This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
I/JjeJj £ CovT for each i € I, then 3 If {Ui -> U} e Cov T and V -»• f/ is any morphism of Cat T, then f/{ x y V exists for any i 6 / and we have {t/j Xy V —> V} € CovT. We say that Cov T gives a Grothendieck topology of Cat T. By abuse of notation, we sometimes denote CatT simply by T. An element {Ui —> U} of CovT is called a covering of U. The set of coverings of U is denoted by covU for U € CatT. Let W = {£/< A [/} and V = {V, % U} be coverings of U £ Cat T. We say that V is a refinement of W if there exist a map between the index sets / : J —> I and morphisms r)j : Vj -» U/j for all j 6 J such that ^ = y>^ o Tjj. If we set V > U when V is a refinement of U, then cov [/ is a (not necessarily small) preordered set. As a category, cov U is filtered. (1.8.4) Let T be a site, and C a category with direct products and finite limits. Let F : T op -> C be a presheaf on T with values in C. We say that F is a sheaf on T with values in C if
i€l
ij€J
is exact, in other words, if the left arrow is a difference kernel of the two arrows on the right for any {C/j —> £/} € CovT, where the arrows on the right are the morphisms induced by the first and second projections, respectively. We denote the set of sheaves on T with values in C by sh(T, C). (1.8.5) We say that T has a small topology if there exists some small full subcategory To of T such that any object U G T has a covering consisting only of morphisms with the source objects in To. If this is the case, covU has a small final subcategory. Moreover, the composite (1.8.6)
sh(T,C) «-> Func(T°p,C) -^> Func(TS>p,C)
is fully faithful. In particular, sh(T, C) is a category with small horn sets in this case. Assume moreover that To is closed under fiber products. Then To is a site with the same topology as that of T, and we have that the functor (1.8.6) induces an equivalence sh(T,C)^sh(T0,C). (1.8.7) From now, until the end of this subsection, let A be a Grothendieck category which satisfies the (AB3*) condition, and T a site with a small topology.
22
I. Background Materials
(1.8.8) For a covering il := {Ut -> U} of U G T and T G Func(T op , A), the Cech cohomology Hl(U., T) is defined as in [71], see [8]. We define H^U,?) := KmH^lX,?), where il runs through cov(U). When we define T+ e Func(T op , .4) by F+(U) := H°{U,F), then we have that JF++ e sh(T,.4), and a functor a : Func(T op , .4) -> sh(T,A) is defined by a{T) := T++. We call a(T) the sheafification of T. Lemma 1.8.9 The functor a is left adjoint to the canonical embedding sh{T,A)->F\mc(T°p,A). For the proof, see [8]. Corollary 1.8.10 The category sh(T, A) is a Grothendieck category which satisfies (AB3*). The sheafification a : Func(T op , .4) —> sh(T, A) is exact. The embedding sh(T, A) —> Func(T op , A) preserves injective objects and projective limits. Proof. Note that S := sh(T, A) is a full subcategory of V := Func(T op , A). We denote the embedding 5 —• V by i. By Lemma 1.2.6, 5 has inductive limits. As projective limits preserve projective limits, any projective limit of sheaves as a presheaf is a sheaf, and hence is a projective limit as a sheaf. Hence, S has projective limits, and i preserves projective limits. In particular, S has kernels and cokernels, and clearly it is additive. As a is a left adjoint, it preserves inductive limits. In particular, it preserves cokernels. As the functor (?) + is left exact, we have that ia is also left exact by construction. Since <S has kernels, both ia and i preserve kernels, and i is faithful, it is easy to see that a preserves kernels. Let
zKercokery> is identified with ia Coker ker iip —> ia Ker coker iip, since the counit ai —> Id is an isomorphism and a preserves both kernels and cokernels. This is an isomorphism as V is abelian. As i is faithful, this shows that S is also abelian. Now it is clear that a is exact. Again by Lemma 1.2.6, i preserves injective objects. The category <S satisfies (AB5). In fact, the inductive limit in S is nothing but the composite alim(z?), and hence it is left exact if filtered. As it is easy to see that (aty is a small family of G-generators of S if is a small family of G-generators of V, we have that 5 is Grothendieck.
•
1. From homological algebra
23
(1.8.11) Let T and V be sites. We say that / : T -> V is a continuous functor if / : CatT -> CatT' is a functor such that for any {£/j ->•[/} G CovT, we have {/[/* -> / [ / } G CovT" and the canonical map
is an isomorphism for any i, j . If / : T -> T" is a continuous functor and .F G sh(T',.4), then / P (.F) is a sheaf. Hence, we obtain a functor fs = fp : sh(T',.4) -> sh(T,.4). Note also that fs := afP : sh(T,A) -> sh(T',.4) is left adjoint to / s . Example 1.8.12 Let B be an abelian category. When we define that {B{ -+ 5 } , 6 / is a covering if / is a finite set (we allow the case / is empty) and ©jBj -» B is an epimorphism (resp. isomorphism), then B is a site, which we denote by B\ (resp. Bo). The category <Si := sh(B\,A) (resp. So := sh(fi0, A)) is nothing but Sex(S op , ^4) (resp. the category of contravariant additive functors from B to A). In particular, as we are assuming that A is Grothendieck and satisfies (AB3*), Sex(6 0p ,.A) is also Grothendieck and satisfies (AB3*). Moreover, Id e : Bo -> B\ is clearly continuous. (1.8.13) Let T be a site with fiber products, and T a subcategory of f such that ob(T) = ob(T). We assume that any isomorphism of T is a morphism of T, and any base change of a morphism of T is again a morphism of T. Moreover, we assume that t/j —» U is a morphism of T for any {U{ -> U\ G Cov(t). For X G f, the full subcategory of f/X consisting of morphisms of T (with the fixed codomain X) is denoted by Tx- Note that Tx is a site with the topology of T. If
A is exact. Note also that B is Grothendieck, which is less trivial. In fact, if U is a generator of A, B € B and / € A(U,B), then we have I m / c B and hence we have I m / € B. So the set (V;) i€/ of quotient objects of U which lies in B (by Lemma 1.7.3, we can take / to be small) is a small family of G-generators of B. As the inductive limit in A of an inductive system consisting of objects in B lies in B by assumption, it is also an inductive limit in B. Hence the (AB5) condition holds in B. As is easily seen from the proof, B is locally noetherian if A is. Lemma 1.10.2 The embedding i : B ji is an isomorphism.
26
I. Background Materials
Proof. For A € A, the set of subobjects of A is indexed by a small set (Lemma 1.7.3). So we may form the sum j(A) of all subobjects of A which lie in B. As B is closed under inductive limits in A, we have j(A) G B. Note that j(A) is the largest subobject of A which lies in B. For B € B and / £ ^4(5, J4), we have that / factors through j(A), as I m / G 5. Hence, we have an isomorphism
In particular, for A, A' £ A and g S A(A, A'), the restriction of g to J(J4) factors through j(A'), and we have an induced morphism j(g) € B(j(A),j(A')). It is easy to verify that j is a functor with this definition, and
j(limAx)
is an isomorphism for any filtered inductive system (Ax) in A. Proof. Note that if B € Bf, then i(B) € A/. Hence, using Lemma 1.9.3, we have isomorphisms of functors on Bf
B(?,\hnj(Ax)) S* lu S A(i(?),\imAx) = B(?,j(\imAx)). As the canonical functor y : B -» Sex(So P ,Ab) is an equivalence and the composite isomorphism above is nothing but y(f) : y(\unj(A\)) —> y(j(\imAx)),
f is an isomorphism.
D
(1.10.4) Let A be a Grothendieck category. We say that A € A is a simple object if there are exactly two subobjects of A. In other words, A is simple if and only if A ^ 0, and any monomorphism into A is either zero or an isomorphism. We say that A £ A is semisimple if A is isomorphic to a direct sum of simple objects. The following is well-known, and is proved in [123] in the case of modules over an algebra. L e m m a 1.10.5 The full subcategory AsS of A consisting of semisimple objects of A is closed under inductive limits, subobjects, and quotient objects in A.
1. From homological algebra
27
By the lemma, the sum of all semisimple subobjects of A G A is the maximum semisimple subobject of A. This object is called the socle of A, and we denote it by soc A. Thus, soc : A —> A s is a right adjoint functor of the canonical embedding AgS •-> A. Note that soc A is also the sum of all simple subobjects of A. As it is a right adjoint, soc preserves projective limits (e.g., kernels). Assume moreover that A satisfies the (AB3*) condition. Then for A £ A, we set
radA:=
f]
B
BcA, v4/£?:semisimple
and call rad^l the radical of A. We denote A/radA by top .A, and call it the top of A. Note that top^l is not necessarily semisimple. However, if A is an artinian object, then top A is semisimple, and hence is the largest semisimple quotient of A. Any non-zero artinian (resp. noetherian) object admits a simple subobject (resp. quotient object). Hence, we have Lemma 1.10.6 Let Abe a Grothendieck category which satisfies the (AB3*) condition. If A is an artinian (resp. noetherian) object of A and soc A = 0 (resp. top A = 0), then we have A = 0. The following is also trivial. Lemma 1.10.7 Let V be a locally noetherian category, and (Dx) a filtered inductive system in V. Then the canonical map limsoc(D^) —> soc(limD.x) is an isomorphism. If A is a locally artinian category and 0 ^ A € A, then we have soc A ^ 0. (1.10.8) We say that a ring A is a division ring if A ^ 0 and any nonzero element of A is a unit. The following is well-known as Wedderburn's theorem [123]. Theorem 1.10.9 Let A be a ring. The following are equivalent. 1 The A-module &A is a semisimple object o/^M. 1* The right A-module AA is a semisimple object ofMA. 2 A is a finite direct product f l ^ i Mat nj (Dj) of matrix rings over division rings. If the conditions above are satisfied, then A is called a semisimple ring. A semisimple ring is both left and right artinian.
28
I. Background Materials
1.11 Full subcategories of an abelian category Let A be an abelian category, and X a subset of ob(,4). We define some full subcategories of A. 1 We denote the full subcategory of A consisting of objects isomorphic to a direct summand of a finite direct sum of objects of X by add-^. If X = 0, then add(A") consists of null objects of A. Obviously, add(X) is a Karoubian additive category. If X is closed under extensions in A, then so is add(A"). 2 The full subcategory of A consisting of objects A € A such that there exists some r > 0 and a filtration 0 = A) C Ax C • • • C Ar = A such that Ai/Ai-x is isomorphic to some object in X for i = 1,2,..., r is denoted by F{X). Note that T{X) is closed under extensions in A. Note also that F(X) is not closed under direct summands in general even if X is so. 3 Let A1 b e a n additive full subcategory of A. The full subcategory of A consisting of A € A such that there exists some exact sequence (1.11.1)
0->Xh^>
> Xx^> XQ-+ A->0
with Xi € X is denoted by X. An exact sequence of the form (1.11.1) is called a finite X-resolution of A. The smallest non-negative integer i such that Xi+l = 0 is called the length of the -^-resolution (1.11.1). For A € X, we call the minimum length of /^-resolutions of A the Xresolution dimension of A, and denote it by #-resol.dim A If A £ X, then we define l :— oo. 3* Let X be an additive full subcategory of A. Then we define X := (X°P)°P. In other words, X consists of A G. A such that there exists some exact sequence (1.11.2)
0^A^X°-^X1^
>Xh^0
with X1 £ X. An exact sequence as in (1.11.2) is called afiniteXcoresolution-oi A. The minimum non-negative integer i such that Xl+1 = 0 is called the length of the /f-coresolution (1.11.2). For A € X, the minimum length of A"-coresolutions of A is called the X-coresolution dimension of A, and we denote it by X -cores.dim A. If A ^ X, then we define X -cores.dim A := oo.
1. From homological algebra
29
4 For A G A, we define *-inj.dim A := sup({z > 0 | Ext^A", A) ^ 0} U {0}), and we call #-inj.dim^l the X-injective dimensionoi A. If A'-inj.dimyl = 0, then we say that A is X-injective. The full subcategory of .4 consisting of AMnjective objects in A is denoted by XL. Similarly, X-projective dimension is defined, and we denote it by A'-proj.dimA We also define an X-projective object in a similar way, and the full subcategory of Xprojective objects of A is denoted by LX. Note that X1 is closed under extensions, direct summands, and monocokernels. Note also that LX is closed under extensions, direct summands, and epikernels.
1.12 .^-approximations and the Auslander-Buchweitz theory Let A be an abelian category. We say that a morphism p : M -> N of A is right minimal if
M of A is called a right X-approximation of M if X e X, and for any X' £ X and any g e A(X',M), there exists some h € A{X',X) such that fh = g. It is equivalent to say that A(?,X) € Func(A', Ab) is representable, and A(7, / ) : -4(?, X) —>• A(f, M) is an epimorphism of the functor category Func(A',Ab). Left X-approximation is the dual notion. A right (resp. left) minimal right (resp. left) A'-approximation is simply called a right (resp. left) minimal ^-approximation. A right minimal Xapproximation of M is unique up to isomorphisms as an object of A/M, if it exists. Let B be a full subcategory of A which contains X. We say that X is contravariantly finite (resp. covariantly finite) in B (or B has right (resp. left) Xapproximations) if any object in B has a right (resp. left) ^-approximation. If any object in B admits a right (resp. left) minimal A'-approximation, then we say that B has right (resp. left) minimal -Y-approximations. Lemma 1.12.1 Let be an exact sequence in A such that X € X, and assume that Ext\(X, Y) = 0. Then p is a right X-approximation of M. Proof. For any X' 6 X, we have that the sequence A{X', X) -> A(X', M) -4 E x t ^ ( * ' , Y) = 0
30
I. Background Materials
is exact.
D
Lemma 1.12.2 (Wakamatsu's lemma) Let
o->r Ux ^M be an exact sequence in A such that p is a right minimal X-approximation of M. If X is closed under extensions, then Ext\(X,Y) = 0. For the proof, see [146, Lemma 2.1.1]. (1.12.3) Let A be a ring. The radical rad^/l of A as a left A-module and the radical rad AA of A as a right Amodule agree, and it is simply denoted by rad A and called the Jacobson radical of A. Note that rad ,4 is a twosided ideal of A li 0 ^ M £ AM is A-finite module, then M ^ (rad A)M (Nakayama's lemma). For basics on Jacobson radicals, see [123]. Lemma 1.12.4 Let A be a ring, and I = rad A The following are equivalent. 1 Any finitely generated A-module has a projective cover (in the category AM).
1* Any finitely generated right A-module has a projective cover (in the category ^opM). 2 The ring A/1 is semisimple, and for any idempotent e of A/I, there exists some idempotent e of A such that e modulo I equals e. We say that a ring A is semiperfect if the equivalent conditions in the lemma are satisfied. Lemma 1.12.5 Let A be a semiperfect ring, I = rad A, and p : P —» M an A-linear map between A-modules. Assume that M is A-finite. Then p is a projective cover if and only if P is A-finite projective and the induced map P/IP —> M/IM is an isomorphism. We say that a ring A is local if A/ rad A is a division ring. A ring A is local if and only if A ^ 0 and the set of non-units of A is closed under addition. We say that a commutative noetherian local ring R is Henselian if any /J-module finite algebra is semiperfect. If moreover the residue field of R is separably closed, then it is called strictly Henselian. Note that a complete local ring is Henselian [110, 111]. If a semiperfect ring A ^ 0 does not have any non-trivial idempotent, then A/ vad A is a division ring by Theorem 1.10.9.
1. From homological algebra
31
Let C be a Karoubian additive category, and C an object of C. We say that C is indecomposable if C ^ 0, and for any split monomorphism i: C" —» C, it holds either i = 0 or i is an isomorphism. If A := Endc C is local, then C is indecomposable, since A does not have any non-trivial idempotent. Conversely, if A := Endc C is semiperfect and C is indecomposable, then A is local. The following Krull-Schmidt theorem holds. Lemma 1.12.6 LetC be a Karoubian additive category. Then the following are equivalent. 1 For any M € C, there exists some decomposition M = Mi © • • •ffiMT (r > 0) such that Endc M{ is local for each i (in particular, each M< is indecomposable). 2 For any object M € C, we have Endc M is semiperfect. Moreover, if these conditions are satisfied, M € C, and there are two decompositions M = Mi © • • • © Mr = A^i © • • • © Ns such that Mt and Nj are indecomposable, then we have r = s, and there exists some permutation a £ S r such that Ni = Ma{ for any i. Here we only sketch the proof of 1=^2. Take a decomposition M = M\ © • • • © Mr and set E := End c M. For each i, set a{ : M{ M to be the inclusion, and TTJ : M —> Mj to be the projection. First prove that J := {
0) iii Ext^(M, w) = 0 (i > 0).
Hence, the morphism p in the short exact sequence in 3, i is a right X -approximation of M. 5 For M £ X, the following are equivalent. iMey ii Ext^X, M) = 0(i>0) iii A'-inj.dimM < oo, and M £ or1.
ii' E x t ^ * , M) = 0
Hence, the morphism L in the short exact sequence in 3, ii is a left yapproximation of M.
34
I. Background Materials
6 For M 6 X, we have A'-resol.dim(M) = y -pro j . dim (M) = w-proj.dim(M). 7 For Y £ y, we have w-resol.dim(y) = A'-resol.dim(y). 8 / / 0 —> Mi -> M2 -> M 3 -» 0 is an exact sequence of A and two of Mi, M2, and M 3 belong to X, then the third also belongs to X. We call an exact sequence as in 3, i an X-approximation. It is called minimal if p in 3, i is right minimal. We call an exact sequence as in 3, ii a y-hull. It is called minimal if L in 3, ii is left minimal. We say that a triple (X, y, u>) of full subcategories of an abelian category A is a weak Auslander-Buchweitz context in A if the conditions AB1—3 in the theorem are satisfied. If moreover X = A, then it is called an AuslanderBuchweitz context. Ii (X,y,cj) is an Auslander-Buchweitz context in A, then any of X, y, and w determines the others. We list some useful lemmas related to (weak) Auslander-Buchweitz contexts. Lemma 1.12.11 Let A be an abelian category, and a :
0 -> M o ->• Mi -> M 2 -> 0,
0o : 0 -> Yo -> X o -> M o -> 0, 02 : 0 -> Y2 -> X2 - ^ M 2 -» 0
exact sequences in A. If Ext2A(X2, Yo) = 0, then we have a commutative diagram 0 0 0
I
I
0
- > Vb - >
0
->
I Xo
I Yi
I ->
->
Mo
I
-¥
-¥ 0
X2
->
0
M2
-•
0
4- V1
-i-*
Y2
I Xi
40
-¥
Mi
I
0
->
I 0
0
exact rows and columns such that the first and third columns agree with 0o and 02, respectively, and the third row agrees with a. If moreover we have ExtiA(X2,Y0) = 0, then such a diagram is unique up to isomorphisms of diagrams. Proof. Set a' : 0 -» M o ^
M[ -> X2 -> 0
1. From homological algebra
35
to be the exact sequence atp, the pull-back of a by ip. From a', we have a long exact sequence Ext1A{X2,Y0)
-
If we have an exact sequence # : 0 -¥ Yo -> Xi ->• Af{ -»• 0 such that #(/3J) = /?o, then letting Y\ be the kernel of the composite morphism X\ -> M[ -» Mj, the construction of a diagram in question is easy. Conversely, if a diagram as in the lemma exists, then taking M[ to be the cokernel of the composite morphism Yo —> Y\ -» -X"i, we obtain /?J such that (p#(P[) = /?o, and giving /3J is the same thing as to construct such a diagram. Such a /3[ does exist if Ext^(Xj, Vo) = 0, and it is unique up to equivalence if Hence, the assertion of the lemma follows.
