s Gr-biregular. 5 . For every m 6 Q (C), Q;-,,(R) i s gr-simple.
4
6. R i s a Zariski extension of C and Qi-,(R)
i s gr-simple f o r a l l
M C Qg(R). I f moreover, R i s f i n i t e l y generated as a C-module then R i s a Zariski extension of C i s and only i f R s a t i s f i e s the l e f t and r i g h t Ore conditions with respect t o G ( P ) f o r a l l graded prime ideals P of R. Let R be a graded ring with regular center C , which
1.5.28. Theorem.
i s f i n i t e l y generated as a C-module then the following statements a r e equivalent : 1. R i s semiprime 2. R i s Gr-regular 3 . R i s Gr-birerjular
4. R i s a Zariski extension of C and a l l graded prime ideals of R a r e idempotent. 5. R i s a n Azumaya algebra over C. Proof.
Easy modification of 2.6. i n
85
1 . Note t h a t already the
weaker homogeneous conditions on C make R i n t o an Azumaya algebra. This i s of course r e l a t e d t o the graded version of the Artin-Procesi theorem
for P . I . rings as expounded i n Section 1.4. 1.5.29 Remark.
The above may be used in describing the "graded Brauer
group" o f a commutative Gr-regular r i n g C . However, the study of graded Azumaya algebras over C reduces e a s i l y t o the study of graded Azumaya algebras over the s t a l k s a t gr-maximal i d e a l s of C . The l a t t e r s t a l k s are graded f i e l d s of the form k[ X,X-
1
]
a n d hence the study of the
graded Brauer groups of C i n the sense of [ 1171 reduces t o the study of t h e graded Brauer groups of graded f i e l d s f o r which we can r e f e r to [ 1 2 1 .
273
1. Non-Comrnutative Graded Rings
1.6.
E x e r c i s e s , Comments, References.
1. E x e r c i s e s .
1. L e t R be a s t r o n g l y graded r i n g o f t y o e G, where G i s a f i n i t e g r o w . Show t h a t : r a d Re = R e fl r a d R. 2. I f R as i n 1 i s s e m i p r i m i t i v e t h e n Re i s s e m i p r i m i t i v e . The converse h o l d s i f t h e o r d e r o f G i s i n v e r t i b l e i n Re. 3. L e t R be s t r o n g l y graded o f t y p e G, i s semi-prime. Show t h a t
iGl= n
0 t h e n G ( g ) ( x the f a c t that Z C F
P+S
L. Thus
i s exact implies t h a t x
x
-
f ( z ) C Fp+s-l
P and
P+S
=
G(f)(zp+s)
y = g ( x ) = !(x-f(
2))
=
g(x’) with
R e p e a t i n g t h i s p r o c e d u r e we f i n a l l y Get t h a t t h e r e i s an
x ‘ c Fp+s-lM. li.
G(*)
) = 0 and P+S f o r some
c F Y such t h a t y P
=
G (u).
3. Take y F F M n I m f . Exactness o f G(*) y i e l d s : G ( g ) ( y p ) = g ( y p ) = 0, P M t h e r e f o r e y = G ( f ) ( x p ( p ) ) f o r some x ( ~ )c F L. Hence y - f ( x ( ’ ) ) F Imf n F P P P- 1 By i n d u c t i o n we f i n d a
c F
x(’-’) y
P-s
-
X(~-’),...,X(~-’)
sequence
L such t h a t :
f(x(P)) -
. .. -
f(x(P-’))
c Imf n F ~ - ~ - ~ Y .
Completeness o f FL a l l o w s t o d e f i n e x = Hence y
-
with
f(x) = y - l i m S-t-
f(x(p)
+
b
s=o
x(~-’)
+
x ( ~ - ’ ) 6 F 1. P
... +
s i n c e FM i s separated, and i t f o l l o w s f r o m t h i s
x ( ~ - ’ ) ) = 0,the l a t t e r t h a t y c f ( F L). P
D. Filtered Rings and Modules
290
F
M fl I m f i s o b v i o u s . P
That f(FpL) c
4. Along t h e l i n e s o f 3. 5. S t r i c t e x a c t n e s s o f (*) i m p l i e s e x a c t n e s s o f G(*)
< M,
suppose t h a t G(*) i s e x a c t and l e t y
because o f 1. Conversely
y # 0 be such t h a t g ( y ) = 0.
and y , I! F f o r some p c Z . P- 1 We o b t a i n t h u s : G ( g ) ( y ) = 0 and so y = C . ( f ) ( x p ( p ) ) = f ( x ( p ) ) f o r P P P some x ( p ) 6 F L. S i n c e FM i s e x h a u s t i v e we have y : F
P
-
Consequently : y
< F
with x(~-’)
P-S
P
M
f(x(p)) 6 F P. By i n d u c t i o n we o b t a i n X ( ~ ) , . . . , X ( ~ - ’ ) P- 1 L and such t h a t :
I f FM i s d i s c r e t e t h e n f o r some i n d e x s, Fp-s-l y = f(x(’)
+
... +
we may t a k e x =
?
s=o
V = 0; hence
and t h u s (*) i s e x a c t . I f FP i s complete t h e n x ( ~ - ’ ) and g e t y = f ( x ) . The f a c t t h a t f and g a r e
s t r i c t morphisms w i l l f o l l o w f r o m 2 . and 3 111.5.
