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Equivalence Theorems and Hopf–Galois Extensions
194, 245]274 Ž1997. JA966994
JOURNAL OF ALGEBRA ARTICLE NO.
Equivalence Theorems and Hopf]Galois Extensions Claudia Menini* and Monica Zuccoli Dipartimento di Matematica, Uni¨ ersita ` di Ferrara, Via Machia¨ elli 35, 44100 Ferrara, Italy Communicated by Susan Montgomery Received August 2, 1996
1. INTRODUCTION In this article we apply equivalence theorems for categories of modules to the category M Ž H . AD of right Ž D y A.-Hopf modules to get a characterization of Hopf]Galois extensions. Indeed let P, Q g Mod-R and T s EndŽ PR ., and assume that P belongs to the category s Ž Q R . of modules subgenerated by Q R . In Theorem 2.3, by modifying a theorem formulated by Dal Pio and Orsatti w5, Theorem 2.6x, we study the situation where, for every M belonging to the category s Ž Q R ., the map CM :
Hom R Ž P , M . m P ª M T
j m x ¬ j Ž x. is an isomorphism in Mod-R. In particular we prove that this holds iff CM is an isomorphism for every M g GenŽ Q R . and T P is flat iff s Ž Q R . s s Ž PR . and PR generates all submodules of PRn for every n g N, n ) 0. Let now H be a Hopf algebra over a field k, let A be a right H-comodule algebra, let D be a right H-module coalgebra, and consider the category M Ž H . AD of right Ž D y A.-Hopf modules, i.e., of those D-comodules which are equipped with a suitable right A-module structure in such a way that *This article was written while the first author was a member of G.N.S.A.G.A. of C.N.R. with partial financial support from M.U.R.S.T. and H.C.M. project n.CHRX-CT93-0091. E-mail: [email protected]. 245 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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MENINI AND ZUCCOLI
these two structures are compatible Žsee Section 3.1.. The starting point for applying the previous result in this setting is that Žsee Lemma 3.9. M Ž H . AD s s Ž AaD* A m D . as remarked by Caeneppeel and Raianu in w3x. H
Moreover, using a grouplike element x g D, A can be endowed with a suitable D-comodule structure in such a way that A g M Ž H . AD Žsee Section 3.13.. Since End l AaD* Ž A. ( A x Žsee Theorem 3.15., we may apply H
the quoted result to the situation where R s A a D*, which is essentially H the smash product introduced by Doi in w6x, PR s AaD* A, Q R s AaD* A m D, H
H
and T s A x . Now if A xA is flat, CM is an isomorphism for every M g GenŽ AaD* A m D . iff CAmD is an isomorphism Žsee Lemma 3.22. and H
CAmD is an isomorphism iff
bx :
Am A ª A m D Ax
amb¬
Ý ab0 m x £ b1
is an isomorphism, i.e., iff A x ; A is a right Hopf]Galois extension. In this way we get Theorem 3.27, which characterizes right Hopf]Galois extensions A x ; A such that A xA is flat, using the structure of A as a left A a D*-module. In particular it is shown that these extensions are exactly H those for which the weak structure theorem in the sense of w7x holds. Also Theorem 3.27 enables us to apply the well-known Fuller theorem on equivalences of modules Žsee Theorem 2.5. in our setting. Thus we get Theorem 3.29, which characterizes right Hopf]Galois extensions A x ; A such that A xA is faithfully flat. In particular we prove that this holds iff A is a quasiprogenerator and s Ž AaD* A. s s Ž AaD* A m D . iff AaD* H
H
H
Žy. x : M Ž H . AD s s Ž AaD* A m D . ª Mod-A x is an equivalence, i.e., the H
strong structure theorem in the sense of w7x holds. In this case when D s H we thus get part of the famous Schneider’s theorem on Hopf]Galois extensions. Moreover, using the adjointness between the induction functor F and the functor G s y I H as described by Caeneppeel and Raianu D w3x, we easily get as a corollary ŽCorollary 3.32. Doi’s theorem on Hopf]Galois extensions w6, Theorem 2.3x. When dim k Ž D . - `, then s Ž AaD* A m D . s A a D*-Mod as remarked H
H
by Doi w6x. Therefore in this case A is a generator of s Ž AaD* A m D . if H
and only if A is a generator in A a D*-Mod. Hence this is equivalent, in H
HOPF ] GALOIS EXTENSIONS
247
the finite case, to A x ; A being a right Hopf]Galois extension and A xA being flat. Theorem 4.2 characterizes this situation. Also AaD* A is a H
quasiprogenerator and s Ž AaD* A. s s Ž AaD* A m D . in the finite case if H
and only if
AaD*
H
A is a progenerator. Theorem 4.3 characterizes this
H
situation. When D s H and dim k Ž H . - `, B s AcoH ; A is a right Hopf]Galois extension if and only if AaH * A is a generator for the category A a H *-Mod. H
H
Theorem 4.7 characterizes this situation. Here we prove that, in this case, A is a Frobenius extension of B. Theorem 4.7 contains essentially the famous result formulated by Cohen, Fischman, and Montgomery for Hopf]Galois extensions w4, Theorem 1.2 and Theorem 1.29x. Finally, using Theorem 4.7, we can prove that the assumption of faithfully flatness can be weakened in the finite case whenever D s H. In fact if B ; A is a right Hopf]Galois extension, then B A is faithfully flat if and only if B A is a weak generator. Theorem 4.8 characterizes such extensions. Part of this result appears in w4, Theorem 2.2x. We have made some efforts to get a paper that even someone not acquainted with Hopf algebra theory could read. Toward this end we have inserted some details that a Hopf-algebra expert would skip. The main results of this article were presented at the AMS-BENELUX meeting, held in May 1996 at Antwerp, Belgium.