D
Proposition 1.12.12 Let A be an abelian category, and X, y, ZQ, and Z be full subcategories of A. Assume that X and y are closed under extensions, and Ext^(A", y) = 0. Moreover, assume one of the following: a Z = T(Z0) b Z = add F(ZQ) and Z C X. If for any Z £ Z§ there exists some exact sequence
o->y ->• x -> z ->o such that Y E y and X £ X, then there exists some exact sequence of the same type for any Z € Z. If for any Z £ ZQ there exists some exact sequence such that Y £ y and X £ X, then there exists some exact sequence of the same type for any Z £ Z. Proof. Assume a. The first assertion follows easily from Lemma 1.12.11, and the last assertion is the dual assertion of the first. The case where b holds is proved easily from a, and is left to the reader. •
36
I. Background Materials
Corollary 1.12.13 Let A be an abelian category, and XQ a full subcategory of A. We set X := add.7r(,fo)- Let LJ be a full subcategory of XC\XX closed under direct sums. If for any X 6 Xo there exists some exact sequence
such that W € LJ and X' £ X, then u> is an injective cogenerator of X. Lemma 1.12.14 Let X andy be full subcategories of A which satisfy A B 1 , AB2. We assume that y C XL. Let ui be an injective cogenerator of X such that ui C X ny and addw = u>. Then (X,y,to) is a weak AuslanderBuchweitz context in A. Proof. By assumption, X fl y is an injective cogenerator of X. Hence, we have that (X,y,X n y) is a weak Auslander-Buchweitz context. By Theorem 1.12.10, 2, we have w = X D y. D Remark 1.12.15 Some important terminology in category theory is not unified in its usage. The definition of Grothendieck category in these notes is the same as that in [33]. In Freyd's book [56], a Grothendieck category means an abelian category which satisfies the (AB5) condition. For the definition of an exact category, we follow Quillen [126]. There is some old literature in which the same expression 'exact category' is used for different meanings. For the definition of generator and cogenerator, we follow Auslander-Buchweitz [10]. A family of generators (resp. generator) in the sense of Grothendieck [63] is called a family of G-generators (resp. G-generator) in these notes for clarity. A family of G-generators and a generator (in our sense) are one and the same thing for an abelian category. Moreover, the words thick and epaisse, and sheaf and faisceau are used for different meanings in these notes. Epaisse is used only for triangulated categories, and a faisceau is a sheaf in the fppf topology. The definition of Grothendieck topology and a site is a little more restrictive than that in [9]. There, a Grothendieck topology in our sense is called a pretopology, and a site in our sense is a site whose topology comes from a pretopology. The definition of sheaf in (1.8.4) is the same as that in [9, (II.6.1)], but we put an unnecessary restrictive hypothesis on the value category C (i.e., the existence of limits) for simplicity. Similarly for the definition of continuous functor (1.8.11), see [9, (III.1.6)]. The definition of contravariant and covariant finiteness of X in (1.12) may be different from that in [13], if
Notes and References. This section is merely a survey of keywords for later use, and there is no new result. For basics on category theory, relative homological algebra, and cobar resolutions, we refer the reader to [144, 105,
2. From commutative ring theory
37
106, 56, 63, 58]. For exact categories, see [126, 141, 120] for more. For triangulated categories and derived categories, see [69, 143, 130, 121]. For Grothendieck topology and sheaf theory, see [8, 111, 9]. For AuslanderBuchweitz theory, see [10, 11, 112, 146, 13].
2
From commutative ring theory
This section is devoted to giving a summary of commutative ring theory used later.
2.1 Flat modules and pure maps An .R-module M is said to be R-fiat if ? ®R M is an exact functor. It is said to be R-faithfully flat if ? ®R M is faithful exact. An .R-algebra A is called i?-flat or fl-faithfully flat if the same holds for A as an .R-module. For any multiplicatively closed subset S of R, the localization Rs is i?-flat. The following is well-known. Lemma 2.1.1 Let M be an R-module. Then the following are equivalent. 1 M is R-flat. 2 For any R-module N and any i > 0, Torf (N, M) = 0. 2' For any finitely generated ideal I of R, Torf (R/I, M) = 0. 3 M is an inductive limit of an inductive system of R-finite free modules parameterized by a directed set. 3' M is a filtered inductive limit of R-flat modules. 4 For any commutative R-algebra S, S H M is S-flat. 4' For any m € Max/?, Mm is Rm-flat. The proof of 1=>3 is due to Lazard [100]. By Lazard's proof, the following holds. Lemma 2.1.2 Let R be a noetherian ring, and F a countably generated Rflat module. Then F is an inductive limit of an inductive system of R-finite free modules parameterized by the ordered set N. The following lemma is also well-known. L e m m a 2.1.3 Let M be an R-module. Then the following are equivalent. 1 M is R-faithfully flat.
38
I. Background Materials
2 M is R-flat, and M ^ mM for any m € Max R. An 7?-linear map / : N —> M of iZ-modules is said to be pure if lv ® / : y ® / V — ^ V ® M i s injective for any .R-module V. If / is a split monomorphism, or / is injective and Coker / is .R-flat, then / is pure. Note that a pure .R-linear map is injective. Let TV be an 7?-submodule of M. We say that TV is a pure submodule of M, if the canonical injection TV •-> M is R-pure. Lemma 2.1.4 Let R be noetherian, P and F be R-flat modules, and f : P —» F be an R-linear map. Consider the following conditions. 1 f is injective and Coker / is R-flat. 2 f is pure. 3 For p £ Spec R, /c(p) / : n(p) ® P ->• «(p) ® F is injective. 4 For any m £ MaxR, /t(m) ® / : «(m) P —> «(m) 0 F is injective. Then 1—3 are equivalent. If moreover P is R-projective, then 1—4 are equivalent. Proof. The direction 1=>2=>3=>4 is easy. First we prove 1, assuming 3, or that P is R-finite projective and 4 holds. We assume the contrary for contradiction. Then there exists some m 6 Max R such that / m is not injective or Coker / m is not .R-flat. Hence, we may assume that (R, m) is local. There exists some ideal / of R such that R/I (8i / is not injective or R/I ® Coker / is not i?/7-flat, because / = 0 is one of them. As R is noetherian, we can take a maximal such 7. Replacing R by R/I, we may and shall assume 7 = 0 is maximal among such. We set C := Coker/ and K := K e r / . Note that R/I <E> / is injective for any non-zero ideal 7 of R. Let 7 7^ 0 be an ideal of R. From the short exact sequence (2.1.5)
0 -> P/K -> F -> C -» 0,
we get a long exact sequence 0 ->• Torf (R/I, C) -¥ R/I ® P/K -> F / 7 F -> C/IC -»• 0. As R/I® f : P/IP -> F/IF is injective, we have R/I®P^ R/I P/7C and TOTf (R/I, C) = 0. Hence, C is 7?-flat. By the short exact sequence (2.1.5), P/K is also R-Hat. Hence, K -> P is pure and Tf is 7?-flat. In particular, 7?/7 ® 7C = 0 for any non-zero ideal 7 of i?.
2. From commutative ring theory
39
Note that K = 0 leads to a contradiction as we already know that C is .ft-flat. We prove K = 0. If R is not an integral domain, then we have R/p ® K — 0 for p € Spec /?, and /? admits a finite filtration
of .R-modules such that for any i, Ri/Ri-i is isomorphic to R/p for some p € Spec R. This shows that K = R K = 0 for this case. So we may assume that i? is an integral domain. First, consider the case that 3 holds. Then the canonical map K —> K(0) K = 0 is injective, and hence K = 0. Next, consider the case that P is .R-finite projective and 4 holds. In this case, we have K/mK = 0, and K is i?-fmite. By Nakayama's lemma, we have K = 0. Finally, we prove that if P is /?-projective and 4 holds, then 1 holds, and this completes the proof of the lemma. We may assume that R is local, and in this case, P is i?-free by Kaplansky's theorem [92]. We fix a basis B of P, and we denote the set of finite subsets of B by A. Note that A is a directed set with respect to the incidence relation. For A € A, we define P\ to be the free summand of P generated by A. We denote the composite map P\ P —> F by f\. Then we have K = limKer/x = 0, and C = lim Coker fx is .R-nat. • Corollary 2.1.6 Let R be noetherian, and P an R-flat module. If P ® K (p) = 0 for anV P £ Spec/?, then we have P = 0. Proof. Applying Lemma 2.1.4 to the map P —» 0, we have that this map is injective. • Lemma 2.1.7 Let A be an R-algebra, M and N be A-modules, and H be an R-module. Then the map p : HomA{M, N) <S> H ->• HomA(M, N ® H)
f h i-> (m >-> fm h)
is an R-linear map (an A-linear map if A is commutative) which is natural with respect to M, N and H. It is an isomorphism if one of the following holds. a H is R-flat and M is A-finitely presented. b M is A-projective and H is finitely presented. c H is R-finite projective. d M is A-finite projective.
40
I. Background Materials
Proof. We only prove a. As both Hom,i(?, N) ® H and Homj4(?, N ® H) are left exact, we may assume that M = An by the five lemma, which case is trivial. D Corollary 2.1.8 Let R be noetherian, and assume that I is an injective R-module, and F a flat R-module. Then I F is R-injective. Proof. Note that the category RM of /^-modules is locally noetherian, and RMJ is nothing but the full subcategory of finitely generated .R-modules. We have an isomorphism Hom(?, / F) =* Hom(?, / ) F of functors on RMJ by the lemma, and hence Hom(?, / F) is exact. By Lemma 1.9.4, / ® F is .R-injective. D (2.1.9) Let T be the full subcategory of i?-flat modules in «M. For an .R-module M, the ^"-resolution dimension (1.11) .F-resol.dimM of M is called the R-flat dimension (or fl-weak dimension) of M, and is denoted by flat.diniR M. Lemma 2.1.10 // P is an R-finitely presented R-flat module, then P is R-projective. Proof. Let / : V —> W be a surjective .R-linear map. We are to prove that Hom(P, / ) : Hom(P, V) -> Hom(P, W) is surjective. By Lemma 2.1.7, a, this is checked after localization at maximal ideals of R, and we may assume that (R, m) is local. As a local ring is semiperfect, P admits a projective cover p : F -> P. Note that F is finite free, and K := Kerp is finitely generated by assumption (see [110, Theorem 2.6]). Since P is flat, 0 -* K/mK -> F/mF - ^ ^ P/mP -> 0 is exact. As pR/m is an isomorphism by Lemma 1.12.5, we have K/mK = 0. Hence, K = 0 by Nakayama. We have P = F is projective. • Lemma 2.1.11 Let R be a commutative ring, P an R-projective module, and M an R-finite pure submodule of P. Then P/M is R-projective, and hence M P splits.
2. From commutative ring theory
41
Proof. As P is a direct summand of an R-free module, we may assume that P is an .R-free module with a basis B. As M is fl-finite, there exists some finite subset Bo of B such that M is contained in the fi-span Po of Bo. If we denote the fl-span of B \ Bo by Pu then we have P/M ^ Po/M © Pi. Hence, replacing P by Po, we may assume that P is infinite free. Then P/M is ^-finitely presented and i?-flat, and hence is /?-projective. D Similarly, the following holds. Exercise 2.1.12 Let R be noetherian, and / : M -> P be an .R-linear map. Assume that P is infinite projective and M is infinite. If / «(m) is injective for any m € MaxR, then / is a split monomorphism. The proof is left to the reader.
2.2
Mittag-Leffler modules
We review the theory of Mittag-Leffler modules after [128]. Throughout this subsection, A denotes a directed set. (2.2.1) We say that a projective system of .R-modules V — (P\, /A^)A€A,/J>A indexed by A satisfies the Mittag-Leffler condition if for any A € A, there exists some /x > A such that for any 7 > fi, we have Im /A 7 = Im f\^. Lemma 2.2.2 (Grothendieck) LetO-*V'^>V^>V"^O be an exact sequence of projective systems of R-modules indexed by A. Then we have 1 IfV' and V" satisfy the Mittag-Leffler condition, then so does V'. ilfV
satisfies Mittag-Leffler condition, then so does V" •
3 Assume that A has a final countable subset. If P' satisfies the MittagLeffler condition, then the sequence 0 -> limP' -> limP -> limT5" -> 0 is exact. For the proof, see [64, Proposition 0.13.2]. From now on, until the end of this subsection, any projective or inductive system is assumed to be indexed by a directed set. Lemma 2.2.3 Let V be a projective system of R-modules which satisfies the Mittag-Leffler condition. If F : RM -¥ Ab is a right exact functor, then F(V) satisfies the Mittag-Leffler condition. In particular, for any R-module M, the projective system M ®V satisfies the Mittag-Leffler condition.
42
I. Background Materials
Proof. Obvious.
D
Lemma 2.2.4 Let (P\) be an inductive system of finite free R-modules such that the protective system {P{) satisfies the Mittag-Leffler condition. If M is an R-module, then the "protective system (HomR (Px, M)) satisfies the Mittag-Leffler condition. Proof. Obvious by Lemma 2.2.3 and Lemma 2.1.7, d.
•
Definition 2.2.5 We say that an i?-module M is R-Mittag-Leffler if there exists some inductive system (FA) of finite free .R-modules such that M = lim FA and the projective system (F£) satisfies the Mittag-Leffler condition. By Lemma 2.1.1, an .R-Mittag-Leffler module is i?-flat. We list some properties of Mittag-Leffler modules. For the proof, see [128]. From Lemma 2.2.4, Lemma 2.1.2 and Lemma 2.2.2, we have the following. Proposition 2.2.6 Let R be noetherian. If M is a Mittag-Leffler R-module of countable type, then M is R-projective. The following criterion for the Mittag-Leffler property of an .R-module is due to Raynaud-Gruson [129]. Proposition 2.2.7 For an R-flat module M, the following are equivalent. 1 M is R-Mittag-Leffler. 2 For any inductive system (F\) of finite free R-modules such that lim F\ = M, the projective system (FJ) satisfies the Mittag-Leffler condition. 3 For any finite free R-module Q and any x € Q ® M, there is a smallest R-submodule Q' of Q such that x £ Q' M. Corollary 2.2.8 A pure submodule of a Mittag-Leffler R-module is MittagLeffler. Lemma 2.2.9 Let (M\) be a family of R-modules. Then($xM\ Leffler if and only if M\ is Mittag-Leffler for any X.
is Mittag-
Corollary 2.2.10 An R-projective module is R-Mittag-Leffler. Lemma 2.2.11 Let (M\) be an inductive system of R-Mittag-Leffler modules consisting of R-pure maps. Then lim M\ is R-Mittag-Leffler.
2. From commutative ring theory
43
Lemma 2.2.12 Let (M\) be a family of R-modules. Then for any finitely presented R-module N, the canonical map
is an isomorphism. Proof. As both sides are right exact with respect to N, we may assume that N = R, which case is trivial. D Corollary 2.2.13 / / moreover R is noetherian in the lemma, then there exists some isomorphism
Tor? (AT, I ] ^A) = I I T o r ? (N, Mx) A
A
for i > 0. Corollary 2.2.14 Let R be noetherian, and (M\) a family of R-modules. Then 17A ^ A is R-flat (resp. Mittag-Leffler) if and only if the same is true of M\ for any A. An argument similar to above shows the following. Proposition 2.2.15 Let R be noetherian, (P\) a projective system of Rflat modules (resp. R-Mittag-Leffler modules) indexed by a countable directed set which satisfies the Mittag-Leffler condition. Then limPx is R-flat (resp. R-Mittag-Leffler). A projective module over a noetherian commutative ring is characterized as follows: Theorem 2.2.16 Let R be noetherian, and P an R-module. Then the following are equivalent. 1 P is a direct sum of countable Mittag-Leffler R-modules. 2 P is R-projective. Proof. 1=>2 follows from Theorem 2.2.6. 2=>1 is well-known as Kaplan• sky's theorem [92]. Exercise 2.2.17 Let R be a noetherian, and F a countably generated flat .R-module. Prove that we have proj.dim^F < 1. Let F :
> Fn -> F n _! -> • • •
be a chain complex of .R-modules. We say that F is an i?-free (resp. Rprojective, /?-flat) complex if each Fn is i?-free (resp. .R-projective, .R-flat). A free complex F is said to be finite free (resp. finite projective) if each Fn is /^-finite free (resp. /J-finite projective) and F is bounded. Sometimes an .R-nnite projective complex is referred as a perfect complex.