Corollary.
Let f : M
--f
N be a morphism i n R - f i l t and suppose t h a t
FF: and FN a r e s e p a r a t e d and e x h a u s t i v e , w h i l e FC i s a l s o complete. Then : G ( f ) i s b i j e c t i v e i f and o n l y i f f i s an isomorphism i n R - f i l t .
111.6.
Corollary.
L e t f c HornFR ( # , N ) .
I f t h e f i l t r a t i o n FV i s s e p a r a t e d
and e x h a u s t i v e t h e n : G ( f ) i s i n j e c t i v e i f and o n l y i f f i s i n j e c t i v e and s t r i c t . 111.7. C o r o l l a r y .
L e t f f HornFR (M,N).
hypotheses i s s a t i s f i e d 1)
M
Assume t h a t one o f
the following
:
i s complete, FN i s s e p a r a t e d and e x h a u s t i v e .
2 ) FF: i s d i s c r e t e and e x h a u s t i v e . Then : G ( f ) i s s u r j e c t i v e i f and o n l y i f
f i s s u r j e c t i v e and f i s s t r i c t .
29 1
111. Filtration and Associated Gradation
Now c o n s i d e r a morphism f :
M
+
N o f t h e category R - f i l t .
f i s said t o
i f f o r every n F Z there e x i s t s a natural
have t h e A r t i n - R e e s p r o p e r t y number h ( n ) F Z such t h a t : f(FnM)
3 Im
n
f
Fh(,,]N.
One can see t h a t i f f i s s t r i c t t h e n f has t h e A r t i n - R e e s p r o p e r t y .
111.8. Theorem.
Consider i n R - f i l t t h e exact
:
sequence
b
L f M
where t h e f i l t r a t i o n s FL, FM and FN a r e e x h a u s t i v e . I f t h e morphisms f and g have t h e A r t i n - R e e s p r o p e r t y t h e n t h e sequence
-
f
L-
M- 9 - i
i s exact. Proof.
F i r s t we c o n s i d e r t h e case where L i s a submodule
i n d u c e d by M and N = V/L w i t h q u o t i e n t f i l t r a t i o n , FiN
=
FiM
+
( i F 2 ) . Thus
L/L
that i s
we have t h e e x a c t sequence
A FI/L--$O.
M
O - L L
with f i l t r a t i o n
Since t h e p r o j e c t i v e l i m i t functor i s l e f t exact, i t i s s u f f i c i e n t t o prove t h a t
i s s u r j e c t i v e . We have
I?
:M
+
xs = y,. where u1 F
t 1
+
F M i s t h e c a n o n i c a l morphism). L e t s F 2 and denote P we may w r i t e y s - l - y, = ul+vl S i n c e ys-l - ys F L + FsM , M /L
L
and v1
c
FsM. L e t xs-l
where u2 c L, v2 F Fss1P. xs-2 - xs-l
lim(M/L + Fi).
,p F Z) be an e l e m e n t o f MIL.
- x1 F FsM. S i n c e yS-*
xs-l
= -
L e t q = (n;lN(yp) (fN
M/L
= v2 € FS-IM.
-
= ys-l
Y,-~
c
L e t xs-2 =
-
ul.
L + Fs-lM,
It i s then
Y ~ - ~ - U ~ - UI t ~ .i
clear that
Y,-~
-
Y,-~
= u 2+v 2
s clear that
F o l l o w i n g t h i s p r o c e d u r e we can
c o n s t r u c t f o r e v e r y t 2 s t h e element xt w i t h t h e p r o p e r t y x ~ - ~xt - F FtM and “(x,)
5
=
= yt.
I f f o r every t
z
s we l e t x t = x,
t h e n t h e sequence
i s an element o f M. C l e a r l y , r i ( c ) = q.
292
D. Filtered Rings and Modules
Consider the following diagram :
91.V/Ker g
f2 f 1 > L/Ker f d Im f = Ker g -M f 3
L
where f
=
f 3 o f2 o f l , g
=
g 3 o g2 o gl;
f l , g1 a r e the canonical
s u r j e c t i o n s ; f 2 , g 2 are b i j e c t i v e and f 3 , g,
a r e the canonical i n j e c t i o n s .