2. PRELIMINARIES 2.1 Throughout this article, all rings have a nonzero identity and all modules are unital. Let R be a ring, PR a right R-module. We denote by GenŽ PR . the full subcategory of Mod-R generated by PR . The objects of GenŽ PR . are the right R-modules MR which are generated by PR , i.e., such that there exists a surjective morphism PRŽ X . ª M ª 0 for a suitable set X. GenŽ PR . is closed under taking epimorphic images and direct sums. Partially following w17x, we denote by s Ž PR . the full subcategory of Mod-R whose objects are the right R-modules subgenerated by PR , i.e., such that there exists an injective morphism M ¨ L for a suitable L g GenŽ PR .. s Ž PR . is closed under taking submodules, epimorphic images and direct sums. Clearly GenŽ PR . s s Ž PR . if and only if GenŽ PR . is closed under submodules.
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MENINI AND ZUCCOLI
2.2. THEOREM. equi¨ alent.
Let PR g Mod-R, T s EndŽ PR .. Then the following are
Ža. For e¨ ery n g N, n ) 0, PR generates all submodules of PRn. Žb. GenŽ PR . s s Ž PR .. Žc. For e¨ ery M g s Ž PR . the map CM :
Hom R Ž P , M . m P ª M T
j m x ¬ j Ž x. is an isomorphism in Mod-R. Žd. For e¨ ery M g s Ž PR ., CM is a surjecti¨ e morphism in Mod-R. Že. T P is flat and the functor Hom R Ž T PR , y . : Gen Ž PR . ª Mod-T is full and faithful. Moreo¨ er if these conditions are fulfilled H s Hom R ŽT PR , y. induces an equi¨ alence between GenŽ PR . s s Ž PR . and ImŽ H .. Proof. Ža. « Žb. follows from Ž1. « Ž2. of Lemma 1.4 in w18x. Žb. « Žc. follows from Ž1. of Lemma 1.3 in w18x. Žc. « Žb., Žc. « Žd., and Žd. « Žb. are trivial. Žb. « Že. follows from Ž2. « Ž3. of Lemma 1.4 in w18x. Že. « Ža. follows from Ž3. « Ž1. of Lemma 1.4 in w18x. Note that the equivalences Ža. m Žb. m Že. and the last assertion appear in w5, Theorem 2.6x. 2.3. THEOREM. Let P, Q g Mod-R, let T s EndŽ PR ., and assume that P g s Ž Q R .. Then the following are equi¨ alent. Ža.
For e¨ ery M g GenŽ Q R . CM :
Hom R Ž P , M . m P ª M T
j m x ¬ j Ž x. is an isomorphism in Mod-R and T P is flat. Žb. For e¨ ery M g s Ž Q R ., CM is an isomorphism in Mod-R.
HOPF ] GALOIS EXTENSIONS
249
Žc. s Ž Q R . s GenŽ PR .. Žd. T P is flat, s Ž Q R . s s Ž PR . and the functor Hom R Ž P , y . : Gen Ž PR . ª Mod-T is full and faithful. Že. s Ž Q R . s s Ž PR . and PR generates all submodules of PRn, for e¨ ery n g N, n ) 0. Proof. Ža. « Žb. Let M g s Ž Q R .. Then there exist a right R-module N g GenŽ Q R . and an injective morphism i: M ª N. By applying the functor H s Hom R Ž P, y. to the exact sequence NrM
6
0,
6
6
p
N
6
i
M
0
where p denotes the canonical projection, we get the following exact sequence: 6
6
HŽ N .
H Žp .
H Ž NrM . .
6
H Ži.
HŽ M .
0
By the flatness of T P, we obtain the following exact sequence: T
HŽ N . m P
h
T
H Ž NrM . m P ,
6
f
6
HŽ M . m P
6
0
T
where f s H Ž i . m id P and h s H Žp . m id P . Consider the following comT
T
HŽ N . m P T
T
CNr M X
p
NrM
0,
6
N
6
i
6
6
6
M
H Ž NrM . m P
CN X
6
CM
0
h
6
f
6
HŽ M . m P
6
0
6
T
mutative diagram:
where CN and CN r M are isomorphisms as both N and NrM g GenŽ Q R .. It follows that CM is an isomorphism too. Žb. « Žc. Since P g s Ž Q R ., GenŽ PR . : s Ž Q R .. As CM is an isomorphism for every M g s Ž Q R ., s Ž Q R . : GenŽ PR .. Žc. « Žd. Since P g s Ž Q R ., we get GenŽ PR . : s Ž PR . : s Ž Q R . s GenŽ PR . : s Ž PR .. Hence GenŽ PR . s s Ž PR . and we can apply Žb. « Že. of Theorem 2.2. Žd. m Že. follows from Že. m Ža. in Theorem 2.2. Žd. « Ža. By Že. « Žc. in Theorem 2.2, CM is an isomorphism for every M g s Ž PR . s s Ž Q R . = GenŽ Q R ..
250
MENINI AND ZUCCOLI
2.4. DEFINITION. Let R be a ring, M a left R-module. R M is called a weak generator if Y m M s 0 for every right R-module Y implies Y s 0. R
Recall that a module PR is called a quasiprogenerator if PR is finitely generated, is quasiprojective, and generates each of its submodules. 2.5. THEOREM w2, 8x. equi¨ alent.
Let
T
PR be a bimodule. Then the following are
Ža. Hom R Ž P, y.: s Ž PR . ª Mod-T and y
m P: T
Mod-T ª s Ž PR .
are in¨ erse category equi¨ alences. Žb. Hom R Ž P, y.: s Ž PR . ª Mod-T is a category equi¨ alence. Žc. y m P: Mod-T ª s Ž PR . is a category equi¨ alence. T
Žd.