44
I. Background Materials
2.3 Faithfully flat morphisms and descent theory (2.3.1) Let / : A —> B be a homomorphism of commutative rings, and assume that / is faithfully flat. Then by definition, F := / # = BA? : ^M —>• B M is faithful exact, and it has a right adjoint G := / * = Hom^-B, ?) : BM —• yjM. Then for an ,4-module V, the cobar resolution (1.6) Cobar F (V) of V with respect to the adjoint pair (F, G) is as follows:
where i+l
® • • • ®fy® u) = ;=0
and e(i>) = 1 u. More generally, if tp : Y -> X is a faithfully flat morphism of schemes, then yM is faithful exact, and y , is its right adjoint. For an Oxmodule M, applying T(X, ?) on Cobar^A-l), we have an exact sequence (2.3.2)
0 -> T(X, M) -> T(Y, ) and (M',') of quasi-coherent sheaves with respect to p, we say that an Oy-module map / : M. —> M! is a map of descent data if ' o p\f = p*2f o TV' is a map of quasi-coherent O^-modules, then p* 0. Proof. Let F be a finite projective resolution of M of length h. Then as Ext^(M, R) = 0 (i =fi h), the complex F*[/i] is a finite projective resolution of ExthR(M, R) of length h. Hence,
and it is 0 for i ^ h, and is M for i = h. As M ^ 0, we have gradeExthR(M,R) = h> proj.dim R Ext£(M,R) > gradeExt£(M,R), and Ext^(M, R) is also perfect of codimension h. Moreover, M, N) S ^(Hom R (F, iV)) = / T ( F N) S Tor^(Ext^(M, ij), TV). D
58
I. Background Materials
Corollary 2.9.2 Let R be a noetherian ring, I a Gorenstein ideal of R of codimension h, and N an R-module. Then we have
ExthR(R/I, R) ®R/I Tor?(R/I, N) * Ext{f (fl/7, N) for any i e Z . The next theorem is called the depth sensitivity of a resolution of a perfect module. Proposition 2.9.3 Let R be a noetherian ring, M a perfect R-module of codimension h, and N a finite R-module. Assume that M N ^ 0. Then we have inf{i | Ext'R(M, N) ? 0} + sup{j | Torf (M, N) ± 0} = h. (2.9.4) There is a nice resolution of a complete intersection ideal, called a Koszul complex. Let a i , . . . , a/, € R. We set F := Rh, and let e\,..., eh be an R-free basis of F. We define Kos(ai,..., o/,) to be the complex 0 -> AA F % A' 1 " 1 F->
>A1^J^>fl->0,
where the boundary map is given by di(eu A • • • A e,,) = ^ ( - l ) ^ 1 ^ ^ , A • •l- • • A e,,). i=\ For an i?-module M, we denote Kos(ai,..., a/,)J{M by Kos(ai,..., O/,; M). The next lemma, which is standard [110, Theorem 16.8], is called the depth sensitivity of a Koszul complex. Lemma 2.9.5 Let R be a noetherian ring, ai,...,ah € R, and M be a finite R-module. Set I := ( a i , . . . , a^) C R. If M ^ IM, then we have depth fi (7, M) = h- inf{z | fT i (Kos(ai,..., afc; M)) ^ 0}. In particular, Kos(oi,...,%; M) is a resolution of M/IM if and only if depth(7, M) = h if and only ifa\,..., a/, is an M-sequence. Assume that a i , . . . , a/, is an ii-sequence so that 7 := ( a i , . . . ,a/,) is a complete intersection ideal. Then F := Kos(ax,..., ah\ R) is a resolution of R/I. As we have F*[/i] = F, we see that 7 is a Gorenstein ideal. The next lemma is called the rigidity of a Koszul complex. Lemma 2.9.6 Let R be noetherian, I a complete intersection ideal of R of codimension h, and M an R-module. If i > 0 and Tor?(7?/7, M) = 0, then we have Torjl(7?/7, M) — 0 for any j > i. In particular, we have depth(7,M) > h-i.
2. From commutative ring theory
59
Note that even the following holds. Theorem 2.9.7 (Lichtenbaum [101]) Let R be a regular ring, M and N finite R-modules, and i > 0. If Tor?(M, N) = 0, then Torf (M, N) = 0 for j > i-
2.10
Dualizing complexes and canonical modules
For dualizing complexes, see [69] and [80]. (2.10.1) Let X be a noetherian scheme. We say that a complex of quasicoherent Ox-modules /* is a dualizing complex of X if /* is bounded, each term of 7* is an injective Ox-module, each cohomology group of /* is coherent, and the canonical map
is a quasi-isomorphism. Usually, a dualizing complex is regarded as an object of the derived category D + ( x M ) , and hence any object isomorphic to a dualizing complex in D+(xM) is also called a dualizing complex. Note that a quasi-coherent Ox-module 7 is an injective Ox-module if and only if its stalk Ix at x is an injective Ox,i-module for any x £ X [69, Proposition II.7.17]. If /* is a dualizing complex of X, for any complex F* of Ox-modules with coherent cohomology groups, the canonical map F* -> H o m ^ H o m ^ F ' , /*),/•) is a quasi-isomorphism [69, Proposition V.2.1]. The dualizing complex is unique, in the following sense. Theorem 2.10.2 ([69, Theorem V.3.1]) Let X be a connected noetherian scheme, 7* a dualizing complex of X, and I" a complex of Ox -modules bounded above with coherent cohomology groups. Then 7'* is dualizing if and only if there exists some invertible sheaf L and some integer n such that I'' is isomorphic to I* ®o x ^[ n ] * n D(xM). In this case, L and n are determined by
(2.10.3) A complex 7* of quasi-coherent Ox-modules is called a fundamental dualizing complex if 7* is bounded with coherent cohomology groups, and
©7'^©J(z) i€Z
x€X
60
I. Background Materials
is satisfied, where J(x) denotes the constant sheaf (ix),{Eox X(K(X)))~, where EOXX(K(X)) denotes the injective hull of the C?x,x-module K{X), and ix : Spec Ox,x —• X is the canonical map. A fundamental dualizing complex is a dualizing complex, and if there is a dualizing complex of X, then there is a fundamental dualizing complex of X [69, V.2.3, V.7.3]. (2.10.4) If there is a dualizing complex of X, then X is finite dimensional. A bounded-below complex /* of quasi-coherent Ox-modules with coherent cohomology groups over a locally noetherian scheme X is called pointwise dualizing if /* is a dualizing complex of SpecO^.i for any x G X. A dualizing complex of a noetherian scheme is pointwise dualizing. Conversely, a pointwise dualizing complex of a finite dimensional noetherian scheme is dualizing. (2.10.5) Let (R, m) be a noetherian local ring, and / ' a dualizing complex of R, that is to say, the complex of Ox-modules associated to / ' is a dualizing complex of X, where X = Spec/?. Then ExtlR(R/m,I') is non-zero for one and only one i, and it is isomorphic to R/m. If the i such that Ext^(i?/m,7') ^ 0 is zero, then we say that /* is a normalized dualizing complex. Note that a normalized fundamental dualizing complex is unique up to isomorphisms of /^-complexes. For a fundamental dualizing complex I' of (R, m), any localization 7* at p £ Spec R is again a fundamental dualizing complex. If moreover / • is normalized, then /*[—dim/?/p] is normalized. If R is a finite dimensional Gorenstein ring, then a minimal injective resolution /* of the .R-module R is a fundamental dualizing complex of R. If R is local and d = dim R, then I'[d] is normalized. The following theorem due to T. Kawasaki is an affirmative answer to Sharp's conjecture for local rings. Theorem 2.10.6 ([94, Corollary 6.2]) Let R be a noetherian local ring. Then R has a dualizing complex if and only if R is a homomorphic image of a Gorenstein local ring. The following theorem is known as the local duality theorem. Theorem 2.10.7 ([69, Theorem 6.2]) Let (i?,m) be a noetherian local ring, I' a normalized dualizing complex of R, and M a finite R-module. Then there is an isomorphism Hlm(M) * Homfi(Ext^(M, / ' ) , ER(R/m)) which is natural with respect to M.
2. From commutative ring theory
61
Proof. As Uom'R(M, I') is bounded with .R-nnite homology groups, it has a free resolution F, with each term finite free. As /* is dualizing, there exist quasi-isomorphisms M -> H o m ^ H o m ^ M , /*), /*) ^ H o m ^ F . , /*). By Corollary 2.1.8, H o m ^ F . , /*) is an injective resolution of M. Hence, we have
H'm(M) * H i (r m (Hom^(F. 1 /'))) £
Hi{HomR{F.,rm(r))).
Note that we have F m (/*) is quasi-isomorphic to Eii(R/m), which is easily seen when we consider the case /* is fundamental. As En(R/m) is an injective module, we have quasi-isomorphisms F., ER(R/m))
* Hom R (Hom^(M, / ' ) ,
ER(R/m)).
Hence, we have the isomorphism in question, as desired.
2.11
•
The duality of proper morphisms and rational singularities
(2.11.1) Let A be an abelian category, and A' its thick subcategory. We denote the full subcategory of D1(A) consisting of objects X such that H^X) € A' for any i by D\,(A), where ? is either b, +, - or 0. Obviously, D^^A) is a triangulated subcategory of D1{A). For a locally noetherian scheme X, we denote DlohX(xM) (resp. Z ^ c o X U M ) ) by Dl(X) (resp. Dlc(X)). Note that the forgetful functor £>?(QcoX) -> D\C(X) is an equivalence for a quasi-compact scheme X for ? = +,0 [25, 6.7]. (2.11.2) Let X be a noetherian scheme. The following was proved by M. Nagata [118]. See also [103]. Theorem 2.11.3 Let f : Y -> X be a morphism of finite type between noetherian schemes. Then f is compactifiable in the sense that there exist some scheme Y, a proper morphism p : Y —> X, and an open immersion i: Y -¥ Y such that pi = / . A factorization pi = f as in the theorem is called a compactification of / . The following is known as the global duality theorem of proper morphisms, see [121] and [102]. See also [69] and its appendix by Deligne [41]. Theorem 2.11.4 Letp :Y -» X be a proper morphism between noetherian schemes. Then the derived functor Rpt : Dqc(Y) —>• Dqc(X) (the unbounded derived functor, see [137]) has a right adjoint p' Dqc(Y). We have p'uh(D+c{X)) C D+C(Y), and the restriction of p''ab to D+.(X), which we denote by p, is a right adjoint functor ofR+pt.
62
I. Background Materials
(2.11.5) Let F(X) denote the category of X-schemes of finite type. Any morphism / : Y -> Y' of F(X) has a compactification pi = f. We define f := i'op1 : D+C(Y') -» D+C(Y), where p ! is the right adjoint of R+pt. Proposition 2.11.6 Under the notation as above, the following hold. 1 The definition of f' is independent of the compactification pi = f of f, up to isomorphisms of functors. 2 For two morphisms f and g in F(X), that go f is defined.
we have (g o / ) ' = / ! o g'-, provided
3 (Residue isomorphism) If h : Y —> Y' is a smooth X-morphism of relative dimension d, then h' is isomorphic to h* := h*? ®£>y wy//y[d]. 4 / / g : Y -» Y' is a finite X-morphism, g'RHom'Cr,(g*Qv,?), where g : (Y,OY) morphism of ringed spaces.
then g- is isomorphic to g^ := ->• (Y',gtOY) is the canonical
5 Let f : Y —> Y' be a morphism of F(X), and g : Z' —» Y' aflat morphism of noetherian schemes. If gf = fg' is a fiber square, then we have a canonical isomorphism (g')* o /• = (/')' o g*. 6 If X has a dualizing complex Ix, then Iy •= fix is a dualizing complex of Y for any morphism f :Y —> X of finite type. Corollary 2.11.7 (Duality for proper morphisms) Let p : Y —» X be a proper morphism between noetherian schemes, F e Dqc(Y) and G £ D*C(X). Then there is an isomorphism 9P : /Zp..RHomg v (F.p l G) = /ZHom^.(flp.F.G). which is functorial on F and G. Proof.
(Sketch) Consider the composite of the canonical maps RHom' ^
(Rp.F,e) >
where the first arrow is the natural map given in [102, (3.5.4)], and e : Rptp- -> Id is the counit of adjunction. It suffices to show that
RT(U,ep) : Rnom'0uy(W\UY, (p'G)\Uy) -» R is an isomorphism for any open set U of X, where Uy = p~l(U). As we may identify (P'G)\UY with (p|£/y)!(G|t/) by 5 of Proposition 2.11.6 and (Rp,F)\u by R(p\uy)*F\uY' w e m a y assume that U = X, after replacing X by U and p by p\uY • Then the assertion is clear, because p! is the restriction of p ! ' ub , and p ! ' ub is the right adjoint of RptD
2. From commutative ring theory
63
(2.11.8) Usually, if the base scheme X and its dualizing complex Ix are obvious from the context, then we call IY := f{Ix) 'the' dualizing complex of Y for any X-scheme / : Y —> X of finite type. If X = Spec /? is affine and R is a d-dimensional Gorenstein local ring, then we always consider that Ix = R[d}. (2.11.9) Let X be a noetherian scheme with a fundamental dualizing complex Ix, and Y a connected X-scheme of finite type. We set the minimal i £ Z such that H'(IY) ^ 0 to be r. We denote Hr{IY) by wy, and call it the canonical sheaf of Y. The coherent Oy-module wy is determined only by Y (not depending on X or Ix) up to the tensor product with an invertible sheaf. If Y is disconnected, then we define u)Y componentwise. Let (R, m) be a complete noetherian local ring. In this case, the fundamental dualizing complex IR of R is uniquely determined up to degree shifting. Hence, U/R is uniquely defined to be the non-zero cohomology group of IR. We call UIR the canonical module of R. For a not necessarily complete noetherian local ring (R, m), if there exists a finite ^-module K such that K = wR, then such a K is unique up to isomorphisms, where R is the m-adic completion of R, and K — R® K. We usually denote this K by KR, and call it the canonical module of R. If R has a dualizing complex, then we have GiR = KR. However, R may not have a dualizing complex even if there is a canonical module of R. Lemma 2.11.10 Let Y be a connected X-scheme of finite type. Then the following are equivalent. 1 For some d 6 Z, we have LJY = IY[—d] in D(Y). 2 wy has a finite injective dimension as an object of YM. 3 Y is Cohen-Macaulay. If the conditions are satisfied, then we have supp wy = Y, and in particular we have wy,y = u>oYjV for any y eY. Proof. 1=^2 is trivial. 2=>3 We set suppwy = Z. We define r to be the minimum i such that H'(Iy) j= 0. Let J be a fundamental dualizing complex which represents IY. Note that Z is the union of all irreducible components YJ of Y such that J(rji) is a direct summand of Jr, where T)t is the generic point of Yt. For y 6 Z, wy y is a non-zero, finitely generated Oy>!(-module of finite injective dimension by [69, Proposition II.7.20]. Hence, by Theorem 2.5.6, Y is Cohen-Macaulay at any point of Z. We denote by Z' the union of all irreducible components of Y not contained in Z. If Z' ^ 0, then as Y is connected, there is a point y G Z D Z'. As Jy is a fundamental dualizing complex of the
64
/. Background Materials
Cohen-Macaulay local ring Gy,y, we have that the positions s at which EoYn (K(rji)) appears as a direct summand of J* are equal for all generic points of irreducible components of Y which contain y. This contradicts y € Z fl Z', and we have that Z' = 0. Hence, we have suppwy = Z = Y, and Y is Cohen-Macaulay. We show 3=>1. An argument similar to above shows that suppwy = Y. For each y € Y, there is at most one i such that H^ (OY,y) ^ 0. By the local duality, there is at most one i such that Hl(IYiy) ^ 0. As we have ujYy 7^ 0, there exists some d G Z, which is independent of y, such that H~d(IY,y) = uy,y 7^ 0. Hence, we have Hl(IY) = 0 (i ^ -d), and we are done. • Corollary 2.11.11 Let f : Y -> Y' be a finite X-morphism between Xschemes of finite type. If Y is connected and Y' is Cohen-Macaulay, then we have where h = codimy Y. Proof. We may assume that Y' is also connected. The morphism of ringed spaces / : (Y, OY) -> (Y1, f,OY) is flat. There exists an integer d such that IY Y' be a smooth X-morphism between noetherian X-schemes of finite type. Then we have U)Y = LJyi ®
Proof. Obvious.
Wy/Yi.
Q
Proposition 2.11.13 Letk be afield, X an x-dimensional Cohen-Macaulay normal k-variety, Y a k-scheme of finite type. Let f : X -> Y be a proper k-morphism, and assume that Oy -> f,Ox is an isomorphism, RlftOx = 0 (i > 0), and there exists an r > 0 such that Rlf,u>x = 0 (i ^ r). Then Y is an (x — r)-dimensional Cohen-Macaulay normal variety, f is surjective, and Rrftux = wy-
2. From commutative ring theory
65
Proof. Note that Y is connected, as X is connected and the support of f,Ox is contained in exactly one connected component of Y. Set y := dimV. As Y is of finite type over Specfc, it is easy to see that u>y = H~V(IY) (we consider that the current base scheme is Specfc). By Corollary 2.11.7 and assumption, we have
H\IY) = Ex£,Y(f,Ox, h) = Ri+Xf^xBy assumption and Lemma 2.11.10, we have Y is Cohen-Macaulay, — y+x = r, and Rrftux = wy. It remains to show that Y is normal and / is surjective. As the question is local on Y, we may assume that Y = Speed, with A of finite type over k. As
A = H°(X,OX) =
f)Ox,x
xex is an intersection of normal domains, A is a normal domain. As / is domi• nating and is a closed map, it is surjective. (2.11.14)
A desingularization / : X —> Y which satisfies the conditions
Rif*Ox = 0 {i > 0), f.Ox = OY, and fff^x = 0 (i > 0) is called a rational resolution of Y. By Proposition 2.11.13, if Y has a rational resolution, then it is a Cohen-Macaulay normal variety. Assume that the characteristic of k is zero, and Y is integral. Then Y has a rational resolution if and only if any desingularization of Y is rational. If the equivalent conditions are satisfied, then we say that Y has (at most) rational singularities.
2.12
Summary of open loci results
Let R be a noetherian ring. Definition 2.12.1 We say that a finite .R-module M is of Gorenstein dimension 0 if M is reflexive (i.e., the canonical map M —> M** is an isomorphism), and Extjj(M, R) = 0 = Extjj(M*, R) for any i > 1. We set Q to be the full subcategory of RMJ consisting of modules of Gorenstein dimension 0. For N € RM^, we call Q -resol.dim N the Gorenstein dimension of N. L e m m a 2.12.2 Let R -> 5 be a homomorphism of commutative noetherian rings, N a finite R-module, M a finite S-module. 1 For each of the following conditions, the subset of Spec R consisting of p € Speci? such that the condition is satisfied is Zariski open: Mp — 0, Mf is Sp-free, Mp is of Gorenstein dimension 0 as an Sp-module.
66
I. Background Materials
2 If S is of finite type over R, then the subset of Spec 5 consisting of P € Spec 5 such that Mp is RRnP-flat is open. 3 If S is essentially of finite type over a complete local ring or S is essentially of finite type over Z, then for each of the conditions CohenMacaulay, Gorenstein, l.c.i., and regular, the subset of Spec 5 consisting of P € Spec 5 such that the condition is satisfied for Sp is open. 4 If S has a dualizing complex, then for each of the conditions equidimensional, Cohen-Macaulay, and Gorenstein, the subset of Spec S consisting of P S Spec 5 such that the condition is satisfied for Sp is open. 5 If S is a homomorphic image of a Cohen-Macaulay ring, then the CohenMacaulay locus and the MCM locus of M are open. In particular, the Cohen-Macaulay locus of S is open. For the proof, see [110], [61], and [65]. Corollary 2.12.3 Let R be a noetherian ring, and M a finite R-module. For each i with 1 < i < oo and each of the following conditions, the subset of Spec R consisting of P G Spec R such that the condition is satisfied is open: proj.dim fip MP < i, Q -resol.dim MP < i and MP is zero or perfect of codimension i. If R is Cohen-Macaulay, then the subset of P such that dim Rp — depth R p Mp < i is also open. Corollary 2.12.4 Let tp : X —¥ Y be a morphism locally of finite type between locally noetherian schemes. Then for each P of the following properties, the subset [/(P, Ii+J/Ii+i+l induced is a graded Gby the product of R. Note that Gr7 M := ©i>0 PM/Ii+1M module in a natural way. If Gr/ M is i?//-flat (in other words, if PM/P+1M is i?//-flat for any i £ No), then we say that M is normally flat along /. Let A" be a scheme, I a quasi-coherent ideal sheaf of Ox which defines a closed subscheme Y of X, and M £ Qco(X). Then we define Qx%M := ©i>0 TM/li+lM. Note that Gr z Ox is a sheaf of Oy-algebras, and Gr z M is a Grj Ox-module. If (Grj M)x is Ox.x/^i-flat for all x £ X, then we say that M is normally flat along I (or along V). We say that X is normally flat along Y if Ox is. Lemma 2.13.2 Let R be a noetherian local ring, and I a proper ideal of R. Then the following are equivalent. 1 / is a complete intersection ideal. 2 proj.dim R / < oo, and I/I2
is R/1-free.