The f i l t r a t i o n s of L/Ker f and M/Ker g are Fi(L/Ker f ) Fi(M/Ker g )
=
FiM
Fi (Im f ) = Im f
9
=
9, 0-
o
9,
-
o
9,
Ker g
=
FiL + Ker f/Ker f ,
+
Ker g/Ker g . The f i l t r a t i o n s o f I m f a n d Im g a r e
n
FiM,
Si ( I m g )
and have t h a t fg
- --+
M
=
-
-
Im g r l FiN. Me have : f = f 3 o f 2 o f I
the sequence :
M/Ker g j 0
i s exact, f l i s s u r j e c t i v e and j, i s i n j e c t i v e . Since f and g have the Artin-Rees property, f 2 and g2 a r e isomorphisms of topological modules a n d therefore f 2 and
i - L M - L N
g2
a r e isomorphisms. Hence the sequence i s exact.
-
+ 92 1%g 93
N
293
IV: Free and Finitely Gemrated Objects of R-filt
L e t R be a f i l t e r e d r i n g , t h e n L R-mod and has a b a s i s (xi)icJ
c
i f it i s free i n
c o n s i s t i n g o f elements w i t h t h e p r o p e r t y
t h a t t h e r e e x i s t s a f a m i l y (ni)iFJ R.xi F L = Z F P i c J P-ni
R-filt i s filt-free
o f i n t e g e r s such t h a t :
8 Fp+. R . x . . icJ 1 Note t h a t f o r any i c J , xi c Fn.L and xi f Fni-lL. =
1
We say t h a t (xi,
ni)ie
i s a f i l t - b a s i s f o r L.
N e x t lemma f o l l o w s from t h i s
v .-
d e f i n i t i o n and Theorem 111.4.
IV.l.
Lemma.
L e t R be a f i l t e F e d r i n g , L
1" L i s f i l t - f r e e w i t h f i l t - b a s i s (xi,
c
R-filt,
i f and o n l y i f L 2' 8 R(-ni).
ni)ieJ
2" I f L i s f i l t - f r e e w i t h f i l t - b a s i s (xi,
then :
ni)icJ
ic J is a free
t h e n G(L)
graded module i n G(R)-gr w i t h homogeneous b a s i s {(xi)?,
icJ}.
3" I f G(L) i s a f r e e o b j e c t i n G(R)-gr w i t h homogeneous b a s i s {(xi)ni,icJ} and i f FL i s d i s c r e t e , t h e n L i s f i l t - f r e e i n R - f i l t w i t h f i l t - b a s i s
nii)icJ' 4" I f M c G(R)-gr i s f r e e graded t h e n t h e r e e x i s t s a f i l t - f r e e L c R - f i l t such t h a t G(L) '2
P.
5" L e t L c R - f i l t be f i l t - f r e e w i t h f i l t - b a s i s (xi,ni)ieJ, If fi {xi.
icJ}
+
M i s a f u n c t i o n such t h a t f ( x i )
let HFR-filt.
F Fs+,,,M,
then there i s
1
a u n i q u e f i l t e r e d morphism o f degree s, g:L
+
M w h i c h e x t e n d s f.
6" L e t M F R - f i l t and suppose t h a t L i s f i l t - f r e e .
I f g:G(L)
+
G(M) i s
a graded morphism o f degree s t h e n t h e r e i s a f i l t e r e d morphism f : L o f degree s such t h a t G ( f ) = g.
+
P1
294
D. Filtered Rings and Modules
c
7" L e t L
R - f i t be f i l t - f r e e ,
and o n l y i f FR
s e x h a u s t i v e ( s e p a r a t e d ) . I f FR i s d i s c r e t e and
c Z, i c J }
ini
t h e n FL i s e x h a u s t i v e ( s e p a r a t e d ) i f
i s bounded below t h e n FL i s d i s c r e t e . I f J i s f i n i t e and
FR i s complete t h e n FL i s complete. 8" I f FR i s e x h a u s t i v e and M
c
R-filt is
then t h e r e i s a f r e e r e s o l u t i o n o f
(*)
...+
L2-
f2
fl
L1
such t h a t FM i s e x h a u s t i v e ,
P in R-filt.
,Lo
fo> K + O
where e v e r y L . i s f i l t - f r e e and e v e r y f . i s a s t r i c t J J
morphism. P o r e o v e r
i f FR and FM a r e d i s c r e t e t h e n we assume t h a t e v e r y FL., j 3 0, i s a l s o J
discrete.
9" I f FR i s e x h a u s t i v e and complete and i f G(R) i s l e f t N o e t h e r i a n and G(M) b e i n g g e n e r a t e d as a l e f t graded R-module t h e n we may s e l e c t t h e L . i n t h e r e s o l u t i o n (*) t o be f i n i t e l y g e n e r a t e d t o o . J Note t h a t an M c R - f i l t i s s a i d t o be f i n i t e l y g e n e r a t e d i f t h e r e i s a f i n i t e f a m i l y (xi,
ni)ieJ
w i t h xi
c
M and ni
or filt-f.g.
c
Z, such
that F P = F R.xi P i = l P-ni
IV.2.