The map CM :
Hom R Ž P , M . m P ª M T
j m x ¬ j Ž x. is an isomorphism for e¨ ery M g s Ž PR . and CNX :
N ª Hom R P , N m P
ž
n¬
ž
CNX Ž n . :
T
/
P ª NmP T
x¬nmx
/
is an isomorphism for e¨ ery N g Mod-T. Že. CM is an isomorphism for e¨ ery M g s Ž PR . and T P is faithfully flat. Žf. CM and CTX r I are isomorphisms for e¨ ery M F PR and e¨ ery I F TT . Žg. PR is quasiprojecti¨ e and generates each of its submodules, T P is a weak generator, and T ( EndŽ PR . canonically. Žh. PR is a quasiprogenerator and T ( EndŽ PR . canonically. Ži. T P is a weak generator, CM is an isomorphism for e¨ ery M g Ž Gen PR ., and T ( EndŽ PR . canonically. Proof. Ža. m Žd. m Žf. m Žh. are in Theorem 2.6 in w8x. Ža. m Žb. m Žc. follow by adjoint properties of Hom R Ž P, y. and y m P. T
HOPF ] GALOIS EXTENSIONS
251
Žh. « Ži. Since we already know that Žh. « Žd., we get that GenŽ PR . s s Ž PR . so that Ž4. « Ž3. of Theorem 10 in w2x applies. Ži. « Žh. By Ž3. « Ž4. of Theorem 10 in w2x. Žf. « Žg. By Žc. « Žd. of Theorem 2.6 in w8x. Žg. « Že. and Žg. « Žf. By Lemma 2.2 in w8x we get s Ž PR . s GenŽ PR . so that, by Lemma 2.1 in w8x, T P is flat and hence faithfully flat, being a weak generator. Now apply Žd. « Žc. and Žd. « Žb. of Theorem 2.6 in w8x. Že. « Ži. Since CP is an isomorphism and T P is faithfully flat it is easy and straightforward to prove that the canonical morphism T ª EndŽ PR . is bijective. We recall now the celebrated Morita theorem. 2.6. THEOREM w12x. are equi¨ alent.
Let PR g Mod-R, T s EndŽ PR .. Then the following
Ža. PR is a faithful quasiprogenerator; T P is finitely generated. Žb. PR is a progenerator. Žc. T P is a progenerator and T PR is faithfully balanced. Žd. The functor map Hom R Ž P , y . :
Mod-R ª Mod-T M ¬ Hom R Ž P , M .
is an equi¨ alence. Proof. See w12, Theorems 3.2 and 3.4x.
3. MAIN RESULTS 3.1 Let H be a Hopf algebra over the field k with comultiplication D, counit « , and antipode S. Let A be a right H-comodule algebra with structure map r , and let B s AcoH s a g A < r Ž a. s a m 1 A 4 Žsee w11x for an explanation and definition.. Recall that a right H-module coalgebra is a coalgebra D together with a right H-module structure m D : D m H ª D: d m h ¬ d £ h such that m D is a coalgebra map, that is, D D Ž d £ h. s
Ý d1 £ h 1 m d 2 £ h 2 ,
«D Ž d £ h. s «D Ž d . « Ž h. . Following Doi we introduce the category of right Ž D y A.-Hopf modules denoted by M Ž H . AD Žor simply MAD . as follows. An object in M Ž H . AD is a
252
MENINI AND ZUCCOLI
right D-comodule with structure map r M , endowed with a right A-module structure such that, for every m g M and a g A, r M Ž ma. s Ým 0 a0 m m1 £ a1. Clearly a morphism in M Ž H . AD is defined to be both a morphism of right D-comodules and of right A-modules. The definition of left H-module coalgebra is given in a similar way. Let D be a left H-module coalgebra, A a right H-comodule algebra. The category of right]left Ž D y A.-Hopf modules denoted by A M Ž H . D Žor simply A M D . is defined as follows. An object in A M Ž H . D is a right D-comodule endowed with a left A-module structure such that, for every m g M and a g A, r M Ž am. s Ýa0 m 0 m a1 © m1. 3.2. EXAMPLE. For every M g Mod-A, M m D g M Ž H . AD via r MmD s id M m D D and m m d ? a s Ýma0 m d £ a1 , m g M, d g D, a g A. Moreover if M g M Ž H . AD it is easy to prove that r M : M ª M m D is a morphism in M Ž H . AD . 3.3. DEFINITION. Let A be a right H-comodule algebra, D a right H-module coalgebra. Let A a D* s A m D* as a k-vector space and let H
aag s a m g for every a g A, g g D*. Given a, b g A, g , x g D*, set aag ? bax s
Ý b0 aa Ž b1 © g . x ,
where ² h © g , d : s ²g , d £ h: for every h g H, d g D, g g D*. By linearity this defines a multiplication in A a D* which becomes a ring with H
identity 1 Aa« D . The ring A a D* will be called the smash product of A H and D* Žover H .. 3.4. DEFINITION. Let A be a right H-comodule algebra, D a left H-module coalgebra. Similarly A a9 D* is defined by setting H
aag ? bax s
Ý ab0 a Ž g £ b1 . x ,
where ²g £ b1 , d : s ²g , b1 © d : for every d g D. Note that this is exactly the smash product introduced in w6x. 3.5. Remark. Assume that the antipode Žof H . S is bijective. In this case the opposite ring H op of H is a Hopf algebra with comultiplication D, counit « , and antipode S, the composition inverse of S. In this case, given a left H-module coalgebra D, D can be regarded as a right H op -module coalgebra and Aop as a right H op -comodule algebra. Then it is easy to check that Aop aop D* s A a9 D*. H
H
HOPF ] GALOIS EXTENSIONS
3.6. PROPOSITION. L:
253
Assume that S is bijecti¨ e. Then the map
ž A a H */
op
H
ª Aop aop H * H
a ag ¬
Ý a 0aŽ S Ž a1 . © g .( S
is a ring isomorphism. Proof. Using the following equalities S Ž h1 g . © Ž Ž h 2 © x . g . s Ž S Ž g 2 . © x .Ž S Ž hg 1 . © g . ,
Ž S Ž h . © g . ( S £ g s S Ž hg . © g ( S, Ž xg . ( S s Ž g ( S . Ž x ( S . , which hold for every h, g g H and g , x g H *, as it is easily checked, it is straightforward to prove that L is a ring morphism. Now it is easy to prove that h© Ž x ( S . s Ž x £ S Ž h . . ( S,
Ž xg . ( S s Ž g ( S .Ž x ( S . . Using these equalities it can be proved that Ly1 :
Aop aop H * ª A a H *
ž
H
a ag ¬
H
op
/
Ý a 0 a Ž g £ S Ž a 1 . . ( S.