2' proj.dimR I < oo, and R is normally flat along I. 3 I/I2 is R/1-free, and the canonical map Sym fl / 7 1/I 2 —• Gr/ R is an isomorphism. For the proof, see [110]. Theorem 2.13.3 ([110, Theorem 15.7]) Let R be a noetherian local ring, and I a proper ideal of R. IfwesetG = Gi[R, then we have dim G = dim R.
68
I. Background Materials Although we will not use it later, the following is also important.
Theorem 2.13.4 Let R be a noetherian local ring, and I a proper ideal of R. If G is Cohen-Macaulay (resp. Gorenstein, regular, normal), then so is R. For the proof, see [34, Theorem 3.9, Theorem 3.13]. This theorem is a corollary to Corollary II.2.4.3, see [16]. Lemma 2.13.5 Let (R, m) be a noetherian local ring, and M a finite Rmodule. Let P be a prime ideal of R, and assume M is normally flat along P. Then the following hold: 1 IfM^O,
then dim M = dim MP + dim R/P.
2 If R is normally flat along P, then M is R-free if and only if Mp is Rp-free. If, moreover, Ext'R(R/Pt M) is R/P-free for all i > 0, then the following hold: 3 We have depth M = depth Mp + depth R/P. In particular, M is CohenMacaulay if and only if both MP and R/P are Cohen-Macaulay. If this is the case, then we have type M = type MP • type R/P, where type denotes the Cohen-Macaulay type, see (2.8.6). In particular, M is Gorenstein if and only if both Mp and R/P are Gorenstein. 4 Assume that R/P i>0.
is Gorenstein.
Then /x£
Proof. 1 As M/PM ^ 0 and M/PM dim M/PM, and hence
lm
is R/P-hee,
(M) = ^lRp{Mp)
for
we have dim R/P =
dim M > dim MP + dim R/P. On the other hand, dim M - dim Gr P M < dim /t(m) Gr F M + dim
M/PM.
As Grp M is /?/P-flat, we have dim K(ITI) Gr P M — dim K(P) Grp M = dim QXPRP MP
since the Hilbert function of K(m)Grp M agrees with that of Hence, dim M < dim MP + dim R/P.
= dim K(P)^)GTP
MP,
M.
2. From commutative ring theory
69
We show 2. The 'only if part is trivial. We prove the 'if part. By assumption, the canonical map 7n
: (Pn/Pn+1)
®R/P M/PM
->
PnM/Pn+1M
is a surjective map of finite free R/P-iree modules for any n. The localized map (jn)p is an isomorphism by [110, Theorem 22.3], as we assume that Mp is .Rp-free. As 7 n is a surjective map between finite free modules of the same rank, it is an isomorphism. Using the local criterion [110, Theorem 22.3] again, M is fl-flat, and it is R-iree. We show 3. We set depth Mp = qQ and depth R/P = p0. There is a spectral sequence (2.13.6) E%« = ExtpR/P{R/m, As ExtqR(R/P, M) is R/P-tee, ExtR(R/P,M)P
ExtqR(R/P, M)) => ExtpR+q{R/m,
M).
ExtqR(R/P, M) = 0 if and only if = ExtRp(K(P),MP)
= 0.
Hence E%'q = 0 for q < q0. On the other hand, as ExtqR{R/P, M) is R/Pfree, we have E^ = 0 for p < p0. Hence Ext^(i?/m, M) = 0 for i < p0 + q0. On the other hand (2.13.7)
Ext%+qo(R/m, M) S E™M ^ E$°'qo
, Ext%(R/P, M)) ± 0. Hence depth M =po + q0 = depth MP + depth R/P. By 1, M is Cohen-Macaulay if and only if both Mp and R/P are CohenMacaulay. Assume that M is Cohen-Macaulay (and hence both Mp and R/P are Cohen-Macaulay), and set typeMp = r and type R/P = r'. Then as we have Extq£{R/P,M) ^ {R/P)Br, , M) S Ext£ / P (i?/m, {R/P))®r
£
by (2.13.7). This shows typeM = rr'. 4 As we have E$'q = 0 for p =£ p0 = dim R/P in (2.13.6), the spectral sequence (2.13.6) collapses, and the assertion is clear. • Remark 2.13.8 In [81] are listed some examples which show that even if R/P is regular, RP is Cohen-Macaulay and R is normally flat along P, R may not be Cohen-Macaulay. The freeness of Ext'R(R/P, R) is really necessary.
70
I. Background Materials
Lemma 2.13.9 Let (R,m) be a noetherian local ring, and P G Spec/?. Assume that R is normally flat along P, and R/P is regular. Then we have: 1 emb.dim R — dim R = emb.dim Rp — dim Rp. In particular, R is regular if and only if Rp is regular. 2 Assume moreover that ExtfR(R/P, R/P) is R/P-free fori>2. Then R is a complete intersection if and only if RP is a complete intersection. Proof. We take bi,... ,bd G m so that the image of b\,... ,bd in R/P forms a regular system of parameters of R/P, where d = dim R/P. We set J := (&i,..., bd). As the image of 6 X ,..., bd in m f l / p / m ^ p = m/(m 2 + P ) is linearly independent, its image in m/m 2 is also linearly independent, and we have emb.dim R/J = emb.dim R — d. As &i,..., bd is an /?/P-sequence, the Koszul complex Kos(6i,..., bd; R/P) is acyclic. As P{/Pi+l is R/P-hee by assumption, Kos(6 1 ; ..., bd; P'/P'+1) is also acyclic. If j > 0 and a is a j-cycle of F := Kos(&i,... ,bd; R), then a G r\i>0(Bj(¥) + P'F 7 ) = Bj(¥), and hence F is also acyclic, where Bj(F) is the set of j-boundaries of F. We have that b\,..., bd is an it-sequence, and F is a free resolution of R/J. In particular, dim R/J = dim R- d. Next, take a free resolution G of R/P. As b\,... ,bd is .R/P-regular, we have Tox?{R/P,R/J) = 0 (i > 0). Hence, G ®R R/J is an R/J-iiee resolution of R/m = R/(J + P). As we have an isomorphism (G ® fi G) K R/J s (G fi R/J) ®R/J (G ® R
R/J),
we obtain a spectral sequence Elq = Tor*(Torf (R/P, R/P), R/J) => Toi%.Jq(R/m, R/m) associated with the double complex (G ®R G) F. As TOT*(R/P, R/P) = R/P and Torf {R/P, R/P) S P/P2 are R/P-hee by assumption, we have isomorphisms Torf/J (/?/m, R/m) s Torf (R/P, R/P) ®H R/J ^Tov«(R/P,R/P)®R/PR/m for i = 0,1,2. As Torf (R/P, R/P) is R/P-hee for i = 0,1, emb.dim R/J = (3*'J(R/m) = dimR/m Torf (R/P, R/P) ® H / P R/m = dim K ( P ) Torf p (/t(P),K(P)) = emb.dim RP. Combining this with Lemma 2.13.5, we have emb.dim R - dim R = emb.dim R/J + d - dim R = emb.dim RP — dim RP,
3. Hopf algebras over an arbitrary base
71
and we have proved 1. We show 2. As G, G) ®fi R/J ^ H o m ^ G ®fi R/J, G ®R R/J) and proj.dim^ R/J < oo, there is a spectral sequence Ep2'q = Tor%(ExtR(R/P, R/P), R/J) =» Ext^« (fl/m, R/m), see Lemma III.2.1.2. As Ext°R(R/P,R/P) S fl/P and ExtJ,(fl/P, R/P) S Hom fl/P (P/P 2 , fl/P), we have that Ext^(.R/P, i2/P) is R/P-hee for g > 0 by assumption. Hence E%q = 0 (p 7^ 0), and we have an isomorphism Ext^iJ/P, fl/P) (gifi/p fi/m ^ Extje/J(i?/m, i?/m). This shows
for i > 0. By Proposition 2.8.4, R/J is a complete intersection if and only if RP is. As J is a complete intersection ideal, R/J is a complete intersection if and only if R is. Hence, R is a complete intersection if and only if Rp is. D Notes and References. There is no new result in this section, except for some of the lemmas followed by proofs. Although some important topics such as Cohen-Macaulay rings and perfect modules are reviewed from the first definitions, this section is merely a glossary on commutative ring theory. For basic notation, terminology and results on commutative ring theory, see [110] and [26]. Undefined terminology on algebraic geometry should be found in [71]. We treat Cohen-Macaulay approximations and related topics in subsection 4.10.
3
Hopf algebras over an arbitrary base
This section is devoted to reviewing Hopf algebras over an arbitrary commutative ring R. All results in this section are basic, and some non-trivial results can be found in [90, 145].
72
3.1
I. Background Materials
Coalgebras and bialgebras
(3.1.1) We say that A is an .R-algebra, if A is a ring, and a ring homomorphism u : R —> A such that u(R) C Z(A) is given, where Z(A) denotes the center of A. This is the same as to say that A is an .R-module, fi-linear maps u : R —> A and m : A ® A —> A are given, and the diagrams A® A® A
m
I
®lA-A®A I
I" A®A
I ^
»A
_ A®R—^—
_ A *—^—R®A
are commutative. In fact, if A is an .R-algebra, then A is an .R-module in a natural way, and if we define m : A ® A ->• A by m(a ® a') := aa\ then it is easy to see that the diagrams above are commutative. Conversely, if u and m are given so that the diagrams are commutative, then A is a ring with the product aa! := m(a®a'), and A is an .R-algebra by u : R —> A. We call m = rriA '• A ® A —> A the product map of A, and u = UA '• R —> A the unit map of A. The notion of R-coalgebra is the dual to that of .R-algebra. Namely, Definition 3.1.2 We say that a triple (C, A, e) is an R-coalgebra if C is an .R-module, and A : C -* C ® C and e : C -» .R are .R-linear maps such that the diagrams ) C < Aig>lc
tA .
®A ?,
g ( g ) g
^^
C
are commutative. We sometimes say that C is an .R-coalgebra if there is no confusion. The commutativity of the first (resp. second) diagram is called the coassociativity law (resp. counit law). We call A = Ac the coproduct of C, and e = £c the counit map of C. If, moreover, C is an .R-algebra and Ac and £c are .R-algebra maps, then we say that C is an R-bialgebra. An .R-coalgebra C is called cocommutative if r o Ac = Ac, where T : C ® C -> C ® C is the ii-linear map given by T(C ® d) = d ® c for c,d e C. Note that cocommutativity is the dual notion of the commutativity of an algebra. Let A and A' be .R-algebras. Then to say that
C is an R-coalgebm map if ip is .R-linear, £ G ° V' = £ c , and Ac o ^ = (ip<S)tp) o A c Let S and 5 ' be i?-bialgebras. Then we say that / : B —> B' is an R-bialgebra map if / is both an .R-algebra map and an /2-coalgebra map. Example 3.1.4 Let B be an .R-bialgebra, and b € B. We say that b is a group-like element if es(b) = 1 and AB(b) = b ® b hold. Let G be a semigroup. Then the group algebra RG is an .R-algebra. It has a unique .R-bialgebra structure such that each g £ G is group-like. Any semigroup homomorphism G -¥ G' is uniquely extended to an .R-bialgebra map RG -> RG'.
3.2
Hopf algebras
(3.2.1) We say that G is an R-semigroup scheme if G is an R-scheme, endowed with R-morphisms e : SpecR —> G and /i : G XRG —t G subject to the semigroup-law Ato(/ix l G ) = / i o ( l G x / x ) ,
fio(lGxe)=pG,
no (e x 1G) = AG,
where pG • G x Spec R = G and AG : Spec i? x G = G are the canonical identifications. We call fi the product of G, and e the unit map of G. A homomorphism of .R-semigroup schemes is an .R-morphism which preserves (j, and e. The category of affine /{-schemes is contravariantly equivalent to the category of commutative /^-algebras. Hence, an affine ^-semigroup scheme G = Spec B, which is completely described in terms of objects and morphisms of the category of affine .R-schemes, can be descried in terms of commutative .R-algebras and .R-algebra maps between them. In fact, B is a commutative .R-bialgebra if and only if G = Spec B is an affine .R-semigroup scheme. A homomorphism of affine .R-semigroup schemes corresponds to an .R-bialgebra map. We say that G is an .R-group scheme if G is an .R-semigroup scheme such that there exists an .R-morphism t : G —> G such that fiG o (1 G x t) o A G = fiG o (t x 1G) o A G = e o uG holds, where uG : G -t Spec R is the structure morphism of G, and A G : G —> G Xft G is the diagonalization. For an .R-semigroup scheme G, such an i is unique, if it exists, and it is called the inverse of G. Considering the case that G is affine, translating the condition for t into the context of commutative bialgebras, and generalizing it to the noncommutative case, we get the following definition.
74
I. Background Materials
Definition 3.2.2 Let B be an .R-bialgebra. We say that 5 : B -> B is an antipode map of B if 5 is .R-linear, and the equality rnB o ( 1 B ® 5 ) o A B = m f i o ( S ® 1B) o A B = uB holds. An .R-bialgebra is called an R-Hopf algebra if it has an antipode map. Remark 3.2.3 The following is known. i If an antipode 5 = 5 B of B exists, then it is unique. ii 5 is an anti-algebra anti-coalgebra map. Namely, the equalities S{bb') = Sb'• Sb,
5(1) = 1,
TO(5®5)OA
= Ao5,
eoS = e
hold, where T : B ® B -> B B is the map given by b ® 6' i-» 6' ® b. iii If B is commutative or cocommutative, then we have 5 2 = id#. iv If B and B' are .R-Hopf algebras and
B ' is an .R-bialgebra map, then we have tp o SB = SB' ° ¥>• Exercise 3.2.4 In Example 3.1.4, G is a group if and only if RG is an .R-Hopf algebra. If G is a group, then S(g) = g~x gives the antipode. Conversely, if 5 is an antipode of RG, then we have g • S(g) = S(g) • g = 1 by the definition of antipode, and it is easy to see that g is invertible in G. In particular, considering the case that G is the trivial group, R is an /J-Hopf algebra. In general, if B is an .R-Hopf algebra and b € B is group-like, then we have 5(6) = b-1. Hence, the set of group-like elements X{B) of B is a subgroup of B*. If G is a group and R has no non-trivial idempotents, then we have X(RG) = G, and we can recover G from the Hopf algebra RG. Let A and A' be commutative .R-algebras, C an ,4-coalgebra, and C" an /l'-coalgebra. Then letting c ®a
Ac
®Ac (c ®A c) ( c ®* c ) (c ® c )
be the coproduct and ec ® £c be the counit map, C <S) C' is an A ® A'coalgebra. If moreover C and C are an .A-bialgebra and an >l'-bialgebra, respectively, then C C is an A ® /l'-bialgebra. If moreover C and C" have the antipode 5 and 5', respectively, then 5 5' is an antipode map of C®C. In particular, considering the case A = A' = R, a tensor product of .R-coalgebras (resp. .R-bialgebras, .R-Hopf algebras) is again an .R-coalgebra (resp. .R-bialgebra, .R-Hopf algebra). Considering the case R = A and C" = A', the base change C A' is an /l'-coalgebra (resp. ^4'-bialgebra, >4'-Hopf algebra).
3. Hopf algebras over an arbitrary base
75
Exercise 3.2.5 Let 5 be an .R-algebra i?-coalgebra. Then B is an Rbialgebra (i.e., both eB and A s are .R-algebra maps) if and only if both UB and rag are fl-coalgebra maps, where the coalgebra structure of the tensor product B M ® C is given, and PM
°
(1M
® £c) ° uM = idM,
( % ® lc) ° w M = ( 1 M A c )
hold, where p M : M .R = M is the canonical identification. We call UJM the structure map of M. Similarly, a left C-comodule is defined. In these notes, a C-comodule means a right C-comodule unless otherwise specified. Note that C itself is a C-comodule with the structure map Ac- Note also that Ac makes C a left C-comodule. Definition 3.3.2 Let M and M' be C-comodules. We say that / : M -> M' is a C-comodule map if / is .R-linear and U>M' ° / = (/ lc) ° W M holds. It is easy to see that taking C-comodules as its objects and C-comodule maps as morphisms, we have an additive category. We denote this category by M c . Similarly, we have the category of left C-comodules, which we denote by C M. Lemma 3.3.3 // C is R-flat (as an R-module), then M c is an R-linear abelian category which satisfies the (AB5) condition. Proof. It is easy to see that the set of C-comodule maps Hom M c(M, M') is an .R-submodule of H o m ^ M , M') for M, M' 6 M c . As ? (g> C preserves inductive limits (in particular, cokernels and direct sums) and kernels (as C is .R-flat), inductive limits and kernels as .R-modules are endowed with structures of C-comodules, and they are inductive limits and kernels in M c , respectively, in a natural way. The lemma follows. D We denote Hom M c(M, M') by Hom c (M, M') for C-comodules M and M'. Let C be an .R-flat coalgebra. For M £ M c and an .R-submodule iV of M, we say that iV is a C-subcomodule of M if LJM(N) C N ®C holds. Note that if ./V is a C-subcomodule of M, then N itself is a C-comodule with the structure map % |JV, and the inclusion Af ^-» M is a monomorphism of M c . Subobjects of M and subcomodules of M are in one-to-one correspondence, and we may identify them.
76
I. Background Materials
3.4
Sweedler's notation
Let C be an i?-coalgebra, and M € M c . For n > 0, we define u/^ M®C®
n
: M ->
by an inductive definition; OJM : = idM and w^j := ( W M ® lc® Hom(K, V ® C ) S Hom(V, V") ® C by w C '(/) :=uv'°
f-
78
I. Background Materials
Lemma 3.5.5 Let C,C',V and V be as above. Then Hom(V, V') is a (C,C')-bicomodule with the structure maps c& anduic- If, moreover, C" is an R-coalgebra, V is a (C",C)-bicomodule, and V is a (C",C')-bicomodule, then Homc"(V, V) is a (C, C')-subbicomodule of Hom(V, V). The canonical map Hom(V, R) ® V -> Hom(V, V) is a (C,C')-bicomodule map. Proof. We only prove the coassociativity with respect to cw. By definition, if we set c^f := 12(f) /i ® /o, then the equality
holds. As the isomorphism in Lemma 2.1.7 is natural on H, it suffices to prove (/)
(/)
(/o)
(/i)
Since we have
(/) (/o)
(/)
(v)
/u 0 = (A ® 1) ^ ui ® /u 0 = (A the assertion follows.