Remarks.
1" If L c R - f i t - f r e e as w e l l as f i n i t e l y g e n e r a t e d i n R-mod t h e n R i s fiIt-f.g. 2" I f M
c
R - f i t i s such t h a t FP i s e x h a u s t i v e t h e n M i s f i l t - f . g .
if
and o n l y i f t h e r e e x i s t s a f i l t - f r e e L w h i c h i s f i n i t e l y generated, and a s t r i c t epimorphism n : IV.3.
L
+
V.
P r o p o s i t i o n . L e t R be a f i l t e r e d complete r i n g and l e t 11 C R - f i l t
be such t h a t FM i s s e p a r a t e d and e x h a u s t i v e . Then V i s f i l t - f . g . i f o n l y i f G(M) i s f i n i t e l y g e n e r a t e d as a graded G(R)-module.
Furthermore,
i f G(M) may be g e n e r a t e d by n homogeneous elements, t h e n 11 may be
g e n e r a t e d as an R-module by l e s s t h a n n g e n e r a t o r s .
and
295
IV. Free and Finitely Generated Objects of R-filt
Proof.
i t follows that, i f i4 i s f i l t - f . g ,
From t h e remark I V . 2 . 2 "
then
G(M) i s f i n i t e l y g e n e r a t e d . Conversely, l e t G(M) be c e n e r a t e d by homogeneous g e n e r a t o r s x ( ~ ) , 1 5 i 5 n. C o n s i d e r L c R - f i l t , L f i l t f r e e Pi . D e f i n e f : L -t Fq by f(yi) = x ( j ) . S i n c e w i t h f i l t - b a s i s (yi, pi)i G ( f ) : G(L)
+
G(F1) i s an epimorphisrn,
Theorem 111.4.5. y i e l d s t h a t
f i s a s t r i c t epimorphism and t h e n Remark IV.2.2" f i l t - f . g and g e n e r a t e d by IV.4.
Corollary.
L e t I1
c
as an R-module. R-filt,
entails that
is
u
such t h a t FFI i s s e p a r a t e d and e x h a u s t i v e
and suppose t h a t FR i s complete. I f 6 ( P ) i s l e f t N o e t h e r i a n as an o b j e c t o f G(R)-gr,
I 5 t h e n 14 i s l e f t N o e t h e r i a n t o o . Moreover, we have : K.dim P R
K.diyR)G(V).
I f K.dimRC = K.dim
G(R)
G(V) = a and E ( V ) i s a - c r i t i c a l ,
t h e n M i s an a - c r i t i c a l R-module. Proof.
L e t N be a submodule o f P I equipped w i t h t h e i n d u c e d f i l t r a t i o n ,
F N = F M n N. S i n c e G(N) c G(ES), F(N) i s f i n i t e l y g e n e r a t e d . P r o p o s i t i o n " P I V . 3 . e n t a i l s t h a t N i s f i n i t e l y generated, so M i s l e f t N o e t h e r i a n . Consider submodules N and P o f P, N c P, b o t h equipped w i t h t h e i n d u c e d { x ( ~ ) ,1 5 i 5 n } i s a Pi s e t o f homogeneous g e n e r a t o r s f o r G(N) where x i i ) c N f o r a l l i, 1 5 ; y n , f i l t r a t i o n , and assume t h a t G(N) = E ( P ) . I f
then, a q a i n f r o m p r o p o s i t i o n IV.3.,
we i n f e r t h a t
g e n e r a t e s N as w e l l as P. Thus N = P.
{x(~),1
5 i 5 n }
T h i s y i e l d s K.dimRI.l 5 K.dimF(R)F(!?)
I n o r d e r t o e s t a b l i s h o u r l a s t c l a i m , l e t N # 0, N c
I.4
and f i l t e r P/N
by p u t t i n g : Fp(M/N) = (N + F F)/N. C l e a r l y , we o b t a i n an e x a c t sequence P i n G(R)-gr : O j G ( N ) -G(M)
C(M/N)--+O.
S i n c e G(N) # 0, t h e K r u l l d i m e n s i o n o f G(M/N) i s l e s s t h a n a K.dimR M/N 2 K.dimG(R)C(M/N) c
a.
T h i s shows e x a c t l y t h a t M i s an a - c r i t i c a l module.
,
hence
296
D. Filtered Rings and Modules
IV.5. Corollary. Let R be a complete f i l t e r e d ring such t h a t F(R) i s a l e f t Noetherian ring, then : 1" R i s l e f t Noetherian and K.dim R
5
K.dim G ( R )
i s l e f t Noetherian and K.dim FOR c K.dirn P ( R )
2 ) Fo R
+ 1.
Proof
1. Follows from the foregoing corollary. 2 . Fo R i s a subring of R and i t i s c l e a r l y complete; Corollary IV.4. together with the r e s u l t s of Section A . I I . 5 . IV.6.
submodules of P are closed, i . e . i f N c F then N Consider submodules N
1. I f G ( N )
=
c
o
< R - f i l t with FP being exhaustive. Assume t h a t
Let M
Proposition.
f i n i s h the proof.