3.7. LEMMA. Let A a right H-comodule algebra, D a right H-module coalgebra. The maps f:
Aop ª A a D* H
a ¬ aa« D and g:
D* ª A a D* H
g ¬ 1 Aag are injecti¨ e ring morphisms. In particular e¨ ery left A a D*-module is a right A-module ¨ ia f.
H
254
MENINI AND ZUCCOLI
3.8 Let M g M Ž H . AD . M becomes a left A a D*-module by setting H
Ž a ag . ? m s
Ý m 0 a ² g , m1 : .
We denote this module by GŽ M .. Moreover, given M, N g M Ž H . AD and an abelian group morphism f : M ª N, f is a morphism in M Ž H . AD if and only if f is a morphism of left A a D*-modules. In this case we denote by H GŽ f .: GŽ M . ª GŽ N . the map f regarded as a morphism in A a D*-Mod. H
The assignments M ¬ GŽ M ., f ¬ GŽ f . define a covariant functor D
G: M Ž H . A ª A a D*-Mod. H
In particular, by Example 3.2, for every M g Mod-A, M m D is a left A a D*-module via H
Ž aag . ? Ž m m d . s
Ý ma0 m d1 £ a1²g , d2 : .
3.9. LEMMA w3, 13x. Im Ž G . s s
ž
AaD*
AmD .
/
H
Therefore G induces an equi¨ alence D
MŽ H.A ª s
ž
AaD*
AmD .
H
/
Proof. Let M g M Ž H . AD . Then r M : M ª M m D is an injective morphism in M Ž H . AD . Let pM :
AŽ M . ª M
Ž am . mgM ¬
Ý mgM
mam .
HOPF ] GALOIS EXTENSIONS
255
Then pM is a surjective morphism in Mod-A and it is easy to prove that pM m id D : AŽ M . m D ª M m D is a surjective morphism in M Ž H . AD . As AŽ M . m D ( Ž A m D .Ž M . in M Ž H . AD , we get that GŽ M . g s Ž AaD* A m D .. H
Conversely let M g s Ž AaD* A m D .. Then there exist a set X, a left H
A a D*-module L and a surjective morphism of left A a D*-modules p H
H
such that Ž A m D .Ž X . p
6
i
M¨
L
As remarked in Lemma 3.7, M is a right A-module via m ? a s Ž aa« D . m. Let us prove that M is a right D-comodule. In fact Ž A m D .Ž X . is a right D-comodule and hence a rational left D*-module. By Lemma 3.7, p is a surjective morphism of left D*-modules so that L is a rational D*-module too. Therefore L is a right D-comodule so that both i and p are morphisms of right D-comodules. Now, by Lemma 3.7, both p and i are right A-module morphisms. As p is surjective and Ž A m D .Ž X . g M Ž H . AD , we get that L itself satisfies the requirement of being a right Ž D y A.-Hopf module, i.e., L g M Ž H . AD . Since i is injective we get that also M g M Ž H . AD . By means of the foregoing theorem, in the following we will often identify M Ž H . AD with s Ž AaD* A m D .. H
3.10 w6x Let D be a right H-module coalgebra and x g D a grouplike element. Let px : H ª D h ¬ x £ h.
px is a morphism of right H-module coalgebras.
3.11. PROPOSITION.
3.12. Remark. Regarding H as a right H-module coalgebra we may consider M Ž H . AH , which will be simply denoted by MAH . Note that A g MAH . In fact for every a, b g A we have r Ž ab. s Ýa0 b 0 m a1 b1. 3.13 Let M g MAH and set
r˜M s Ž id M m px . ( r M :
MªMmD m¬
Ý m 0 m x £ m1 .
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MENINI AND ZUCCOLI
M can be regarded as an object of M Ž H . AD via r˜M . We will denote this object by Mp x. Given a morphism f : M ª N in MAH , it is easy to prove that f is a morphism Mp x ª Np x in M Ž H . AD . In this way we get a covariant functor D
MAH ª M Ž H . A M ¬ Mp x .
In the following for a given M g MAH we will simply write M instead of Mp x, whenever no confusion will arise. 3.14. DEFINITION.
For every M g M Ž H . AD set M x s m g M < rM Ž m . s m m x 4 .
In particular A x s a g A <Ýa0 m x £ a1 s a m x 4 . In the particular case where D s H and x s 1 H , M x will be denoted by M coH . 3.15. THEOREM. Let A a right H-comodule algebra, D a right H-module coalgebra, and x g D a grouplike element. Then Ž1.
For e¨ ery M g M Ž H . AD ,
n M : Hom AaD* Ž A, M . ª M x H
f ¬ f Ž 1A . is an isomorphism and its in¨ erse is VM :
M x ª Hom AaD* Ž A, M . H
m¬ Ž2.
ž
V M Ž m. :
AªM a ¬ Ž aa« D . ? m
/
The map
nA : End lAaD* Ž A . ª A x H
is a ring isomorphism. Therefore A is an A a D* y Aop x -bimodule. H
Ž3.
For e¨ ery M g M Ž H . AD , Hom AaD* Ž AaD* A A opx , H
AaD*
H
A x-module and n M is an isomorphism of right A x-modules.