D
Lemma 3.5.6 ie< C and C be R-coalgebras, M a (C',C)-bicomodule, and V a left C-comodule. Then the map $ : Homc «(M, V) -»• Hom (c -, C) (M, V ® C) defined by $(/)(m) := E(m)/ m (o) ® "i(i) *5 well-defined, and an isomorphism which is natural with respect to M and V, where S(m) 7?io<E>mi is the image of m by the structure map of M as a C-comodule. Proof. It is easy to verify that $ ( / ) is a ( C , C)-bicomodule map, and hence $ is well-defined. We define * : Hom(c Hom C '(M, V) by *(p)(m) := Eg(m), where £ : V ® C -> V is denned by E(v ® c) := e(c)w. It is also straightforward to check that ty(g) is a C'-comodule map. When we set gm = Yli Vi ® Q, then we have
(m)
3. Hopf algebras over an arbitrary base
79
This shows $\I> = id. On the other hand, we have
) = E{*(f)(m)) = e(ml)f(mQ) = fm, and hence * $ = id. Thus, $ is an isomorphism. Let C be an fl-coalgebra. Then the functor F right adjoint G =? C by Lemma 3.5.6. For M € resolution Cobar;r(M) (1.6) the cobar resolution of Cobar c (M). More explicitly, we have Cobar c (M) i = boundary map is given by
• c
: M -» RM has the M c , we call the cobar M, and denote it by MC®(i+1), and the
t=0
By definition, we have: Lemma 3.5.7 Let V be an R-module, and M a C-comodule. Then we have VCobarc(M) = Cobarc(Vr(giM) as complexes ofC-comodules. Moreover, we have
Cobarc (M)' S Cobarc,(M') for any homomorphism of commutative rings R —t R!', where (?)' denotes the functor ? ® R'. We also have the following, by the definition of the C-comodule structure of Hom(V.M). L e m m a 3.5.8 Let C be an R-flat R-coalgebra, V a finitely presented Rmodule, and M a C-comodule. Then we have a canonical isomorphism Hom(V, Cobar c (M)) ^ Cobarc(Hom(Vr, M)). Lemma 3.5.9 Let C be an R-flat R-coalgebra. Then the set ofC-comodules of the form J®C with J an injective R-module, is an injective cogenerator of Mc. In particular, M.c has enough injectives. If, moreover, R is noetherian, then any C-injective comodule is R-injective. Proof. As G = ? ® C has an exact left adjoint F, it preserves injectives. Hence, G(J) = J®C is injective as a C-comodule for any injective .ft-module J. Let M e M c , and take an injective hull of M as an .R-module i: M ^> J. Then \I>i: M -» J ® C is an injective C-comodule map, and the first assertion follows, where * is the map in the proof of Lemma 3.5.6. The last assertion is obvious by Corollary 2.1.8. D
80
I. Background Materials
Corollary 3.5.10 Let C be an R-flat R-coalgebra, V an R-finitely presented C-comodule, M a C-comodule, and R' an R-flat commutative Ralgebra. Then the canonical map Hom c (V, M) ® R! ->• Hom C 0 f f {V ®R',M®
R')
is an isomorphism. Proof. As both Hom c (V, ?)®R' and Homc®R>{V®R', ?®R') are left exact, we may assume that M is of the form J ®C with J an injective /?-module by Lemma 3.5.9 and the five lemma. In this case, as the canonical map Hom(V, M)®R'
->• Hom ff {V ®R',M®
R')
is an isomorphism, the assertion follows from Lemma 3.5.6.
D
L e m m a 3.5.11 Let C be an R-flat coalgebra. Then for any C-comodule M and any R-module V, we have Ext l c (M, V ® C) S ExVR(M, V). Proof. Obvious by Lemma 1.6.12.
D
Lemma 3.5.12 Let C be an R-flat R-coalgebra, V a finitely presented Rmodule, and M and N be C-comodules. Then the canonical map Hom(V, N) ® V ->• N
{f®v^fv)
is a C-comodule map. The isomorphism Hom(M V, N) =* Hom(M, Hom(V, N)) induces an isomorphism Hom c (M ® V, N) S Hom c (M, Hom(Vr, N)). Proof. Easy.
•
If R is noetherian, then for an /?-flat .R-coalgebra C, any C-injective comodule is /?-injective (Lemma 3.5.9). Hence, for any finite .R-module V, Ext'R(V, ?) is the derived functor of Hom(Vr, ?) in the category M c . Hence, Ext'R(V, M) has a canonical C-comodule structure for a C-comodule M. Proposition 3.5.13 Let R be noetherian, C an R-flat R-coalgebra, V an R-finitely presented module, and M and N be C-comodules. If M orV is R-flat, then there is a spectral sequence
Ep2'q = Extpc{M, Ext9fi(y, N)) =* Ext£ +9 (M ® V, N).
3. Hopf algebras over an arbitrary base
81
Proof. By Lemma 3.5.12, we have an isomorphism of left exact functors Hom c (M V, ?) = Hom c (M, ?) o Hom(K, ?). For an injective ^-module J and i > 0, we have Extj^M, Hom(V, J C)) = Exf c (M, Hom(V, J) C) S Ext' fl (M, Hom(V, J)) = 0. Hence, Hom(V, /) is Homc(M, ?)-acyclic for an injective C-comodule /, and the assertion follows from Grothendieck's spectral sequence theorem. •
3.6
The restriction and the induction
(3.6.1) Let B and C be .R-coalgebras, and
C a coalgebra map. For M € M B , letting
be the structure map, we have M e M c . It is also easy to see that this defines a functor res B : M B -> M c . Note that res B M is nothing but M itself, as an i?-module. For a left C-comodule L and a right C-comodule N, we define the cotensor product of N and L, denoted by N K c L, by the exact sequence 0 —> N K c L —» N <S> L
PN L
' ) N C ® L ,
where pN]L := wN ® 1L - 1N ® LJL. If C" and C" are i?-flat i?-coalgebras, N is a ( C , C)-bicomodule, and L is a (C, C")-bicomodule, then N ® L and N®C®L are ( C , C")-bicomodules in a natural way, and pniL is a ( C , C")bicomodule map. Hence, N^FL has a (C',C")-bicomodule structure. Note that the definition of cotensor product is dual to that of tensor product. Almost by definition, the next lemma holds. Lemma 3.6.2 Let C be an R-coalgebra, L a left C-comodule, and N a right C-comodule. Then we have the following. 1 ? S c L and Nfflc? preserve any filtered inductive limits. 2 LetF:Mc^> 3IfC
RM
be the forgetful functor. Then ? E c L is F-left exact.
and L are R-flat, then 1MC L is a left exact functor from Mc to RM.
L e m m a 3.6.3 Let V be an R-module, N a right C-comodule, and L a left C-comodule. Then there are isomorphisms of R-modules N S c (C <S> V) = N V and (V C) G3C L = V L, which are natural with respect to V, N and L. If C is R-flat, then these isomorphisms are isomorphisms of C-comodules.
82
I. Background Materials
Proof. The cobar resolution Cobar c (iV) : 0 -t N - ^ N ®C —> N C C -> • • • of N is a split exact sequence of .R-modules (if, moreover, C is .R-flat, then it is an exact sequence of C-comodules which i?-splits). As we have d° Hom B (M, V 0 C ® B) B
is 0 for M 6 M and V € M c . As the sequence 0 -> Hom B (M, indg(V)) - ^ Hom B (M, V ® B) - ^ Hom B (M, V ® C ® 5 ) is exact, we get a map q : Hom c (resg(M), V) -> Hom B (M,indg(V)), which is natural with respect to M and V. To prove that q is an isomorphism, we may assume that V is of the form V = VQ ® C with Vo an .R-module by the five lemma, as the sequence is exact. In this case, we may identify ind B (V) = Vo ® B, while u : ind B (^) -> V ® £ is identified with
Hence, as we have J2(b) £c(fh)b2
= b for b G B, q is given by
q( C = R. Then we have ind5. V = V ® B for V ® B is endowed with a B-comodule structure by The counit of adjunction M -4 M ® 5 is nothing is a B-comodule map.
Definition 3.6.7 Let C be an .R-coalgebra. We say that B is an Rsubcoalgebm of C if B is a pure i?-submodule of C, and Ac(B) C B ® B holds. A subalgebra subcoalgebra of a bialgebra is called a subbialgebra. (3.6.8) Assume that C is i?-flat. Note that B C C is an it-subcoalgebra of C if and only if it is a (C, C)-subbicomodule of C, which is also a pure i?-submodule. If this is the case, B is also R-R&t. If B is an .R-subcoalgebra of C, then ind^(iV) is identified with the C-subcomodule {n e N | wjv(n) G N ® 5 ( c W ® C)} of TV for iV € M c , via the injective map indc(A0 = N®cB
C N®CC
^ AT.
Remark 3.6.9 In particular, if C is an .R-flat .R-coalgebra and 5 C C is an i?-subcoalgebra of C, then the counit of adjunction (ind^ ores^)(M) -> M is an isomorphism. Hence, res§ is fully faithful (and is obviously exact, as res^(M) = M). Thus, M B is identified with the full subcategory of M c consisting of C-comodules Af such that wN{N) C N ® B. Note that M B is closed under subobjects, quotients, and direct sums. The converse holds in the following weak form. Lemma 3.6.10 Let k be a field, C a k-coalgebra, and B a full subcategory of Mc closed under subobjects, quotients, and direct sums. We denote the inclusion B t-> M c by i, and denote its right adjoint (it does exist, see Lemma 1.10.2) by j . Then there is a unique k-subcoalgebra B of C such that any B-comodule is in B, and res5. : M B —> B is an equivalence. In fact, B is given as j(C). Proof. Let V be a /c-vector space, and M S M c . Then the canonical map V ® jM —> j(V ® M) is an isomorphism. This is trivial when V = k, and hence also for the case dim V < oo, as j is additive. As j is also compatible with filtered inductive limits by Lemma 1.10.3, the general case follows. Now let N £ B. Then the coaction ui^ : N —> N C is a C-comodule map, and hence it factors through j(N ®C)=N<E> j(C). Considering the case Af = j(C), we have that j(C) is a fc-subcoalgebra of C. This also shows that any object of B is a j(C)-comodule. Conversely, as j(C) € B, any j(C) comodule N is in B, as N is a subobject of N ®j(C). Hence, the existence of B is proved.
84
I. Background Materials
We show the uniqueness. of j . We show j(C) C B. we have that any object of 5-comodule. For c € j(C), uniqueness is proved.
As B G B, we have B C j{C) by the definition As M B is closed under isomorphisms in Mc, B is a B-comodule. In particular, j(C) is a we have c = E(c) £c(c(i))c(2) 6 B, and the D
Lemma 3.6.11 Let C be an R-coalgebra, C and C" be R-flat coalgebras, P an R-finite R-projective (C,C')-bicomodule, N a (C", C)-bicomodule. Then there is a (C"', C')-bicomodule isomorphism
which is natural with respect to P and N. Proof. It is easy to see that the kernel of the map N ®P -> N ®C P agrees with the image of Hom c (P*, N) ^ Hom(P*, N) = N®P" = N®P.
a Corollary 3.6.12 Let B —» C be an R-coalgebra map. If B is R-finite projective, then there is an isomorphism of functors ind^ = Hom c (S*,?). (3.6.13) From now on, until the end of this subsection, we assume that C is an R-flat .R-coalgebra. Note that M c is abelian, and the forgetful functor F : MP —> ijM is faithful exact, and its right adjoint G = ? <S> C is also exact. Definition 3.6.14 For M 6 M c and a left C-comodule N, we denote R'F(? S c N)(M) by Cotor^M.TV), and call it the ith cotorsion module of M and N. By definition, we have Cotor'c(M, TV) = /T(Cobarc(M) ®c N). As Cobarc(M) Mc N is the complex of the form
it is symmetric with respect to M and N, up to sign change. We denote it by Cobar c (M, N). We have Cobarc(M,AT) ^ CobarCop(7V,M). Hence, we have Cotor'c(M, TV) S /T(Cobar c (M,A0) ^ ifi(CobarCoP(iV, M)) ^ Cotor'cop(N, M).
3. Hopf algebras over an arbitrary base
85
Lemma 3.6.15 For a homomorphism of commutative rings R —• R', we have
Cobarc(M, N)' Si CobarC'(M;, N') in a natural way, where (?)' = ? R' •
Proof. Obvious. By Lemma 3.6.11 and Lemma 1.6.12, we have the following. Lemma 3.6.16 Let M be an R-finite R-projective C-comodule, and N a C-comodule. Then we have natural isomorphisms
Ext'c(M,JV) S ^ i ( £* H C) S c N) Si H\r
® N).
As N is .R-flat, we have that J' <S> N is quasi-isomorphic to V N, and the assertion follows. • Corollary 3.6.19 Let B be an R-flat coalgebra, and B -> C an R-coalgebra map. Then we have
R{ ind£ S // i (Cobar c (?, B)) = Cotor^?, B).
86
I. Background Materials We have the following.
Proposition 3.6.20 Let R-> R' be a flat morphism of commutative rings. We denote the functor ? ® R! by (?)'. For i > 0, we have 1 For M e Mc and N e CM, we have Cotor^(M, N)' £ Cotorjy(M#, N'). 2 Let M be a C-comodule, and B -> C an R-coalgebra map with B R-flat. Then we have 3 Assume that R is noetherian. Then for any R-finite C-comodule V and any C-comodule M, we have Ext^V, M)®R' = Extj^V, M'). 4 Assume that R is noetherian. For any R-finite C-comodule V and any C-comodule N, we have
The proof is straightforward. We give a remark on 4. If 7* is a Cinjective resolution of M, then /* (C ® C)* -> C*. The unit map is e*c. Using Sweedler's notation, we can write ( 6 V ) ( c ) = £ 6 * c ( 1 ) • c*c{2) (c)
We call the i?-algebra C* the dual algebra of C.
(b*,c'£C',ceC).
88
I. Background Materials
Definition 3.8.2 Let R be a field. Let V be an .R-space, and W a subspace of V*. We say that W is a dense subspace of V*, if V —> W (v >-> (w >-> (w,v))) is injective. Let R be a general commutative ring, and V and W be iZ-modules. We say that an .R-linear map / : W —> V" is universally dense, if the .ft-linear map 9u : 1/ V -> Hom(H/, £/)
u ® u H> (w H (to, t>)u)
is injective for any .R-module f/. In the definition above, the pairing W ® V -> i? which corresponds to / by the isomorphism Hom(W, K*) = Hom(VK ® V, R) is denoted by (—,—)• For a homomorphism R —)• R' of commutative rings, we still denote the pairing W ®RI V —> R' obtained by the base change by the same symbol (—,—). From the pairing (—,-),
f':RomRI(W',HomK(V',R')) is induced. Note that the associated map V -> Homfl'(Vy', R') (v' t-^ (w' H* {w',v'))) is nothing but the composite map
V = R' ® V ?£> Rom{W, R') S Note also that 6V in the definition above is natural with respect to U. Lemma 3.8.3 Let R be a field, V an R-vector space, and f : W —¥ V* an R-linear map. Then the following are equivalent. 1 f is universally dense. 2 OR : V -¥ W* is injective. 3 Imf
is a dense subspace of V*.
Proof. 1=>2 is obvious. As OR : V —> W* is the composite of V -4 (Im/)* and the injective map (Im/)* —• W*, 2-O3 is obvious. We show 2=>l. It is obvious that Qu is injective for finite dimensional U. Now consider the case dim fl U = oo, and assume that 0[/ is not injective. As there is a finite dimensional .R-subspace Uo of U such that Ker 6y (~\ (Uo ® V) ^ 0 and 0 is D natural, 6u0 is not injective, and this is a contradiction. Lemma 3.8.4 Let f : W —> V be an R-linear map. Consider the following two conditions.
3. Hopf algebras over an arbitrary base
89
1 f is universally dense. 2 V is R-flat, and the induced map
is universally dense for p £ Spec R, where ?(p) denotes the functor K(P)
K ( P ) M.
In general, we have 1=>2. //, moreover, R is noetherian, then we have 2=>1. Proof. We prove 1=>2. For any injective ^-linear map g : U -> U', we prove that 5®lyr : t/®V -> t/'®V is injective. As g, : Hom(VK, £/) -> Hom(W, [/') is injective, 0[/ is injective, and 0 is natural so that 6u> ° (g 1. Assume the contrary. Then we have an i?-module U such that 8u is not injective. Take a non-zero element X^Ui (8)i»i in Kerfly, and set UQ to be the i?-submodule of U generated by all ttj's. Then 8u0 is not injective, and we may assume that U is fi-finite, replacing U by Uo. As V is i?-flat, if 0 -> f/j ->• t/2 -> I/3 is an exact sequence of fl-modules and 6Vl and ^u3 are injective, then 6y2 is also injective by the five lemma. As any finitely generated module over R has a finite filtration whose successive subquotients are of the form R/p with p £ Spec./?, there exists some p € SpecR such that 6R/V is not injective. So we may assume that U = R/p. Replacing R/p by R, we may assume that R is an integral domain, and OR : V —• W* is not injective. We set K = K(0) to be the quotient field of R. As r : R K is injective and V is fi-flat, r i g i l v ^ - ^ / ^ i E i V i s injective. On the other hand, 6K • K V -> rlomK(K (8) W, K) is injective. As the injective map 6K ° {r <S) lv) agrees with the composite map V -> W* -> rlomK{K ® W, K), we have that 0 fi is injective, and this is a contradiction. •
Lemma 3.8.5 Let W -> V* be a universally dense R-linear map. Then for any R-module U, we have that 6V : U ® V —> Hom(W, U) is R-pure. In particular, 6R:V -¥ W* is R-pure.
90
I. Background Materials
Proof. Let X be an /^-module. Then the composite map
X&UQV
±^% X ® Hom(Wl U) -»• Hom{W,X ® U)
agrees with $x®u, which is injective. Hence, lx 9u is also injective for any X, and this shows that 9y is i?-pure. D Exercise 3.8.6 Prove that if V and W are fl-projective modules, then V* ® W* —» (V <E> VF)* is universally dense. Considering the case W = R, i d y : V* —> V* is universally dense, if V is /?-projective. Utilizing this, prove that if C is an .R-projective i?-coalgebra, then C is cocommutative if and only if C* is commutative. Let M be a right C-comodule. Then M is a (left) C*-module with the action c'm := 5Z( c * m (i)) m (o) (c* £C*, m€ M). (m)
Any C-comodule map is a C*-linear map. Exercise 3.8.7 Prove the assertions above. Thus, for an i?-algebra map A —» C*, the exact functor $ : M c -> C'M. —» ^M is denned in an obvious way.
3.9
The dual coalgebra of an algebra
Let A be an i?-algebra. In general, A* does not have a canonical .R-coalgebra structure even if R is a field. This is because even if / 6 A*, iri*A(f) £ (A A)* does not belong to A* A* in general. However, if R is a field, then there is a dense subspace of A* which is endowed with a canonical coalgebra structure. In this subsection, we assume R — k is a field. We set
A°:={fE
A'\mrA(f)eA*®A*}.