=
fl ( N PCZ
+
F H). P
P c Fc, then :
G(P) then N
P . I n p a r t i c u l a r i f the Krull dimension o f
=
G(M) i s well defined then the Krull dimension of C i s defined a n d K.dimRM
5
K.dim
G( R )
2 . I f K.dimRM = K.Jim i s an
G(F).
G(M)
F( R )
= a
and i f G ( M ) i s a - c r i t i c a l then
a - c r i t i c a l module.
3. I f G ( M ) may be generated by n homogeneous generators then F? may be generated by n generators. Proof.
1. Take x F P , then x c FiM f o r some i , x d F i - l P . Xi c G(P)i = G(N)i = ( P there i s a y1 c N
n
FiM) + Fi-lb'/Fi-l
Pl = ( H
Because :
n FiFI) + Fi-lM/Fi-lF Hence x c y1
n F i F such t h a t x-y1 c Fi-lM.
+
PnFi-l!l.
Repeating t h i s process we and up with elements y l , y 2 , . . . , y o < N such
t h a t x-(yl + y p +...+ys) i s i n Fi-s M n P , s Thus x c N
N
=
+
Fi-,F f o r s
P follows.
>
0. I t follows t h a t x
0.
c n (N+F M ) pcz
= N , whence
I V. Free and Finitely Generated Objects of R-filt
2 . Proceed as i n the proof of Corollary.
x ( ~ ) 1, c i 5 n } be a family of homogeneous aenerators f o r Pi G(M). Write rl' f o r the submodule of M generated by the elements x ( ~ ) ,
3. Let
{
1 5 i 5 n . Obviously x ( i ) 6 E ( M ~ )
Pi
Pi
=
(171
n F M)/(MI n F ~ , - ~ M ) . Pi 1
Therefore G ( M ' ) = G ( M ) and by 1. t h i s y i e l d s PI'
IV.7.
=
!I.
Remark. The above proposition i s generally applied i n case F!
i s exhaustive
and d i s c r e t e .
297
298
V: The I-adic Filtration
The I-adic f i l t r a t i o n i s very important f o r the study of
commutative
rings ( s e e [ 0 . 1 I ) . I-adic non commutative f i l t r a t i o n i s useful, f o r example in the
The study
of Lie Algebras
or f o r the study of the integral group ring
Z(G) where G i s a nilpotent group ( s e e exercises).
Let R be a r i n g , I a n ideal of R, Fc c R-mod. Using the I-adic f i l t r a t i o n as introduced before, we obtain the I-adic completion b o f P a s
fi
=
1jm M/I~M.
n Let i : M
+
M be the canonical f i l t e r e d morphism, iInC)n20is the
f i l t r a t i o n of
i.
I n case M
=
R- l i n e a r
I\ '\
c
t h a t the diagram
R>M-
: R b~~
B!
+
i s a ring homomorphism,
as an R-bimodule.
by means of which we regard The
R, then R
--f
El, a(5
C% m) =
5 i(m) has the property
: j
RR
F1 a
where j ( m )
=
160 m
i s commutative. We s h a l l now present some of the important properties of the I-adic filtration.
1" I f M i s of f i n i t e type then a i s s u r j e c t i v e . I n p a r t i c u l a r and M i s a n ;module of f i n i t e type. Indeed, there e x i s t s a f r e e module L of f i n i t e type such t h a t LLM-+O
(TT
surjective)
i(F1) =
299
V. The I-adic Filtration
We have t h e comnutati ve diagram
i s surjective, since
The morphism n
TI
i s strict.
As t h e completion f u n c t o r comnutes w i t h f i n i t e d i r e c t sums,
then
p
i s an isomorphism. Hence u i s s u r j e c t i v e . 2.
?
i s an i d e a l and
if E
t o proof t h a t
Let
5.1 c
i
0. Indeed, i t i s s u f f i c i e n t
E ~ E~, ,...,5,
c
n I
.
R + R / I p i s t h e canonical morphism). Since P .P P a (nP: i i then a; c I f o r 1 i i c n. On t h e o t h e r hand, s i n c e a2 - al C I
i
F I.B u t a3
a;
i
therefore
c1
-
i
2
.
i
a2 F I y i e l d s a3 F I, 1 s i 5 n. F i n a l l y , 1
F I, f o r a l l 1 5 i 5 n. Hence an
we o b t a i n an
7
2
= (n ( a ' ) )
we have
3.
in c Inf o r a l l n ~ E~ , ,...,5, F I , then
c2...En
6
n I
series 1 + E
^I.
+ 52
As gn F t
a:
C In and
.
i s contained i n t h e Jacobson r a d i c a l o f
Indeed l e t 5 F
2 a,...
in c In , lim
... converges.
n+m
i.