H
M . is a right
HOPF ] GALOIS EXTENSIONS
257
Proof. Ž1. Let f g Hom AaD* Ž A, M .. Then H
r M Ž f Ž 1 A . . s Ž Ž f m id D . ( r˜A . Ž 1 A . s Ž f m id D . Ž 1 A m x . s f Ž 1 A . m x. Note that the first equality holds as f : A ª M is a morphism of left D*-modules and hence a right D-comodule morphism. Therefore we get f Ž1 A . g M x . Let m g M x . We have
Ž n M ( V M . Ž m . s Ž V M Ž m . . Ž 1 A . s Ž 1 Aa« D . ? m s m « D Ž x . s m,
Ž Ž V M ( n M . Ž f . . Ž a. s Ž V M Ž f Ž 1 A . . . Ž a. s Ž aa« D . ? f Ž 1 A . s f Ž Ž aa« D . 1 A . s f Ž a . . Ž2. We prove that V A is a ring morphism. Let b, b9 g A x , a g A. We have
Ž V A Ž bb9. . Ž a. s Ž aa« D . ? bb9 s Ž bb9. a, Ž V A Ž b . ( V A Ž b9. . Ž a. s V A Ž b . Ž Ž aa« D . ? b9 . s V A Ž b . Ž b9a. s Ž b9aa« D . ? b s b Ž b9a . . Ž3. Let b g A x . We will prove that n M Ž fb . s n M Ž f . b. We have
n M Ž fb . s f Ž b . , by Section 3.8 and
n M Ž f . b s f Ž 1 A . b s Ž ba« D . f Ž 1 A . s f Ž Ž ba« D . 1 A . s f Ž b . . 3.16 Let V g Mod-A x . Since A is an A x y Ž A a D*. op -bimodule, V m A is a H
left A a D*-module, where for a given a ag g A a D* we have H
H
Ž a ag . ? Ž ¨ m a. s ¨ m Ž Ž a ag . ? a. s
Ý ¨ m a0 a ²g , x £ a1 :
Ax
258
MENINI AND ZUCCOLI
for every ¨ g V and a g A. Clearly V m A g GenŽ AaD* A. : s Ž AaD* A Ax
H
H
m D . s M Ž H . AD . It is easy to prove that r VmA s id V m r˜ and Ž ¨ m a. ? b Ax
s ¨ m ab for every ¨ g V, a, b g A. Let M g M Ž H . AD s s Ž AaD* A m D .. H
We set
Mx m A ª M
FM :
Ax
m m a ¬ ma. FM is a morphism in A a D*-Mod. H
Moreover let
Hom AaD* Ž A, M . m A ª M
CM :
H
Ax
j m a ¬ j Ž a. . Note that this is exactly the map introduced in Theorem 2.2 for R s A a D*, T s A x , and P s A. H
For e¨ ery M g M Ž H . AD the diagram
3.17. PROPOSITION.
H
Ax
n Mmid A
FM
cM
66
Hom AaD* Ž A, M . m A
M
6
Mx m A Ax
is commutati¨ e. In particular CM is an isomorphism if and only if FM is an isomorphism. Proof. Let a g A and f g Hom AaD* Ž A, M .. We have H
Ž FM ( Ž n M m id A . . Ž f m a. s FM Ž f Ž 1 A . m a. s f Ž 1 A . a s f Ž a . s CM Ž f m a . , as f is a morphism of right A-modules by Section 3.8. 3.18. LEMMA.
For e¨ ery M g Mod-A the map LM :
M ª Ž M m D. x
m¬mmx is an isomorphism of right A x-modules and its in¨ erse is the map GM :
Ž M m D. x ª M n
n
Ý mi m di ¬ Ý mi «D Ž di . . is1
is1
HOPF ] GALOIS EXTENSIONS
259
Therefore QM sGM ( n MmD :
Hom AaD* Ž A, M m D . ª M H
n
f¬
Ý yi « D Ž d i . , is1
where f Ž1 A . s Ý nis1 yi m d i , is an isomorphism. 3.19 Let M g Mod-A and let à M s QM m id A . Then
à M : Hom AaD* Ž A, M m D . m A ª M m A H
Ax
Ax
n
Ý yi « D Ž d i . m a,
fma¬
is1
where f Ž1 A . s Ý nis1 yi m d i , is an isomorphism. Let b xM s Ž FMmD .(Ž L M m id A ., we have
b xM :
Mm A ª M m D Ax
m m a ¬ Ž m m x. a s
Ý ma0 m x £ a1 .
For e¨ ery M g Mod-A the diagram
3.20. PROPOSITION. H
Ax
ÃM
cMm D b xM
MmD
6 6
Hom AaD* Ž A, M m D . m A 6
Mm A Ax
is commutati¨ e. In particular CMmD is injecti¨ e Ž resp. surjecti¨ e . if and only if b xM is injecti¨ e Ž resp. surjecti¨ e .. In the following b xA will be simply denoted by b x so that
bx :
Am A ª A m D Ax
amb¬
Ý ab0 m x £ b1 .
3.21. LEMMA. The map
m:
A a D* ª A H
aag ¬ aag ? 1 A
260
MENINI AND ZUCCOLI
is a surjecti¨ e morphism of left A a D*-modules. In particular A is a cyclic left H
A a D*-module. H
Proof. aa« D ? 1 A s a « D Ž x . s a. Assume that
3.22. LEMMA.
CAmD :
A xA
is flat and that
Hom AaD* Ž A, A m D . m A ª A m D H
Ax
j m a ¬ j Ž a. is an isomorphism. Then for e¨ ery M g s Ž AaD* A m D ., CM is an injecti¨ e H
morphism. In particular CM is an isomorphism for e¨ ery M g GenŽ AaD* A. H
= GenŽ AaD* A m D .. H
Proof. Note that, as CAmD is an isomorphism, A m D is generated by A in A a D*-Mod so that H
Gen
ž
AaD* H
A m D : Gen
/
ž
AaD* H
/
A .
Let M g s Ž AaD* A m D . s M Ž H . AD . As M g Mod-A, there exists an exact H
sequence h
AŽY . ª AŽ X . ª M ª 0, where X and Y are suitable sets. By applying the functor ym D we get that the sequence AŽY . m D ª AŽ X . m D ª M m D ª 0
Ž ).
is exact in A a D*-Mod. Recalling now that, by Lemma 3.21,
AaD*
H
A is
H
cyclic, we get the following isomorphisms of right A x-modules: Hom AaD* Ž A, AŽ X . m D . ( Hom AaD* Ž A, Ž A m D . H
ŽX.
H
( Hom AaD* Ž A, A m D .
ž
H
/
.
ŽX.
.
Now it is easy to prove that, given a nonempty set X, CAŽ X .mD is an isomorphism, CAmD being an isomorphism. Let H s Hom AaD* Ž A, y.. H
HOPF ] GALOIS EXTENSIONS
261
From the exact sequence Ž). we get the following diagram: f
H Ž A, M m D .