Lemma 3.9.1 For f £ A*, the following conditions are equivalent. i / € A°, that is, mA(f) £ A* ® A' ii mA(f) G A0 A° iii There exists some ideal I of A such that dimjt A/1 < oo and / ( / ) = 0. Hence, we have mA(A°) C A° ® A°, and A° is a fc-coalgebra with the coproduct mA. We call A° the duai coalgebra of A If ,4 and 5 are fc-algebras, then we have A°®B° = (A®B)°. If dim* A < co, then we have A* = ^4°, and A = A". In this case, $ : M'4* —• yiM is an equivalence.
3. Hopf algebras over an arbitrary base
3.10
91
Rational modules
In this subsection, R denotes a general commutative ring again. Let C be an i?-coalgebra, and A -> C* a universally dense i?-algebra map (hence, C is R-ftat). We denote the canonical functor M c —» ^M by $. As an .R-module, we have $(M) = M. Let V be an A-module. We denote the .A-action A ® V —» V by ay- The canonical isomorphism Hom(.4 V, V) £ Hom(V, Hom(v4, V)) is denoted by pv- By the universal density assumption, 6V : V ® C -> Hom(>4, V) is injective. We define the rational part of V to be the pull-back (pvav)~l(lTn(6v)) of Im(0y) by the map pyay • V -* Uom(A, V), and denote it by Vnt. Lemma 3.10.1 With the notation above, Vrat is an A-submodule of V. Moreover, Vrat 4 ^ - > Hom(yl, V) factors through Vrat ® C ^
Uom(A, Vrat) -»• Hom(A, V),
the coaction U)VTM : Vrat -> Vrat ® C is canonically defined, and Via% has a C-comodule structure. The A-module structure of $(Kat) agrees with that o/Vrat 05 an A-submodule ofV. Proof. For v € Vrat> we may write {pva,v){v) = £< ^KC^t^Ci)- For o, a' £ /I, as we have
({pvav){av))(a!) = a'(av) = (a'a)t;
= 5^(a'a, Ci)vi = 53 53< i
t
( Ci )
we have
(pvav)(av) = 5ZS V is clearly an isomorphism, hence $ is fully faithful. • If V is an .4-module such that Vrat = V, then we say that V is rational. For a C-comodule M, we have that M = $ M is rational. Conversely, if V is rational, then V = Viat is a C-comodule. So we identify rational ^-modules with C-comodules. Exercise 3.10.4 Rational .4-modules are closed under submodules, factor modules, and inductive limits in ^M. Exercise 3.10.5 If M is a flat /t-module and V is an i4-module, then the canonical map Vrat M -> (V M)rat is an isomorphism. Exercise 3.10.6 If there is a universally dense .R-algebra map A -* C*, then the intersection of (infinitely many) C-subcomodules of a C-comodule is again a C-subcomodule.
3.11
FPCP coalgebras and IFP coalgebras
Throughout this subsection, R denotes a noetherian commutative ring. Let C be an i?-coalgebra. Definition 3.11.1 We say that C is ind-finite projective (IFP, for short) if for any i?-finite i?-submodule M of C, there exists some /2-finite Rprojective /?-subcoalgebra of C containing M. By definition, we have L e m m a 3.11.2 If C is an IFP R-coalgebra, then a base change R' (g> C is an IFP R'-coalgebra for any commutative noetherian R-algebra R! of R. Lemma 3.11.3 If C is IFP, then C is R-Mittag-Leffler. In particular, C is R-fiat. If R is noetherian and C is R-countable, then C is R-projective. Proof. Follows immediately from Lemma 2.2.11 and Proposition 2.2.6. D Now we assume that C is /?-flat. Definition 3.11.4 We say that C satisfies the finite projective cover property (resp. projective cover property), FPCP (resp. PCP) for short, if for any i?-finite C-comodule (resp. any C-comodule) M, there exists some surjective C-comodule map P —> M with P /?-finite projective (resp. i?-projective). Lemma 3.11.5 If C satisfies FPCP, then C satisfies PCP. Conversely, if C satisfies PCP and gl.dim R MA with PA infinite projective for each A. Then the composite map
®P A ^0M A ->lunM A = M \
A
is a surjective C-comodule map, and ® A PA is fi-projective. This shows that C satisfies PCP. Assume that C satisfies PCP, gl.dimi? < 2, and M is an it-finite Ccomodule. As C satisfies PCP, there exists some surjective C-comodule map / : P -> M with P it-projective. We can take an it-finite C-subcomodule N of P such that f(N) = M, by Lemma 3.7.1. We can also take an it-free module F with a basis B which contains P as its direct summand. Then we may consider N C P C F, and there exists some finite subset Bo of B such that Fo := R • Bo contains N. Now we define Q to be the kernel of the composite of the C-comodule maps p:P^P®C^>F®C^
F/Fo ® C.
For q € Q, we have q = £( 9 ) £c(c(i))c(o) € FQ, and hence Q C Fo is ii-finite. As both P and F / F o C are i?-flat and gl.dim R < 2, we have that Q is .R-fiat, and hence is .R-finite projective. As we have p(N) = 0, it follows that f(Q) D f{N) = M, and hence the restriction / | Q : Q —> M is a surjective C-comodule map. Hence, C satisfies FPCP. Lemma 3.11.6 If C is IFP, then C satisfies FPCP. Proof. Let M be an .R-finite C-comodule. As the image of w^ : M —> M ® C is .R-finite, there exists an .R-finite i?-projective .R-subcoalgebra D of C such that Im u M C M ® £>, by assumption. This shows that M is a Dcomodule, and hence is a D*-module. As D* is fl-finite .R-projective, there exists some surjective D'-linear map / : P —» M such that P is .R-finite .R-projective. As a D*-module is always a Z)-comodule and a DMinear map is a £>-comodule map, we have that C satisfies FPCP. • Lemma 3.11.7 Let R be a noetherian ring, R-t K an injective homomorphism of commutative rings, P an R-projective module, and MK a finite K-submodule of PK := K P. Then MKC\P is R-finite. Proof. Replacing P by an it-free module which contains P as its direct summand, we may assume that P is an .R-free module with a basis B. As MK is if-finite, there exists some finite subset So of B such that MK is contained in the if-span K • BQ of BO- Replacing P by R • Bo, we may assume that P is .R-finite, which case is trivial. •
3. Hopf algebras over an arbitrary base
95
Corollary 3.11.8 Let R be a reduced noetherian ring, P an R-projective module, and M an R-submodule of P. If Mp is a finite dimensional Rpvector space for any p G Min(/Z), then M is R-finite. Proof. Set K to be the total quotient ring npgMin(fl) ^p O I R- By the lemma, MK n P is fi-finite, where MK := M ®K. As M C MK n P, M is also fi-finite. D Lemma 3.11.9 Let R be a hereditary {i.e., gl.dim/? < 1) noetherian ring, and C an R-projective R-coalgebra. Then C is IFP. That is to say, for any R-finite R-submodule M of C, there exists some R-finite protective Rsubcoalgebra D of C which contains M. Proof. First, we consider the case gl.dim R = 0. As the image of M by A'2) : C —> C®C®C (c i-> Y.(c) c(i)®c(2)®C(3)) is .ft-finite, there exists some .R-finite .R-submodule N of C such that A (2) (M) is contained in C®N®C. then we have M C D. It is easy to If we set D := ( A ^ ) " 1 ^ ®N®C), check that A(D) CD®D. As gl.dimR = 0, D is an fi-subcoalgebra of C. If d € D, then d = E(d)e(^i)e(^3)^2 £ N, and hence D C N. This shows that D is .R-finite. It is .R-projective, as gl.dim R = 0. Next, we consider the case gl.dim R = 1. Let K be the total quotient ring of R. As gl.dim K = 0, there is a A'-finite /C-subcoalgebra D ^ of KC which contains K®M. Set L> := DKC\C. By Lemma 3.11.7, D is fi-finite. As C/D is torsion-free, it is i?-flat, since gl.dim R = 1. This shows that D is a pure submodule of C, and D is .R-nnite projective. Moreover, the composite map D ^ c A c ® C 4 i C ® {C/D ®C®C® C/D) is zero by the choice of DK. subcoalgebra of C.
3.12
Hence, A(£>) C D ® D, and Z) is an RD
and Horn of modules and comodules over a Hopf algebra
In this subsection, R denotes an arbitrary commutative ring again. (3.12.1) Let U be an fl-Hopf algebra. For V,W G y M , we define the (/-module structure of V ® W by u£U,veV,w€
W),
96
I. Background Materials
and the [/-module structure of Hom(V, W) by
(«/)(«) = Euw(f((Su(2))v))
( u € U , f € KomR(V,W),
v e V),
where S = Su denotes the antipode of U. Exercise 3.12.2 Check that the definitions above do give [/-modules. (3.12.3) An J?-module M endowed with the [/-module structure given by urn := e(u)m (u 6 U, m 6 M) is called a trivial [/-module. If we want to emphasize the trivial [/-structure, we denote it by M t n v . However, by the [/-module R we always mean the trivial module Rtnv, unless otherwise specified. Lemma 3.12.4 Let V, W, and X be U-modules. maps (3.12.5) (3.12.6) (3.12.7) (3.12.8) (3.12.9)
The standard R-linear
Hom(V, W) ® Hom(X, V) -> Hom(X, W) Hom(V ® X, W) S Hom(V, Hom(X, V<S)R = V^ R®V (v®l >-> v (V®W)®X &V®(W®X) ({v®w)®x>-*v®{w®x)) X Hom(V, W) -> Hom(V, X W) (x ® f .-> (« K> X ® fv))
are U-linear, and natural with respect to V, W, and X, where the map in (3.12.6) is given by /4(()H(li-j/(lJ® X))).
Proof. Straightforward.
D
(3.12.10) See Lemma 2.1.7 for sufficient conditions for the map (3.12.9) to be an isomorphism. If W = R, then (3.12.9) is nothing but the map X ® V -)• Hom(K,X)
(x®ip^{v^(ip,v)x)).
If U is cocommutative, then r :V ®W = W ®V (r{v ® w) = w ® v) is also [/-linear and natural with respect to V and W. (3.12.11) We denote the functor Ext\j{R, ?) by H{(U, ?). For [/-modules V and W, we have V, W) s Hom c/ (fl, HomR(V, W)) = H°(U, Hence, taking H°(U, ?) of both sides of (3.12.6), we have (3.12.12)
Homu(V ®X,W)^
Homt/(Vr,Homfl(X, W)).
3.
Hopf algebras over an arbitrary base
97
This shows that ? ® X is left adjoint to Hom^A", ?). In particular, considering the case that V is [/-projective and X is i?-projective, as the left-hand side is an exact functor on W, we have that V X is [/-projective. Considering the case V = Hom/e(A", W) in (3.12.12), the element of the left-hand side corresponding to idy is nothing but the evaluation map ev: KomR(X,W)®X->W
(f®x^fx).
Hence, ev is [/-linear. If U is cocommutative moreover, then the [/-linear map which corresponds to ev or by the isomorphism Homy (A" ® Homfi(A:, W), W) S Homy (A, Hom /J (Hom fi (X, W), W)) is nothing but the duality map n - » ( / 4 fx). Hence, the duality map is [/-linear, if U is cocommutative. (3.12.13) Let H be an /2-Hopf algebra. For //-comodules M and TV, we define the //-comodule structure of M ® N by M ® N -> M ® JV #
(
(3.12.14) We have seen that in the following cases, Hom(M, ?) H = Hom(M, ?H) is an isomorphism for i?-modules M and H, see Lemma 2.1.7. (3.12.15) H is fl-flat and M is of finite presentation. (3.12.16) M is .R-projective and H is of finite presentation. (3.12.17) H is fl-finite projective. Assume one of the conditions above is satisfied. Let iV be an //-comodule. Then we define the //-comodule structure of Hom(M, iV), defining w(/) € Hom(M, N H) by TO
) := 5Z S (/ (o))(o) (m)
for / £ Hom(M, A^). It is left to interested readers to check that these definitions do give //-comodules. (3.12.18) For an fl-module M, the //-comodule M with the structure map u>(m) = m ® 1 is again denoted by M. If necessary, it is denoted by M t n v . The functor (?) triv is nothing but the restriction via the /?-coalgebra map UH '• R -»• H, and its right adjoint is the induction ? ® H. The Hcomodule R means Rtm, unless otherwise specified. We denote the functor Ext^H (R, ?) by / / ' ( M " , ?). Note that the functor ^ ( M " , ? ) = Hom M «(/?,?) is nothing but the induction via UH, and it is called the H-invariance.
98
I. Background Materials
3.13
The dual Hopf algebra
(3.13.1) Let R be a commutative ring, H an i?-bialgebra, and U an Rbialgebra, and (—,—) : U <S> H —> R an i?-linear map. We assume the following conditions. 1 ( - , - ) is a pairing of i?-bialgebras. Namely, the induced map U -t H' is an i?-algebra map, and H -> U* is also an it-algebra map. 2 U -t H* is a universally dense injective map. We call such a pair U and (—, —) a generalized hyperalgebra of H. (3.13.2) When is there a generalized hyperalgebra of HI As a necessary condition, H must be it-flat by Lemma 3.8.4. In the following two cases, a generalized hyperalgebra of H exists. Example 3.13.3 Let R be a field, and H an it-Hopf algebra. As H is an it-algebra, U := H° is an it-coalgebra. It is easy to verify that H° is an /Z-subalgebra of H*, and is an i?-Hopf algebra. We call H° the dual Hop} algebra of H. The canonical map H -4 U* is an algebra map. If H is commutative, then H° is cocommutative [1, Corollary 2.3.17]. If H is commutative and of finite type over R, then the inclusion H° —>• H* is universally dense, and hence H° is a generalized hyperalgebra of H. We prove the last assertion. By Lemma 3.8.3, it suffices to show that the canonical map 0 : H -> (H°)* (6(h)(h*) = (h*, h)) is injective. Assume that h e Ker#. For any maximal ideal m of H and n > 1, as H/mn is a finite dimensional /t-space by the Hilbert Nullstellensatz (see [110, Theorem 5.3]), the image of h in H/mn = (H/mn)** is 0 by Lemma 3.9.1. Hence, we have • m ^ supp Hh. As m is arbitrary, we have Hh = 0. Example 3.13.4 Assume that H is i?-finite projective. Then as we have = H' (and idy) is a generalized hyperalgebra of H' ®H* ^ (H®H)\U H. If H is commutative, then U is cocommutative. (3.13.5) Let U be a generalized hyperalgebra of H. If M is an Hcomodule, then M is an i/*-module, hence is a [/-module, and an exact functor $ : MH -» yM is induced. As U -» H* is a universally dense algebra map, there is a right adjoint (?) rat : (/M -> MH of $, and $ is fully faithful (Corollary 3.10.3). In this situation, more is true. The functor $ also preserves tensor products. That is, for M,N e MH, the identity map $M®$N = M®N = $ ( M ® N) is a [/-isomorphism. Moreover, $ preserves Horn. That is, if
3. Hopf algebras over an arbitrary base
99
M,N e MH and one of (3.12.15-3.12.17) is satisfied, then the identity map Hom($(M),$(TV)) = $(Hom(M, TV)) is a [/-isomorphism. Moreover, the functor $ preserves trivial representations. That is to say, $(M t r i v ) = M t r i v for any fl-module M. Hence, rational [/-modules are closed under tensor products, and if moreover one of the conditions (3.12.15-3.12.17) is satisfied, then they are also closed under Horn.
3.14
Module algebras and comodule algebras
The reference for this subsection is [114]. (3.14.1) Let R be a commutative ring, and U an i?-Hopf algebra. We say that A is a U-module R-algebra if A is an .R-algebra and a [/-module, and the product TUA '• A A —> A is [/-linear. In this case, UA '• R —> A is also [/-linear. For [/-module .R-algebras A and B, we say that tp : A —> B is a [/-module i?-algebra map if tp is [/-linear and is an fl-algebra map. (3.14.2) Let U be an .R-Hopf algebra, and A a [/-module algebra. We say that M is a (U,A)-modu\e if M is both a [/-module and is an ,4-module, and the ,4-action A ® M —> M is [/-linear. For ([/, /l)-modules M and TV, we say that / : M —> TV is ([/, >l)-linear if / is both [/-linear and j4-linear. We denote the category of (U, j4)-modules and ([/, yl)-linear maps by y^M. Note that the [/-module .R-algebra R is a [/-module algebra. (3.14.3) Let A and [/ be as in (3.14.2). We define the smash product A#U of A and U as follows. As an i?-module, A#U is A U. The product of A#U is given by (a®u)(b®v) = ^2a(u(i)b) ®U(2)U
(a, b € A,
u,v€U).
It is easy to see that A#U is an .R-algebra. Note that both A —> A#U (a i-> a(8)l) and U -* A#U (u i-> l®u) are i?-algebra maps. So any >l#[/-module is in a natural way a [/-module >l-module, which is also a (U, .4)-module. Conversely, if M is a (U, .4)-module, then defining (a <S) u)(m) = a{wm) for a®u £ A#U and m £ M, M is an j4#[/-module. These correspondences are quasi-inverse to each other, and an A#U-module and a ([/, /l)-module are one and the same thing. Thus, we have that ^ y M and y ^ M are equivalent. We always identify an A#U-modu\e and a ([/, i4)-module. As i?#[/ ^ [/, we have UiRM = VM. Note that H°(U, A) is an fl-subalgebra of A, and H°(U, ?) is a left exact functor from y ^ M to 7/°(y,/i)ML (3.14.4) Let /f be an fi-flat Hopf algebra. We say that B is an Hcomodule R-algebra if B is an .R-algebra //-comodule, and the product
100
/. Background Materials
TUB : B <S> B —> B is an //-comodule map. By an //-comodule /?-algebra map we mean an //-comodule map which is also an /?-algebra map. We say that M is an (H, B)-Hopf module if M is an //-comodule B-module, and the action B M is an //-comodule map. A 5-linear //-comodule map between (//, 5)-Hopf modules is called an (//, B)-linear map. The category of (//, B)-Hopf modules and (//, 5)-linear maps is abelian, and we denote it by flMfl. Note that B M H satisfies the (AB5) condition. Note also that R is an //-comodule algebra, and we have RMH = MH. We have that BH = H°(M":B) is an fl-subalgebra of B, and H°(MH,?) is a left exact functor from B M H to B H M . (3.14.5) Let U be a generalized hyperalgebra of H. If B is an //-comodule algebra, then $ 5 = B is a [/-module algebra in a natural way. If M is an (//, B)-Hopf module, then M is a (U, 5)-module. Thus, we have an exact functor $ : BMH -> y p M . Conversely, if A is a [/-module algebra, then ^4^ is an /?-subalgebra of A, and AjaX, is an //-comodule algebra. If M is a ([/, >l)-module, then M rat is an (//, >lrat)-Hopf module, and we obtain a functor (?) rat : y.^M —>• Ar^MH. If B is an //-comodule algebra (and hence B = BraX), then (?) rat : y #B Wl —> g M " is right adjoint to $ , and preserves injective objects. Note also that $ is fully faithful in this case.