En = 0 and t h e r e f o r e t h e
We have ( l - < ) - '
= 1t
+ 52
t
and t h e r e f o r e 5 belongs t o t h e Jacobson r a d i c a l .
4. I f I i s o f f i n i t e type as a l e f t and as a r i g h t i d e a l , then : i )
in = i
=
i
f o r a l l n 2 0. I n p a r t i c u l a r
i i ) I f M i s an R-module o f f i n i t e t y p e t h e n : I"M Proof. -__
= I ~ M=
in ti, f o r
a11 n z 0.
For t h e a s s e r t i o n i ) we apply 1.
i i ) Since I n M i s o f f i n i t e type we have :
rn
=
in.
..
300
D. Filtered Rings and Modules
1% =
i
i (I'M)
=
i
I n i(M) = I n
h(M)
=
In
i. i f f o r any
The ideal I i s sa i d t o s a t i s f y the l e f t Artin-Rees property
f i n i t e l y generated l e f t R-module M , any submodule N of M and any natural number n , there
e x i s t s an i n t eg er h ( n )
3
0 such t h a t I h ( n ) M n N c I n N
I n other words I s a t i s f i e s the l e f t Artin-Rees property i f the I-adic topology of N coincides with the topology induced i n N by the I-adic topology of M 5 ) If I has the Artin-Rees property then f o r every module M of f i n i t e
type : Ker i = n
n> 1
InM = {x 6 M, (1-r)x = 0 f o r an a r b i t r a r y r
6
I}
In p a r t i c u l a r , i f I i s contained i n the Jacobson radical of R, then
n I ~ =M 0.
n? 1
Obviously we have the inclusion :
{x
€
M,(l-r)x
=
0 f o r an a r b i t r a r y r
Conversely, i f x c
n
n? 1
I'M,
6
I} c rl
n? 1
I'M.
then because I has the Artin-Rees property,
there e x i s t s a n 6 N such t h a t I n M fl Rx c I(Rx) = I x . Hence Rx c I x and therefore there ex i s t s an r c I such t h a t x = rx; therefore (1-r)x = 0 . 6 ) I f I has the Artin-Rees property and R i s a domain then
n In n? 1
= 0
7) I f R i s a l e f t Noetherian ring, I a n ideal of R sa tisfying the l e f t Artin-Rees property, then the following statements a r e equivalent : i ) I i s contained i n the Jacobson radical o f R . submodules of M a r e
i i ) I f M 6 R-mod i s f i n i t e l y generated then the closed i n the I-adic topology. i ) = i i ) We apply 5 ) i i ) = i ) Let A Hence R
=
A
be a maximal l e f t i d e a l . Assume t h a t I
+ 1.R
= A
+
I(I+A) = A
+
I* = A
+
2
I (I+A)
Q A. Then R =
A
+
I3 =
=
...
I + A. =
A+In =
...
V. The Ii+dic Filtration
Since A i s closed, A = Hence I c A
I7
n? 1
(A
+
301
I n ) and t h e r e f o r e A = R; c o n t r a d i c t i o n .
finishes the proof.
8) L e t R be a l e f t Noetherian r i n g and I an i d e a l s a t i s f y i n g t h e l e f t A r t i n-Rees p r o p e r t y . f u n c t o r t a k i n g M t o M i s exact i n t h e category o f l e f t R-modules
i ) The
o f f i n i t e type. i i ) I f M C R-mod i s f i n i t e l y generated t h e n
iz
@R M.
i i i ) R i s a r i g h t f l a t R-module. Proof. __ For i ) we apply Theorem 111.8. i i ) There e x i s t s an e x a c t sequence i n 0-K-LM-0 where L i s a f r e e module o f f i n i t e type and K i s o f f i n i t e type.
- -
From t h i s we deduce t h e f o l l o w i n g comnutative and e x a c t diagram : R@K->
R@L
11
0-K-
.
!p
i
L-M-0
R@M
ri
0
la
Since p i s an isomorphism and y i s s u r j e c t i v e , i t f o l l o w s t h a t u i s i n j e c t i v e and u i s an isomorphism.
,
Obviously
ii)
iii)
9 ) L e t R be a l e f t Noetherian r i n g . Assume t h a t I i s c o n t a i n e d i n t h e Jacobson r a d i c a l o f R and has t h e l e f t Artin-Rees p r o p e r t y . Then R i s faithfully flat.
Proof. Since
-
i
It
i s s u f f i c i e n t t o prove t h a t
i
BRM = 0 i m p l i e s M = 0.
i s a r i g h t f l a t R-module, i t i s s u f f i c i e n t t o
-
case M = R scRM = 0. Since M/I^M M = I M , t h e r e f o r e M = 0.