6
H Ž A, AŽ X . m D .
6
H Ž A, AŽY . m D . QAŽY . X
QAŽ X . X
QM X
AŽY .
AŽ X .
M
6
6
6
6
6
6
0.
It is easy to prove that this diagram is commutative; moreover its vertical arrows are isomorphisms by Lemma 3.18. It follows that the sequence H Ž A, AŽY . m D . ª H Ž A, AŽ X . m D . ª H Ž A, M m D . ª 0 Ž )) . is exact. Let us prove that CMmD is an isomorphism. By applying the functor ym A to the exact sequence Ž)). we get the diagram
CAŽ Y .m D X
H Ž A, M m D . m A CMm D
6
t
6
AŽ X . m D
MmD
0,
6
l
0
Ax
CA Ž X .m D X
6
6
AŽY . m D
h
Ax
6
Ax
6
H Ž A, AŽ X . m D . m A
6
g
6
Ax
H Ž A, AŽY . m D . m A
whose rows are exact. It is straightforward to prove that this diagram is commutative. Since CAŽY .mD and CAŽ X .mD are isomorphisms we get that CMmD is an isomorphism. By Example 3.2 and Section 3.8, r M : M ª M m D is an injective morphism in M Ž H . AD s s Ž AaD* A m D .. Therefore, since H
A xA
is flat, we get the exact sequence 0 ª H Ž A, M . m A ª H Ž A, M m D . m A Ax
Ax
so that from the commutative diagram Ax
H Ž A, M m D . m A Ax
X CMm D
CM
6
6
M
6
H Ž A, M . m A
6
6
0
MmD
we get that CM is injective. 3.23. DEFINITION. Let A a right H-comodule algebra, D a right Hmodule coalgebra, x g D a grouplike element. We say that A x ; A is a right Hopf]Galois extension if the map
bx :
Am A ª A m D Ax
amb¬ is bijective.
Ý ab0 m x £ b1
262
MENINI AND ZUCCOLI
3.24. DEFINITION. Let A a right H-comodule algebra, D a left Hmodule coalgebra, x g D a grouplike element. We say that A x ; A is a left Hopf]Galois extension if the map
b xX :
Am A ª A m D Ax
amb¬
Ý a0 b m a1 © x
is bijective. 3.25. Remark. Taking D s H, we have that H can be considered both as a right and as a left H-module coalgebra with respect to the natural H-module structure of H. Moreover x s 1 H is a grouplike element of H. In this case we simply set b s b 1 H and b 9 s b 1X H . Whenever the antipode S of H is bijective with composition inverse S, the map
u:
AmHªAmH amh¬
Ý a0 m a1 S Ž h .
is bijective and its inverse is the map
u 9:
AmHªAmH amh¬
Ý a0 m S Ž h . a1 .
Moreover it is straightforward to prove that b 9 s u ( b Žsee w10, Proposition 1.2x.. 3.26. Remark. If A s D s H, then H is a right Žleft. Hopf]Galois extension. In fact it is easy to prove that H coH s k ? 1 H and that the map
a:
HmHªHmH hmg¬
Ý hS Ž g 1 . m g 2
is the two-sided inverse of the map
b:
HmHªHmH hmg¬
Ý hg 1 m g 2 .
3.27. THEOREM. Let k be a field, H a Hopf algebra, A a right H-comodule algebra, D a right H-module coalgebra, and x g D a grouplike element.
HOPF ] GALOIS EXTENSIONS
263
Then the following are equi¨ alent. A x ; A is a right Hopf]Galois extension and For e¨ ery M g M Ž H . AD s s Ž AaD* A m D .,
Ža. Žb.
A xA
is flat.
H
Mx m A ª M
FM :
Ax
m m a ¬ ma is an isomorphism of left A a D*-modules, i.e., the weak structure theorem in H the sense of w7x holds. Žc. The map CAmD :
Hom AaD* Ž A, A m D . m A ª A m D H
Ax
j m a ¬ j Ž a. is an isomorphism of left A a D*-modules and H
A xA
is flat.
For e¨ ery M g GenŽ AaD* A m D .,
Žd.
H
CM :
Hom AaD* Ž A, M . m A ª M H
Ax
j m a ¬ j Ž a. is an isomorphism of left A a D*-modules and H
A xA
is flat.
For e¨ ery M g s Ž AaD* A m D .,
Že.
H
CM :
Hom AaD* Ž A, M . m A ª M H
Ax
j m a ¬ j Ž a. is an isomorphism of left A a D*-modules. H
Žf.
s Ž AaD* A m D . s GenŽ AaD* A..
Žg.
Ž . Ž . A xA is flat, s AaD* A s s AaD* A m D , and
H
H
H
H
Hom AaD* Ž A, y . :
Gen
H
ž
AaD* H
A ª Mod-A x
/
M ¬ Hom AaD* Ž A, M . H
is full and faithful. Žh. s Ž AaD* A. s s Ž AaD* A m D . and H
H
ules of A , for e¨ ery n g N, n ) 0. n
AaD* H
A generates all submod-
264
MENINI AND ZUCCOLI
Proof. Ža. m Žc. follows from Proposition 3.20. Žb. m Že. follows from Proposition 3.17. Žc. « Žd. follows from Lemma 3.22. Žd. « Žc. is trivial. Žd. m Že. m Žf. m Žg. m Žh. follow from Theorem 2.3 applied to R s A a D*, T s A x , P s A, and Q s A m D, in view of Theorem 3.15, ReH
mark 3.12, and Section 3.13. 3.28. Remark. Clearly analogous results hold in the category A M Ž H . D , where A a9 D* and b 9 play the role of b and A a D*, respectively. H
H
3.29. THEOREM. Let k be a field, H a Hopf algebra, A a right H-comodule algebra, D a right H-module coalgebra, and x g D a grouplike element. Then the following are equi¨ alent. Ža.
A x ; A is a right Hopf]Galois extension and
Žb.
The map
Ž y. x :
D
M Ž H . A ss
ž
AaD*
A xA
is faithfully flat.