3.15
Coalgebras and comodules over a scheme
Let A" be a scheme. We say that C is an Ox-coalgebra if C is a quasicoherent Ox-module, Ox-module maps e : C —> Ox and u> : C —> C ®Ox C are given, and the coassociativity and the counit laws are satisfied. Similarly, Ox-algebra, Ox-bialgebra, and Ox-Hopf algebra are defined, replacing an /^-module by a quasi-coherent Ox-module. In [71], an Ox-algebra is called a quasi-coherent Ox-algebra.
Notes and References. For basics on Hopf algebra theory, see [140], [1], [93], and [114]. As the base ring R in our text is not restricted to a field, we have discussed some difficulties arising from this point. In particular, the notion of universal density and related results on rational modules and generalized hyperalgebras, and the notion of IFP, FPCP, and PCP are new here. Some of the important properties of a flat coalgebra and its comodules in this section are proved in [145].
4. From representation theory
4 4.1
101
From representation theory Group schemes as faisceaux
(4.1.1) Let X be a scheme. The category of X-schemes Sch/X with the fppf topology is a site, and a sheaf with respect to the Grothendieck topology is called an X-faisceau, see Example 1.8.14. For an X-scheme Y, y(Y) = Homsch/x('i Y) ' s a set-valued X-faisceau. An X-faisceau is called representable if it is isomorphic to y(Y) for some Y £ Sch/X, see (1.1.7). (4.1.2) We denote the full subcategory of the category of X-schemes Sch/X consisting of affine X-schemes by X-aff. Note that X-aff is also a site with the fppf topology. If F is an X-faisceau, then F is completely determined by its restriction to X-aff. Hence, F can be viewed as a faisceau over X-aff, and is a covariant functor from the category of X-algebras X-alg to Set, where an X-algebra is a commutative ring A together with a morphism Spec A -> X (note that X-alg is contravariantly equivalent to X-aff). We call a covariant functor on X-alg an X-functor. The functor which maps an X-faisceau F to the X-functor F has a left adjoint (?) by (1.8.8). For an X-functor P, the sheafification P is called the associated faisceau of P. For more, see [43]. We remark the following. Lemma 4.1.3 Let F be a subfunctor of an X-faisceau G. Then F is the subfunctor of G given by F(A) = {x£ G{A) | 3 fppf A-algebra B such that x £ F{B)} for A € X-alg, where fppf means faithfully flat of finite presentation. Definition 4.1.4 A group (semigroup)-valued X-functor G is called an Xgroup scheme if G, viewed as a set-valued functor (with composing the forgetful functor), is a representable X-faisceau. By Yoneda's lemma (Lemma 1.1.6), to say that an X-scheme G is a semigroup-valued functor is the same as to say that X-morphisms HG • G xx G -> G and e : X -¥ G are given, and the semigroup laws HG ° (1 G x no) = ficiVG x 1G), MG ° (e x 1G) o AG' = 1 G = /xG o (1 G x e) o p^1 are satisfied, where AG : X x * G -> G and pG : G xx X -> G are the canonical identifications. Further, G is group-valued if and only if there is an X-morphism t G : G -> G such that He o (1 G X ( C ) o i G = e o u G = | i C o (t G x 1G) o A G is satisfied, where uG : G —• X is the structure map, and A G : G —• G xxG is the diagonalization. Thus, we see that the definition above agrees with
102
I. Background Materials
that in (3.2.1), when X is affine. In particular, if both X = Spec/? and G = Spec H are affine, then to give an X-group scheme (resp. X-semigroup scheme) structure to G is the same as to give an fl-Hopf algebra (resp. i?-bialgebra) structure to the commutative .R-algebra H. (4.1.5) We say that a semigroup X-scheme G acts on an X-scheme Y (from the right) if the X-functor G acts on Y from the right. Translating this situation in terms of Yoneda's lemma, an action of G on Y is nothing but an X-morphism a : YxxG —> Y such that ao(ax 1G) = ao(l y x^iG) and the unit element acts as the identity morphism. The left action is defined similarly. Unless otherwise specified, an action of a semigroup scheme on a scheme is a right action. However, if G is a group scheme, then a right action a :Y xxG —>Y is sometimes viewed as the left action G xxY ->Y given by g - y : = y g ~ l . The quotient of Y by G, denoted by Y/G, is the associated faisceau of the X-functor F defined by F(A) := X(A)/G(A). We say that a subscheme Z of Y is G-stable if Z(A) is a G-stable subset of Y(A) for any X-algebra A.
4.2 (4.2.1)
Rational representations of an algebraic group Let R be a commutative ring.
Definition 4.2.2 Let G be an affine i?-semigroup scheme with H = R[G]. We call an //-comodule a G-module or a rational G-module. There is an alternative definition, which is more natural. Let X be a scheme, and G an X-semigroup scheme. For a quasi-coherent C?x-module M, we define an X-semigroup functor End(A^) (resp. X-group functor
GL(M)) by End(7W)(y) := EndOy(/*7W)
(resp. GL(M)(Y) := End Oy (/*7W) x )
for each X-scheme / : Y —> X. We say that M is a G-module, if A^ is a quasi-coherent Ox-module, equipped with a morphism G —> End(M) of X-semigroup functors. This definition looks more like that of group representation. If G is an X-group scheme, then the representation G —> End(.M) factors through GL(M). If, moreover, M is locally free coherent, then we have End(A4) = Spec(SymHornOY(.M, M)v), and both End(M) and GL(M) are representable. In this case, the representation G —> End(A^) or G-> GL(M) is a morphism of X-schemes, by Yoneda's lemma.
4. From representation theory
103
(4.2.3) We briefly review the correspondence between the two definitions provided in the last paragraph in the case where both X = Spec R and G = Spec H are affine [901. If M is an H-comodule, then we have a morphism G ~ GL(M) given by 9 f-t (a ® m
f-t
L ag(m{l)) ® m(O) (m)
for each A and 9 E G(A) = HOmR-alg(H, A). Conversely, assume that a morphism h : G ~ GL(M) of R-group functors is given. Then as idH E G(H) = HOmR-alg(H, H), we have hH(id H) E EndH(M ® H). It is easy to see that M is an H-comodule, letting the composite map
be its coaction. A little more generally, if G = Spec 1£ is affine over X, then 1£ is an Ox-Hopf algebra, and a G-module and an 1{-comodule are the same. In the sequel, we only consider group schemes G = Spec 1£ affine over the base scheme X. --
(4.2.4) Let X be a scheme, and G an X-group scheme affine over X. Let M and M' be G-modules. We say that If : M ~ M' is a G-linear map if cp is an Ox-module map, and for any morphism f : Y ~ X, r(Y,/*cp): r(Y,/*M) ~ r(Y,/*M') is G(Y)-linear. We denote the category of G-modules and G-linear maps by eM!. Note that eM! is equivalent to the category M!1i.
(4.2.5) For G-modules M and M', we define the G-module structure of M ®ox M'. For f : Y ~ X with Y affine and 9 E G(Y), 9 acts on /*(M ®ox M') ~ f*M ®Oy /*M' so that the action on the right hand side is given by 9 ® g. This definition agrees with the tensor product of 1£-comodules.
Lemma 4.2.6 Let A be both a G-module and an Ox-algebra. Then the following are equivalent. 1 The coaction WA : A
~
A ®ox 1£ is an Ox-algebra map.
2 The product map A ®ox A
~
A is G-linear.
104
I. Background Materials
(4.2.7) If the conditions above are satisfied, then we say that A is an Kcomodule algebra or a G-algebra. Applying the functor Spec to the coaction w,4, we get a right action az : Z xx G -> Z, where Z = Spec A. Conversely, if an affine morphism / : Z —> X and a right action az : Z xx G -> Z are given, then f,Oz is a G-algebra in a natural way. Thus, a G-algebra and a right G-action affine over X are one and the same thing. Lemma 4.2.8 Let R be a noetherian commutative ring, and G = Specif an affine R-group scheme of finite type. Assume that H is IFP. Then the coordinate ring H of G is R-projective. Moreover, there exists some n such that G is a closed subgroup of GLn(R). Proof. Note that H is countably generated as an it-module. The first assertion is obvious by Lemma 3.11.3. As H = k[G] is of finite type over R, there exists some it-finite projective it-subcoalgebra D of H, which generates if as an it-algebra, by the definition of IFP group. Note that the dual algebra D* is an impure subalgebra of (EndD*) op = EndD, via the right multiplication. This is trivial when R is a field, and the general case follows easily from Lemma 2.1.4. This shows that there is a surjective coalgebra map (End D)* -4 D. The composite coalgebra map (EndD)* -> D '-* H is uniquely extended to a fc-algebra map Sym(EndD)* —>• H, which is obviously an it-bialgebra map. This map is surjective, because the image of this map contains D, which generates if as an it-algebra. Taking the corresponding geometric morphism, we have a closed immersion it-semigroup homomorphism G -¥ End D. As D is a direct summand of Rn for some n, there is a closed immersion itsemigroup homomorphism G -> End it". Because G is a group, this map factors through G t-> GLn, which is also a closed immersion. • (4.2.9) Let tp : H -> G be a homomorphism of it-flat affine semigroup schemes. Then we have a bialgebra map R[G] -> R[H]. The restriction res fl//i an y~1xy). Thus, H = R[G] is a G-module. As the unit element e is fixed by the action, the defining ideal I := Ker£ W of e is a G-submodule of if. As the product of H is
4. From representation theory
105
G-linear, I/I2 is also a G-module. The Zariski tangent space (I/I2)* of the unit element is denoted by Lie(G), and called the Lie algebra of G (it is an i?-Lie algebra, in fact). The G-module Lie(G), as (I/I2)*, is called the adjoint representation of G.
4.3
Algebraic tori
(4.3.1) For a positive integer n, the X-group scheme GLO®" is denoted by GL{n,X) or GLn(X). We denote O$ = GL(1, X) by Gm,x or G m . The direct product G£, of G m is called the n-fold split torus. An X-group scheme which is isomorphic to the n-fold split torus for some n is also called a split torus. An X-group scheme T is called an n-torus if T is affine flat of finite type over X, with its all geometric fibers n-fold split tori. An n-fold split torus is an n-torus, but the converse is not true. (4.3.2) Consider the case X = Spec R is affine. Then we can express /?[Gm] = Jty.f"1 ]) with t group-like. Hence, if G is an affine i?-group scheme, then to give a rank-one i?-free representation of G is the same as to give a homomorphism of R-group schemes G —> G m , and it is the same as to give a bialgebra map R[t, t'1] —> R[G), which is given by a group-like element of R[G]. Thus, the set of isomorphism classes of rank-one i?-free representations X(G) of G, and the set of group-like elements X(/?[G]) of R[G] are identified. Moreover, the canonical bijection X(G) —y X(R[G]) is an isomorphism of abelian groups, where the product of X(G) is given by tensor products, and the product of X(R[G]) is the product of R[G). However, it is common to view X(G) as an additive group, and express its product by '+'. The group X(G) or X(R[G\) is called the character group oiG. (4.3.3)
As the coordinate ring of T := G£, R is expressed as
with each tt group-like, it is easy to see that X(H) = {tx \ A G Z"}, where as usual tx := tXl ••• tx" for A = (A x ,..., An) e Z n . By the map given by tx y-¥ A, we have an isomorphism of additive groups X(H) = Z n . When T is an i?-split torus, we call n = rankzX(T) the rank of T. Note that X(H) above is an i?-basis of H, and H is a direct sum of rank-one i?-free i?-subcoalgebras. (4.3.4) Let T = Spec H be as in (4.3.3). If V is a T-module, then we have a direct sum decomposition V = ©Agx(//) Vx> where (4.3.5)
Vx =
{veV\wv{v)=v®\).
106
I. Background Materials
If Vx 7^ 0, then A is called a weight of V. If / : V —> V is a T-homomorphism, then obviously we have f(V\) C {Vx'). Thus, we have a canonical functor from j-M to the category of X(T)-graded i?-modules. Conversely, letting (4.3.5) be the definition, we have its quasi-inverse, and we see that a Tmodule is nothing but an X(T)-graded module. Even if the base scheme X is not affine, for a split torus T = G£, we have X(T) = Z" by the same reasoning, and X(T)-graded quasi-coherent Ox-modules and T-modules are the same thing.
4.4
Maximal tori, Borel subgroups, and reductive groups
(4.4.1) Let k be an algebraically closed field, and G a reduced affine algebraic fc-group scheme. In this situation, G and the group G(fc) are sometimes identified. Any Zariski-closed subset of G(k) is considered as a reduced closed subscheme of G. G has a maximum connected normal solvable subgroup, which is a closed subgroup of G, called the radical of G. We say that G is reductive if G is connected and non-trivial, and the radical of G is a torus. If G is reductive, then the connected component Z(G)° of the center Z(G) of G containing the unit element agrees with the radical of G. For example, a torus, GL(n, k), SL(n, k), SO(n, k), Sp(n, fc), and their direct products are reductive. If G is reductive, then the derived subgroup [G, G] and G/Z(G) are also reductive. (4.4.2) Let k and G be as in (4.4.1). A maximal connected solvable subgroup of G is called a Borel subgroup of G. A Borel subgroup is Zariski closed. It is not unique, but is unique up to conjugacy. For a closed subgroup P of G, P contains some Borel subgroup of G if and only if G/P is a fc-projective variety. If the equivalent conditions are satisfied, then P is called a parabolic subgroup of G. If, moreover, there is no closed subgroup Q of G such that P C Q C G , then P is called maximal. A subgroup which is maximal among closed subgroups which are tori is called a maximal torus. Note that maximal tori are conjugate to one another, hence their ranks are equal. We call the rank of a maximal torus of G the rank of G. By Lemma 4.2.8, G is a closed subgroup of some GL(n, k). We say that x £ GL(n, k) is unipotent if its eigenvalues are 1 only. As this is equivalent to (x - 1)" = 0, the set of unipotent matrices in GL(n, k) is Zariski closed. Hence, the set of unipotent elements Gu in G C GL(n, k) is a closed subset of G. Note that G u is independent of the embedding G G is defined over Z. In particular, (LieG)* is .R-free for any A £ X(T). For a G-module V, a weight of V as a T-module is called a weight of the G-module V. A non-zero weight of the adjoint representation Lie G is called a root of G. The set of roots of G is denoted by E G = £. Note that we have (Lie G) o = LieT, and we have a direct sum decomposition LieC = L i e T ® ® (Lie G) a . If a e £, then (LieG) a is rank-one .R-free. If a G £, then - a 6 £. For any Z?-scheme X, Hom fl . sch (X, A}j) = T(X, Ox) as an additive group forms a representable .R-group. Thus, A}j can be viewed as a commutative .R-group scheme (by addition), which we denote by G o. For a S E, there exists some .R-group homomorphism xa : Ga —> G such that the conditions
108
I. Background Materials
1 For any commutative /?-algebra A, any t € T(y4) and any a € A = Ga(A), txa(a)t~l = xa(a(t)a). 2 The tangent map dxa is an isomorphism LieGa = (LieG)Q. are satisfied. Note that xa is unique up to isomorphisms of Go by the action of R* by multiplications. We always assume that xa is the base change of an xa defined over Z (and hence is uniquely determined up to sign change). Note that xa is a closed immersion. We denote the scheme-theoretic image xa(Ga) by Ua, and call it the root subgroup of G with respect to a. The group functor Ua represents A i-> Im(a;Q(yl)). (4.5.4)
Let R, G, T and E be as in (4.5.3). The set y(T):=HomR. groU p(G m ,T)
is an abelian group. If T ^ G^, then Y(T) * Z n . For f,g e Y{T), the sum / + g is nothing but the composite A2 > • • • > An.
4.7
Representations of reductive groups over an algebraically closed field
Let R be a noetherian commutative ring, G an .ft-split reductive group, T a split maximal torus of G defined over Z, and let A be a base of the root system E of G. We recall that U is a normal subgroup of the negative Borel subgroup B of G, and B is a semidirect product of U and T, with U normal. Let A G X(T), namely, A is a rank-one i?-free T-module. Then letting U act on A trivially, A is a rank-one R-iree B-module, whose restriction to T is the original A. We denote this rank-one i?-free J5-module by RxDefinition 4.7.1 For A G X+, we denote ind£(i?A) by V(A) = V G (A), and call it the induced module of highest weight A. The G-module V(A*)* is denoted by A(A) = Ac(A), and called the Weyl module of highest weight A. (4.7.2) From now on, R = k denotes an algebraically closed field. We denote R\ by k\. The following is well-known. (4.7.3) The set {k\ \ A G X(T)} is a complete set of representatives of the isomorphism classes of simple 5-modules. Any simple [/-module is trivial.
112
I. Background Materials
(4.7.4) If M is a finite dimensional B-module, then •R'indg(M) is also finite dimensional. If i > dimG/B, then we have R'md^(M) = 0 (this vanishing holds also for infinite dimensional B-modules, see Lemma 3.6.17). In particular, for A e X+, we have that VG(A) and AG(A) are finite dimensional. (4.7.5)
For A e X(T), we have ind^(kx) ^ 0 4=> A G X%.
In the representation theory of the reductive group G, induced modules and Weyl modules play important roles. Theorem 4.7.6 ( K e m p f s vanishing) IfX £ X+, then for i > 0 we have #ind|(A: A ) = 0. (4.7.7) For A £ X+, we have soc(V(A)) = top(A(A)), and they are simple. We denote this simple G-module by L(X) = LG(X), and call it the simple G-module of highest weight A. As a result, we have that {L(X) | A £ X+} is a complete set of representatives of isomorphism classes of simple Gmodules. Moreover, if A, fi E X+ and L(/x) is a subquotient of rad A(A) © V(A)/socL(A), then fi< X. Theorem 4.7.8 (Cline-Parshall-Scott-van der Kallen, [39]) For any dominant weights X,fx£ X Q , we have
(4.7.9) Let V be a finite dimensional G-module. Then it is a T-module by restriction, hence we have a decomposition V = 0* e x(r) ^A- The element
ch(V):=
Y,
is called the formal character of V. It is easy to see that ch(V) € (ZX(T))W, where W is the Weyl group of G. As a consequence of Weyl's character formula [90, p. 250], we have that for any A 6 X£, ch(Ac(A)) = ch(Vc(A)), and they are determined only by A, and independent of A;. In particular, we have dim/t AG(A) = dim^ VG(A) are independent of k. (4.7.10) Another important property of weights of induced (or Weyl) modules is: if VG(A) M ^ 0, then we have u;0A < n < X- Moreover, VG(A)A = 1. Similarly for AG(A) and LQ{X). The name 'highest weight' comes from this fact. It follows that, if V is a G-module, A € XQ, dim^ VA = 1, and
dim* VM = 0 for n € X£ \ {A}, then V £ LG(A).
4. From representation theory
113
(4.7.11) If Jfc is of characteristic 0, then we have A(A) =* V(A) for For the results above, we refer the reader to [90].
4.8
XeX+.