E
c o n s i d e r the
M/IM we o b t a i n M/IM = 0, hence
302
0. Filtered Rings and Modules
10) Let R be a r i g h t a n d l e f t Noetherian ring. Assume t h a t I has the
r i g h t and l e f t Artin-Rees property. I f
then the image of a i n Proof. $,(A)
k
i s also a nonzero-divisor.
The homomrphism R =
a c R i s a nonzero d i v i s o r i n R ,
>R,
cpa
(X) =
Xa and
JIa
R
>
R,
ah a r e i n j e c t i v e . Then we apply 8 ) , i i i ) .
Definition.
Let R be a ring and a E R. We say t h a t a i s a normalizing
element of R i f a R = R a
. The
s e t of normalizing elements of R i s
denoted by N ( R ) . Definition.
Let R be a ring. The elements al , a2, ...,a n a re a normalizing
s e t of elements of R i f i )a l E N ( R )
i i ) ai
+
( a l , a 2 ,..., a 1. - 1 ) CN(R)Xal, a 2 ,..., a i - l )
f o r i = 2 ,...n .
The elements a l , a*, ...,a n form a cen t r al i zi n g s e t of elements of R i f i ) a l E Z ( R ) ( t h e cen t er of the ring R ) i i ) ai
+
(al,
...,
a i - l ) c Z(R)/(al. a 2 ,..., a i - l )
for i
=
2 ,...,n .
A generating s e t of a n ideal I of R which i s al so a normalizing s e t of elements of
R i s cal l ed a normalizing
s e t o f generators of I . Similarly
one defines a c e n t r al i zi n g s e t of generators f o r an ide a l. V . 1.
Proposition.
Let R be a l e f t Noetherian ring, I an ideal generated
by a central system, then I s a t i s f i e s the l e f t Artin-Rees property. Proof.
s
C
Let N be a submodule o f a f i n i t e l y generated
N and consider a submodule M ’ of
t h a t M i fl N = I’N.
M
c R-mod. Let
M maximal w i t h the property
This choice makes M/M’ i n t o an essential extension
of N/ISN, where Is (N/IsN)
=
0 . I f we a r e able t o e sta blish t h a t under
t
the circumstances M/M‘ may be annihilated by some I t then I (M/M’) = 0 y ie l d s 1% n N c ISN. Note a l s o t h a t i t i s s u f f i c i e n t t o do t h i s f o r s = 1,
V. The I-adic Filtration
303
because f o r a r b i t r a r y s i t wi l l follow from t h i s case applied to
IN,
...,
instead o f N together with the conjunction of the inclusions
IS-'N
t h u s obtained a t each s t e p . I f I = Rcl a central system, l e t
p :
M
+
+...+ Rc,,
M be the map given by m
Since M i s l e f t Noetherian, there ex i s t s r e N for all k c N
i.e.
Ker pr and N i s an
where (cl
Im pr fl Ker kr = 0 .
an i n j e c t i v e R-morphism M/Ker 1
+
Ker
clm f o r a l l m c M. r+k such t h a t Ker pr = Ker p
However N i s contained i n
pL
pt-'.
Let
pr = 0 .
= 0 , then p gives r i s e to
To prove our original claim
i t will therefore be s u f f i c i e n t t o prove t h a t both Ker p and Ker
kt-l
can be annihilated by l ar g e powers of I . Now c1 being c e n t r a l , Ker i s an R/Rcl-mdule and we apply f o r some p Ker $/Ker
p +
p
induction on n t o deduce t h a t I p Ker p = O
N. Since there i s an i n j e c t i v e morphism
C
is
--t
es s en t i al submodule of M, t he re fore Im
t be the smallest natural number such t h a t
,..., c n )
Ker pm-',for 1 c m
5
:
t , the f a c t t h a t Iq Ker
f o r some q t N willfollow by induction on t .
kt-'
= 0
o
I t i s easy t o see t h a t (exactly as in Proposition V . l . )
V.2. Remark.
one may deduce the following as s er t i o n : Let R be a r i n g and I be an ideal of R which i s generated by elements of N ( R ) ( I i s not necessarily generated
by a normalizing s e t of generators). Then I has the Artin-
Rees property. Proposition.
V.3.
and denote
R
Let R be a r i g h t Noetherian ring and l e t J be a n ideal
= R/J.
i ) I f J C I i s a n ideal and
T -
=
I / J , then the completion of
R
f o r the
adic topology i s i d e n t i f i e d w i t h R/RJ.
i i ) I f I i s generated by a cen t r al i zi n g system ( a l , a*, ..., then
an)
I i s generated by the cen t r al i zi n g system ( i ( a l ) , i ( a 2 ),...,i ( a n ) )
where i : R i s the canonical morphism.