A m D ª Mod-A x
/
H
M ¬ Mx is an equi¨ alence, i.e., the strong structure theorem in the sense of w7x holds. Žc.
The map CAmD :
Hom AaD* Ž A, A m D . m A ª A m D H
Ax
j m a ¬ j Ž a. is an isomorphism and Žd.
A xA
is faithfully flat.
The map CM :
Hom AaD* Ž A, M . m A ª M H
Ax
j m a ¬ j Ž a.
HOPF ] GALOIS EXTENSIONS
265
is an isomorphism for e¨ ery M g Ž AaD* A m D . and H
N ª Hom AaD* A, N m A
CNX :
H
n¬
ž
Ax
A ª Nm A Ax
a¬nma
/
0
is an isomorphism for e¨ ery N g Mod-A x . Že. The map Hom AaD* Ž A, y . : H
s
ž
AaD*
A m D ª Mod-A x
/
H
M ¬ Hom AaD* Ž A, M . H
is an equi¨ alence. Žf. The map y m A: Ax
Mod-A x ª s
ž
AaD*
AmD
H
/
N ¬ Nm A Ax
is an equi¨ alence. Žg. AaD* A is quasiprojecti¨ e and generates each of its submodules,
A xA
H
is a weak generator, and s Ž AaD* A m D . s s Ž AaD* A.. H
H
A is a quasiprogenerator and s Ž AaD* A m D . s s Ž AaD* A.. AaD*
Žh.
H
H
H
A xA is a weak generator, CM is an isomorphism for e¨ ery M g GenŽ AaD* A., and s Ž AaD* A m D . s s Ž AaD* A..
Ži.
H
H
H
Proof. Recall now, by Theorem 3.15, that End l AaD* Ž A. ( A x . H
Ža. m Žc. follows from Proposition 3.20. Žb. m Že. follows from Theorem 3.15. Žc. « Žd. By Žc. « Že. in Theorem 3.27, CM is an isomorphism for every M g s Ž AaD* A m D .. Moreover, by Žc. « Žh. in the same theorem, H
s Ž AaD* A m D . s s Ž AaD* A.. So we may apply Že. « Žd. of Theorem 2.5. H
H
H
H
Žd. « Žc. and Žd. « Že.. By Že. « Žg. in Theorem 3.27 we get that s Ž AaD* A m D . s s Ž AaD* A.. Now Žd. « Žc. follows from Žd. « Že. in
266
MENINI AND ZUCCOLI
Theorem 2.5 and Žd. « Že. follows from Žd. « Žb. in the same theorem. Že. « Žf. Note that for every N g Mod-A x we have N m A g Ax
GenŽ AaD* A. : s Ž AaD* A mD .. Therefore ImŽym A. : s Ž AaD* A mD .. H
Since y
Ax
H
m A is a left adjoint of Hom AaD* Ž A, y. we get Ž f .. Ax
H
H
Žf. « Žg. We have
s
ž
AaD* H
A m D s Im y m A : Gen
ž
/
:s
ž
/
Ax
AaD*
ž
AaD* H
A :s
/
ž
AaD* H
A
/
AmD .
/
H
Now apply Žc. « Žg. of Theorem 2.5. Žg. « Žc. Since A m D g s Ž AaD* A m D . s s Ž AaD* A., by Žg. « Že. H
H
in Theorem 2.5, we get that CAmD is an isomorphism and A xA is faithfully flat. Žg. m Žh. m Ži. follow from Žg. m Žh. m Ži. in Theorem 2.5. 3.30. Remark. Clearly analogous results hold in the category A M Ž H . D , where A a9 D* and b 9 play the role of b and A a D*, respectively. H
H
3.31. Remark. The equivalence Ža. m Žf. for D s H is part of the famous Schneider’s theorem w14, Theorem 1x. 3.32. COROLLARY w6, Theorem 2.3x. Let H be a Hopf algebra, A a right H-comodule algebra, D a right H-module coalgebra, and x g D a grouplike element. Assume that H is a faithfully coflat left D-comodule ¨ ia px , i.e., h ¬ ÝŽ x £ h1 . m h 2 . Then A x ; A is a right Hopf]Galois extension and A xA is flat Ž resp. faithfully flat . if B ; A is a right Hopf]Galois extension and B A is flat Ž resp. faithfully flat .. In this case Theorem 3.27 Ž resp. Theorem 3.29. applies. Proof. Consider
px :
HªD h ¬ x £ h.
px is a coalgebra morphism. Therefore by Theorems 1.1 and 1.3 in w3x the induction functor F: M Ž H . AH ª M Ž H . AD defined by F Ž M . s M endowed with the right D-comodule structure r F Ž M .Ž m. s Ým 0 m px Ž m1 . has, as a
HOPF ] GALOIS EXTENSIONS
267
right adjoint, the functor G: M Ž H . AD ª M Ž H . AH defined by GŽ M9. s M9 I H, with structure map rGŽ M 9.ŽÝmXi m h i . s ÝmXi m h i, 1 m h i, 2 and D ŽÝmXi m h i . ? a s ÝmXi a0 m h i a1. Since H is a faithfully coflat left D-comodule, G is a faithful and exact functor. Assume now that B ; A is a right Hopf]Galois extension and B A is flat. Then, by Theorem 3.27, A is a generator of M Ž H . AH . Let g: M1 ª M2 be a nonzero morphism in M Ž H . AD . Then GŽ g .: GŽ M1 . ª GŽ M2 . is a nonzero morphism in M Ž H . AH and hence HomŽ A, GŽ g ..: HomŽ A, GŽ M1 .. ª HomŽ A, GŽ M2 .. is a nonzero morphism, A being a generator of M Ž H . AH . Thus we get that Ap x s F Ž A. is a generator in M Ž H . AD so that GenŽ AaD* A. s s Ž AaD* A m D . and A x ; A is right H-Galois and A xA is H
H
flat. Assume now that B A is faithfully flat. Then, by theorem 3.29, Hom M Ž H . AH Ž A, y. s Žy. coH : M Ž H . AH ª Mod-B is an equivalence. It follows that the functor Hom M Ž H . AD Ž F Ž A., y. is exact, so that Ap x s F Ž A. is a projective object of s Ž AaD* A m D . s GenŽ AaD* A.. In particular AaD* A H
H
H
is quasiprojective. Since A is a cyclic left A a D*-module we conclude that AaD*
H
A is a quasiprogenerator.