Universal module functors
(4.8.1) Let X be a scheme. We say that U — (UA) is a universal family over X if for any X-algebra A (i.e., a morphism Spec>l -» X), a full subcategory UA of ^M closed under isomorphisms corresponds, and for any X-algebra map A -> B (i.e., a morphism Spec B —> Spec .4 of X-schemes), M € UA implies M ®A B £ UB- For example, the family Vx = (PA), where PA is the full subcategory of ^M consisting of finite projective /1-modules, is a universal family. Definition 4.8.2 Let s, t > 0, andZ^i,... ,Us+t, and V be universal families. We say that M = ((MA), (pf)) : Wi x • • • x U3 x U°^ x • • • x U%t -> V is a universal functor of type (r, s), if for each commutative X-algebra A, MA : (U,)A X • • • X ([/,)* x ( ^ + 1 ) 7 x • • • x (f/s+f^p -» V^ is a functor, for each X-algebra map / : A —> 5 , p/ : (B?) o M^ ->• M B O ((B®?)S
X ((5®?) 0 p )')
is a natural isomorphism, and for any composable X-algebra maps
the diagram
C®BB®A
MA
C
®Bpf
A
. C®B MB((B®A?y, (B®A?)S) MC((C ®B (B®A?)Y, (C ®B \Mc(ar,as)
C ®A MA
^Z
. M
is commutative, where a : C®B(B®A7) -> C®^? is the usual identification. If Id = • • • = Us+t = Vx and V = (AM), then we say that M is a universal module functor of type (r,s). If Wi = • • • = Us+t = V = Vx, then we say that M is a universally projective functor of type (r, s). If it happens that X = Spec/? with R a PID, then a universally projective functor is sometimes referred as a universally free functor.
114
I. Background Materials
This definition could be made as a special case of part of the theory of fibered categories and pseudo-functors [66, VI]. In the sequel, we only treat universal module functors, for simplicity. Definition 4.8.3 Let M = ((MA), (/>,)) and M = ((NA), (p'f)) be universal module functors of type (r,s) over X. We say that
NA is & natural transformation, and for any X-algebra map f : A-> B, p'f o {(B®A?)B((fl fq is an isomorphism for N G yA, and M
-+HomA{KA,KA®M)
given by m *-* (q >-> q ® m) is an isomorphism for M G VA. Proof. Let
For M G AA and p 6 Spec,4, we set cM(p) := dimylp — depthM p . F
: • • . - > F2 % Ft i > Fo ^ M -> 0
be a free resolution of M with each Ft .A-finite. We set fi*M := lmd{. Then it is easy to see that (fijM)p is a maximal Cohen-Macaulay .Ap-module if and only if CM(P) < i. By Corollary 2.12.3, CM is an upper semicontinuous function over Spec A A similar argument applied to PM(P) := proj.dim^ M p instead of CM yields that PM is also upper semicontinuous. As Spec>l is quasi-compact, we have d := maxcM(p) < °°) a n d by definition, Sl^M is maximal Cohen-Macaulay. Hence, we have M e XA, and AA = XA. Similarly, an object M of AA belongs to VA if and only if proj.dim^ Mp < oo for any p G Spec A Next, we show that XA = ±KA. To prove this, we may assume that A is local. In this case, as KA is a dualizing complex of A, the assertion follows easily from the local duality (Theorem 2.10.7). Also, for any M e XA, we have that RomA(M, KA) G XA and M^EomA(}iomA(M,KA),KA) is an isomorphism. This is also checked after localization. Hence, (4.10.15) holds. Next, we show that uiA is an injective cogenerator of XA. We already know that UJA = &ddKA is #-injective, and u>A C XA. When we take an exact sequence 0 -> N -> F -> EomA{M, KA) -> 0
4. From representation theory
119
with F .A-finite free, then the sequence is an exact sequence in XA. Applying the functor Hom^?, KA) to the exact sequence, we have an exact sequence 0 -> M -> EomA(F, KA) -» EomA(N, KA) -+ 0 in XA again. As B.omA(F, KA) G u>A, we have that wA is an injective cogenerator of XA. XAnyA, Next, we show that XAf\yA C CJA- To verify this, we take M e and it suffices to show that HomA(M, KA) e VA. Hence, we may assume that (>l,m) is local. As KomA(M,KA) is maximal Cohen-Macaulay, it suffices to show that pTO).dimAUomA(M,KA)
< co
by Theorem 2.5.3. Let F be the minimal free resolution of A/m, and /* the minimal injective resolution of KA. Then as we know that M £ ±KA, we have
REom'A(F,M) £ REomA{¥,REom'A{REomA(M,I'),I')) £ REomA{¥ ®LA Hom^(M, I'),I') s RHom^(F ®J Hom j 4 (M,^),/'). Note that /2Hom^(F, M) has bounded homology groups, since M € Hence, F ®A EomA{M, KA) S i? EomA(R Hom^(F ®$ Hom yl (M, ^ ) , /*), / ' also has bounded homologies. This shows proj.dim^ Hom^(M, KA) < CXD. Hence, we have that (4.10.14) holds. The assertions (4.10.12) and (4.10.13) are consequences of Theorem 1.12.10. The assertion (4.10.16) is well-known. It was first proved by Sharp [136], and is generalized by Avramov and Foxby [18, Corollary 3.6]. We prove the first assertion of (4.10.16). As N G y& = &A, there is a finite o^-resolution W of iV. Since KA ®A EomA{KA, KA) —> KA is an isomorphism, we have that KA®A EomA(KA, W) -> W is an isomorphism of complexes. As the augmented complex W —> N —> 0 is a bounded exact complex consisting of objects of yA C KA, we have that EomA(KA, W) is a resolution of EomA(KA, N). By the five lemma, KA®A EomA(KA, N) -> N is also an isomorphism. The second assertion is proved similarly, utilizing Theorem 4.10.19 below. D As in the proof of the theorem, the following is easy to prove. Lemma 4.10.17 Let R be a regular ring, and V E RMJ. Then we have proj.dim fi V < oo.
120
I. Background Materials
(4.10.18) Let R be a noetherian commutative ring. We say that an Rmodule N is locally of finite flat dimension if flat.dimfip Np < oo for p G Spec R. Note that if N is of finite flat dimension, then it is locally of finite flat dimension. The converse is true if the Krull dimension of R is finite, see [55, Corollary 3.4]. Related to the proof of (4.10.16), the following holds [18, Corollary 3.6]. Theorem 4.10.19 Let R be a Cohen-Macaulay ring, and M a maximal Cohen-Macaulay R-module. If N is an R-module locally of finite flat dimension, then we have Torf (M, N) = 0 for i > 0. Note that the proof is easily reduced to the case that (R, m) is a complete local ring, and we may also assume that M = QR by (4.10.14). Corollary 4.10.20 Let R and M be as in the theorem, and let N be a perfect R-module of codimension h. Then we have Ext'R(N, M) = 0 (i ^ h). By Lemma 2.9.1, we have proj.dim R Ext^(iV, R) < oo, and Ext^N,
M) S T o r j ^ E x t * (JV, R), M).
By the theorem, the assertion follows.
•
Proposition 4.10.11 is well-known as the Cohen-Macaulay approximation. The name 'approximation' comes from the fact that XA is contravariantly finite (see Theorem 1.12.10). In Theorem 4.10.8, we have that any object of y is of finite injective dimension. However, in the situation of Proposition 4.10.11, this is not true any more if dim .4 = oo. (4.10.21) For a finite dimensional algebra A over a field, AuslanderBuchweitz contexts in (/iM)/ and basic cotilting modules of A are in one-toone correspondence. This beautiful result was proved by Auslander-Reiten [11]. Let A be a ring, and M an yl-module. We say that M is basic if M does not have any direct summand of the form N © N, where N is a non-zero i4-module. Theorem 4.10.22 Let k be a field, and A a finite dimensional k-algebra. We set AA •= CiM)/. Consider the following. a A full subcategory u in AA such that u/ = adda> C u>L and Xu = AA, where Xu is the full subcategory of AA consisting of X G x u such that there exists an exact sequence
such that T ' e w and Im/' G x w.
4. From representation theory
121
b A full subcategory X of AA which is closed under extensions, epikemels, and direct summands, and has an injective cogenerator, such that X = AA. c A covariantly finite full subcategory y of AA closed under extensions, monocokernels, and direct summands such that AA* G y and any object in y is of finite injective dimension. d An Auslander-Buchweitz context (X,y, w) in AAe The isomorphism class of a basic cotilting A-module T. The objects a—e above are in one-to-one correspondence by the following correspondences. a=>b u) to Xu. b=>c X to Xx. c=>a y
to-Lyny.
a,b,c=>d Obvious correspondence. a,b,c,d=>-e As we have AA* G X1 = u, there is an exact sequence 0 - » u)n - » • • • - > wi - > w 0 -> A-A* - > 0 withuji G u. Starting with the Krull-Schmidt decomposition ofu0®---(B u)n, we get a basic module T, removing N whenever we find N © TV in the decomposition. The isomorphism class ofT is uniquely determined by (X,y,ui), and T is the corresponding cotilting module.
e=>a T to addT. Corollary 4.10.23 Let A be a finite dimensional algebra over a field k, and (X,y,u) an Auslander-Buchweitz context in ,jM/. Then the number of isomorphism classes of indecomposable objects in u is equal to the number of isomorphism classes of simple A-modules. Proof. By the theorem, the number of isomorphism classes of indecomposable objects in w agrees with the number of indecomposable direct summands of T in the theorem, which agrees with the number of indecomposable direct summands of T". As T* is a tilting j4op-module, these numbers agree with the number of simples of Aop, which is equal to the number of simples of A by (4.9.9). D Notes and References. There is no new result at all in this section. In this section, we listed basic results in the representation theory of algebraic groups and algebras. For more, we refer the reader to [87], [90], [42] for algebraic groups, and to [12], [11], [112], [113] for algebras. Although we assume that a reductive group (over an algebraically closed field) is connected, note that it is not always assumed in the literature.
122
5
I. Background Materials
Basics on equivariant modules
5.1 Cocommutative Hopf algebra actions (5.1.1) Let R be a commutative ring, U a cocommutative .R-Hopf algebra, and A a commutative [/-module /^-algebra. Let M be an A#U-module, and N a [/-module. The /^-module M N is an A#U-module by (a ® u)(m ® n) = ^ a ( u ( i ) m ) 4#[/ denotes the smash product (3.14.3). If M is a [/-module and N an .4#[/-module, then letting A act on iV we get an A#U-module M ® N. If both M and TV are A# [/-modules, then there are two different ways to see that M <S> N is an A#U-module. Unless otherwise specified, we understand that A acts on M (so we take the former definition). Let M and N be yl#[/-modules. When we define an fi-linear map
d: M ®(A®N)
-¥ M ®N
by d(m B®U
= B#U
to be (p®\du- It is easy to see that y # [ / is an .R-algebra map. In particular, any 5#[/-module is an >l#[/-module by restriction. Let M be a S#[/-module, and V an A#U-modu\e. Then M ®A V, V ®A M, UomA(M,V) and UomA(V,M) are B#U-modules in a natural way, and as yl#[/-modules they agree with the ones defined in (5.1.1) and (5.1.2).
5. Basics on equivaiiant modules
123
(5.1.4) Let R, U, and
B be as in (5.1.3). Let M and N be jB#[/-modules, and V and W be ^4#t/-modules. Then the standard maps (5.1.5) (5.1.6) (5.1.7) (5.1.8) (5.1.9) (5.1.10) (5.1.11) (5.1.12) (5.1.13) (5.1.14) (5.1.15) (5.1.16) (5.1.17) (5.1.18) (5.1.19)
M —> EomA(A,M) ( m 4 ( a H am)) UomA(V, W) ®A Hom / i(M, K) -> Hom i4 (M, IV) Hom^lV, M) <S)A Hom/i(Vr, W) -^ Hom A (y, M) Hom/i(Vr ®yi W, M) ^ Hom/i(K! Ho m / i(iy, M)) M O.4 V, W) S Hom j4(M, Homj4(Vr, V (8.4 M, W) s Homii(Vp, Ho my i(M, Hom B (V ^ M, TV) ^ Hom^V, Hom B (M, N)) EomB(M ®A V, N) S Hom B (M, Hom,i(Vr) N)) HomA{M ® B -?V, V) ^ Hom B (M, Hom^(A^, V)) (M ®^ V) (8)4 IV «* M ®>i (V ®^ IV) M ®,i V = V ®A M M -4 Hom/i(Hom/ 4(M, V), V) (the duality map) r M ®A Homj4(V, IV) -> Hom/i(V , M ®^ IV) M ®B Hom/i(Vr, AT) -> ftomA{V, M ®B N)
are B#t/-linear maps, which are natural with respect to M, N, V, and W. As a special case, we will use the case A = R or A = B frequently. Taking the invariances H°{U,?) of both sides of (5.1.11), (5.1.12), and (5.1.13), we get natural isomorphisms (5.1.20) * : Homfl#C/(Vr ®A M, N) (5.1.21) V:1iomB#u(MAV,N) (5.1.22) * : HomA#u(M ® B N, V)
HomA#u(V, Hom B (M, N)) HomB#u(M,HomA(V,N)) Hom B # u (M, Homyl(A^, V)),
respectively. In particular, we have Lemma 5.1.23 Let R, U, f : A-> B, M, N, and V be as in (5.1.4). Then the following hold. 1 If N is B#U-injective injective. 2 If N is B#U-injective
and M is A-flat, then UomB(M,N) and V is A-flat, then Hom/i(Vr, N) is
3 IfN is B-flat and V is A#U-injective,
then YiomA(N, V) is
4 If M is B-projective and V is A#U-projective, projective.
is
A#U-
B#U-injective. B#U-injective.
then V ®A M is
B#U-
124
I. Background Materials
5 If M is B#U -projective and V is A-projective, then M A
B is
B#U-
Moreover, since B#[/ =* B®A{A#U) 2* (A#U)®AB as 4#[/-modules, we have Corollary 5.1.24 / / B is A-projective, then any B#U -projective module is A#U-projective. When we consider the case M = B in Lemma 5.1.23, 1, we have the following. Corollary 5.1.25 If B is A-flat, then any B^U-injective injective.
5.2
module is A#U-
Tor/4 and Ext^ as A#C/-modules
(5.2.1) Let R be a commutative ring, U a cocommutative /?-Hopf algebra, A and B commutative [/-module algebras, and ip : A -> B a [/-module algebra map. We assume that U is ii-projective. Lemma 5.2.2 Any A#U-projective injective module is A-injective.
module is A-projective.
Any
A#U-
Proof. We prove the first assertion. It suffices to show that A#XJ is Aprojective. As the action of A on A#U is given by a(b ® u) = ab u and U is /Z-projective, A#U is .A-projective. We prove the second assertion. It is enough to show that any injective . A ® H
(a® h H-> ^ a
0
® (Scn)h).
(a)
For an i?-algebra C, we consider the trivial G-action on C, and C is a G-algebra. However, if C has another G-algebra structure, we denote the trivial G-algebra C by C, to avoid confusion. For a C-module M, when we consider the trivial G-action, then M is a (G, C')-module. We
132
II. Equivariant Modules
denote this by M', to avoid confusion. Note that a : A —> A ® H and 7 : A ® H -> A' ® H are G-algebra maps, whence so is 0 : A —> A' ® H. Note also that a : A' -> A' ® H is a G-algebra map, which we denote by a', to avoid confusion. Lemma 1.1.3 Let C and D be G-algebras, and V a (G,C)-module. Let a : C -» C ® D be the G-algebra map given by a(c) — c 0 1. Then a#V is isomorphic to the (G, C D)-module V D, where the C D-action is given by (c ® d)(v a") := cv ® dd!. Proof. When we define
ip : a#V = V®C{C®D)-*V®D by v (c <S> d) i-> cu ® d, then it is a G-isomorphism. It is C ® ZMinear as well, by the definition of C ® D-action on V (8) Z). D By the lemma, a#V = V ®H.
Lemma 1.1.4 Let V be a (G,A)-module. Then (3#V is isomorphic to the (G, A'®H)-module V®H with the A'®H-action (a®h)(v®h') := £( o) aov® Proof. We have /3# = 7 # o a # ^ ( 7 ~ 1 ) # ° a # . Hence, @#V is the (G, /l' module
(j-lf(V
®A (A ® H)) ^ ( 7 " 1 ) # (^ ® H)
by Lemma 1.1.3. Now the assertion is trivial.
D
L e m m a 1.1.5 Let A be a G-algebra, and M a G-module A-module. Then M is a (G, A)-module if and only ifuiM '• M —> M'®H is a G-linear A-linear map from M to M ® H = /3 # a' # M'. Proof. As M is a G-module, w M is G-linear. For a £ A and m € M, we havew M (am) = H{am)(arn)o® (am)u and aw M (m) = E(o),(m)aomo®onmi, D and we have u)M is .A-linear if and only if M is a (G, i4)-module. L e m m a 1.1.6 We have the following. 1 Let M be a' (G,A)-module. When we define UM : M ® H -> M' ® H by OM{m ® h) := Y,(m)mo ® fn\h, then we have that D : (3# —> a# is a natural isomorphism between functors from G,A^ to /t®//M. Moreover, the composite map KP ® 1H) O 0\#M
= [(lA ® A H ) O (3}#M
(U8AH)#
° M ) [{lA ® AH)
1. Homological aspects of (G,A)-modules
133
agrees with the composite map [(/? \H) O (S]#M
{m )
" *°M)
= [o12 o f3]#M
[(/3 ® iH) o a]#M (ai2)# M
° ) [a12 o a ] # M = [(U ® A w ) o a)#M,
where a ] 2 : A® H -* A® H ® H is given by an(a® h) := a® h® 1. 2 Conversely, if an A-module M and an isomorphism D M : (3#M -> a # M suc/i t/io< A' ® H are G-algebra maps, we have /? # (a' # (V")) is a (G,A)module for any >l-module V. More explicitly, /? # (a^(V")) is the G-module V'®H, equipped with the ,4-module structure by a(v®h) := £(„) aov®aih. If M is a (G, ,4)-module, then uM:M-*M'®H = /3*{a'#(M')) is 4-linear (Lemma 1.1.5) and G-linear (Example 1.3.6.6), and hence it is (G, .4)-linear. Lemma 1.1.8 Let M 6 G ^ M , and V G ^M. Then the map $ : HomA(M,V)
->
EomG>A{M,0*(a'#(V')))
defined by $(/)(m) := £( m ) frriQ ® m\ is an isomorphism which is natural with respect to M and V. In particular, the forgetful functor C,A^ —> >iM has SA '•— /8*a^(?') as its right adjoint.
134
II. Equivariant Modules
Proof. As we have $ ( / ) = /? # (a' # /') o u, $ ( / ) is certainly a [G,A)linear map by the remark above. We define \& : Homc^M,/? # (a' # (V'))) —> H o m ^ M , V) by $(g)(m) := E(gm), where E : V ® 77 -> V is given by E(v /i) := e(/i)t>. As can be checked easily, E is ,4-linear, and hence so is \J/(