304
D. Filtered Rings and Modules
-P -roof.
in = J + In/J, t h e n t h e completion o f t h e l e f t R-module
i ) Since
t h e I - a d i c topology i s i d e n t i f i e d w i t h t h e c o m p l e t i o n o f
R
R
for
-
f o r the
a d i c- t o p o l ogy . From t h e e x a c t sequence o f l e f t R-modules :
R-0
O j J - R A we o b t a i n
t h e exact sequence o f l e f t R- modules :
0-j-i
-
L
P
i -0
- -
B u t ir i s a r i n g homomorphism. Since J = RJ
ii)I f I = (al)
then s i n c e al
t
i t follows t h a t
Z(K) and i ( R ) i s dense i n
R
-
^
^
R/RJ.
R, i(al)
€ Z(R).
Then we procede by recurrence, t a k i n g i n t o account i ) .
If I i s an i d e a l o f a r i n g R, such t h a t R / I i s l e f t
V.4. Lemma_,
Noetherian and I = ( a ) i s generated by a s i n g l e element o f t h e center o f R, then R i s l e f t Noetherian.
--Proof.
We see t h a t GI(R)
. Since
ring R/I [ X I
r i n g . Hence GI(R) C o r o l l a r y IV.5.)
i s isomorphic t o t h e q u o t i e n t r i n g o f t h e
R / I i s a b e t h e t i a n ring, R / I [ X I
i s a Noetherian r i n g . Since Gi(R)
%
i s a Noetherian GI(R)
(from
i t f o l l o w s t h a t R i s a l e f t Noetherian r i n g .
V.5. Theorem.
L e t R be a r i n g and I an i d e a l w i t h a c e n t r a l i z i n g s e t
of generators
I f R i s l e f t Noetherian, t h e n R i s l e f t Noetherian.
Proof.
Suppose t h a t I = (al,
a*,
...,
an) where al,a2
,...,an
is a
c e n t r a l i z i n g s e t o f generators. The p r o o f i s given by i n d u c t i o n on t h e number o f generators o f I . I f I = (al)
then R
i s l e f t Noetherian by Lemma V.4..
From statement
8) and P r o p o s i t i o n V.3. i t f o l l o w s t h a t t h e r e e x i s t s an exact sequence :
OjRi(al)+
R-R-0
(* 1
305
V. The I-adic Filtration
where
R
=
R/Ral
generators o f I
and t h e i(al) ,...,:(an)
are a centralizing s e t o f
From ( * ) i t f o l l o w s t h a t
R i s l e f t N o e t h e r i a n by t h e
induction hypothesis. ^
^
Hence R/Ri(al) J = Ri(al)
i s a l e f t - N o e t h e r i a n r i n g . I f we denote by J t h e i d e a l
-
-
t h e n f r o m Lemma V.4. t h e J - a d i c c o m o l e t i o n R o f R i s a l e f t
N o e t h e r i a n r i n g . Because J c I we have a commutative diagram
where
U,
i and j a r e t h e c a n o n i c a l morohisms. S i n c e j i s an isomorohism,
i t f o l l o w s t h a t a i s a s u r j e c t i v e rnorohism. S i n c e R i s l e f t N o e t h e r i a n ,
R i s l e f t Noetherian.
306
VI: The Functor HOM&,-).
Filt-projective (-injective) Modules
The p r o o e r t i e s o f HOMR mentioned e a r l i e r make i t i n t o a f u n c t o r
(R-filtf+Z-filt;
and i t i s c l e a r t h a t , by d e f i n i t i o n o f t h e f i l t r a t i o n i s e x h a u s t i v e f o r any M, N C R - f i l t .
i n HOMR(-,-),
F HOMR (M,N)
VI.l.
L e t R be a f i l t e r e d r i n g and l e t M, N E R - f i l t be such
Lemma.
that M
i s a f i n i t e l y generated o b j e c t w h i l e F N i s exhaustive, then
HOMR(M,N) Proof. f
c
=
HomR(M,N).
L e t { x ( ~ ) ,ni}
HornR (F1,N).
be a system o f g e n e r a t o r s f o r F1 and t a k e
There e x i s t s s 6 Z such
that f(x(i))
E Fni+s
N for
any 1 5 i 5 n. I t f o l l o w s f r o m t h i s t h a t f C Fs HOPR(M,N) and hence f E HOMR (M,N).
Note t h a t , i f FN i s n o t e x h a u s t i v e and i f N ' i s t h e
exhaustion o f N then HOMR(M,N) = HOMR (M,N') VI.2.
Remark.
=
HornR ( M , N ' )
I n g e n e r a l HOMR(C,N)
# HomR(M,N). F o r e x a m p l e l e t R be a
d i s c r e t e l y and e x h a u s t i v e l y f i l t e r e d r i n g such t h a t R = FiR
f o r a l l i when
Fi(R) # O . T a k e M t o b e f i l t - f r e e w i t h f i l t - b a s i s {X(~),O} w'ith lcicm, a n d t a k e N = R . Define f : M
+
N by
f(x(i))
= a(i)
f i s p. l e t io 6 Z be such t h a t FiN
i < io -p. VI.3.
hence f ( x ( i ) )
Proposition.
Let
= a(i)
M, N