H
3.33. Remark. We insert the following well-known facts for the generic reader’s sake. By applying Theorem 3.29 for D s A s H Žsee Remark 3.26. we get ‘‘the fundamental theorem of Hopf-modules.’’ The functor
Ž y.
coH
MHH ª k-Mod
:
M ¬ M coH is an equivalence. Its inverse is the functor ym H :
k-Mod ª MHH V ¬ V m H.
I
We recall now that H s H * g with respect to the right H-module structure defined by setting ²g h, x : s ²g , xSŽ h.: for every g g H * rat , h, x g H Žsee w15, Theorem 5.1.2x. and Ž H I . coH s x g H * < g ? x s ²g , 1 H :x for every g g H *4 s HHl * s the space of left integrals in H * Žor left integrals on H *.. Therefore we get that FH I :
MHH
rat
l
HH * m H ª H gmh¬g
is an isomorphism.
I
h
268
MENINI AND ZUCCOLI
If dim k Ž H . - `, H I s H * so that we get that FH * :
l
HH * m H ª H * gmh¬g
h
is an isomorphism. Since dim k Ž H . s dim k Ž H *. we get, in particular, dim k Ž HHl * . s 1. Assume now that h g kerŽ S . and let 0 / T g HHl * . Then for every x g H we have ²T h, x : s ²T, xSŽ h.: s 0. Hence FH * ŽT m h. s T h s 0, by injectivity of FH * , we get T m h s 0 and hence h s 0. Therefore S: H ª H is injective and hence also bijective. 3.34 Let now T / 0 be a fixed element of map
w:
l
HH *. Then, by the foregoing, the
H ª H* h¬T
S Ž h.
is an isomorphism. Note that, for every x g H, we have ²T
S Ž h . , x : s ²T , xS Ž S Ž h . . : s ²T , xh: s ² h © T , x:.
Therefore, for every h g H, T g H *, we have
w Ž h. s h © T . 4. THE FINITE CASE 4.1. LEMMA.
Assume that dim k Ž D . - `. Then
s
ž
AaD* H
A m D s A a D*-Mod.
/
H
Proof. See w6, page 375x. Using the above result, from Theorems 3.27 and 3.29 we get the following results. 4.2. THEOREM. Let A be a right H-comodule algebra, D a finite-dimensional right H-module coalgebra, and x g D a grouplike element. Then the
HOPF ] GALOIS EXTENSIONS
269
following are equi¨ alent. A x ; A is a right Hopf]Galois extension and A is a generator in A a D*-Mod. AaD*
Ža. Žf.
is flat.
H
H
Ži.
A xA
AaD*
A A opx is faithfully balanced and A A opx is finitely generated and
H
projecti¨ e. Moreo¨ er each of these conditions is equi¨ alent to each of the conditions Žb., Žc., Žd., Že., Žg., and Žh. of Theorem 3.27 with s Ž AaD* A m D . s A a D*H
H
Mod.
Proof. Ža. m Žb. m Žc. m Žd. m Že. m Žf. m Žg. m Žh. follow from Theorem 3.27 and Lemma 4.1. Žf. m Ži. follows from Theorem 17.8 in w1x. 4.3. THEOREM. Let A be a right H-comodule algebra, D a finite-dimensional right H-module coalgebra, and x g D a grouplike element. Then the following are equi¨ alent. Ža. Žf.
A x ; A is a right Hopf]Galois extension and A xA is faithfully flat. A is a faithful quasiprogenerator, A xA is finitely generated. AaD* H
Žg.
AaD*
A is a progenerator.
H
Žh.
A xA
is a progenerator and
AaD*
A A opx is faithfully balanced.
H
Moreo¨ er each of these conditions is equi¨ alent to each of the conditions Žb., Žc., Žd., and Že. of Theorem 3.29 with s Ž AaD* A m D . s A a D*-Mod. H
H
Proof. Apply Lemma 4.1, Theorem 3.29, and Theorem 2.6. 4.4. DEFINITION Žsee w9x.. Let B be a subring of a ring A with identity 1 A Žsuch that 1 A g B .. A is a Frobenius extension of B if A B is finitely generated and projective and moreover
B
A A (B Hom B Ž A B , BB . A .
4.5. Remark Ž see w9, Bemerkung 1x.. A is a Frobenius extension of B if and only if B A is finitely generated and projective and A
A B (A Hom B Ž A B , BB . B .
270
MENINI AND ZUCCOLI
4.6. Remark. Let H be a finite-dimensional Hopf algebra and A a right H-comodule algebra. Regarding H as a left H-module coalgebra, we have that A gA M Ž H . H s A a9 H *-Mod. Now, by Remark 3.5, A a9 H * s H
H
Aop aop H * and, by Proposition 3.6, H
L:
ž A a H */
op
H
ª Aop aop H * H
aag ¬
Ý a 0aŽ S Ž a1 . © g .( S
is a ring isomorphism. Therefore A becomes a right A a H *-module with H
respect to a ? Ž a ax . s L Ž a ax . ? a s Ž x ( S . ? Ž a a. ,
a g A, a ax g A a H *. This is the right A a H *-module structure of A H
considered in the following theorem.
H
4.7. THEOREM Žw4, Theorems 1.2 and 1.29x, w10, 1.7x, w16, 1.1x.. Let H be a finite-dimensional Hopf algebra and A a right H-comodule algebra, B s AcoH . Then the following are equi¨ alent. Ža. Ža9. Žb.
B ; A is a right Hopf]Galois extension. B ; A is a left Hopf]Galois extension. The map
b:
Am A ª A m H B
amb¬
Ý ab0 m b1
is surjecti¨ e. Žb9. The map
b 9:
Am A ª A m H B
amb¬
Ý a0 b m a1
is surjecti¨ e. Žc. The map CAmH :
Hom AaH * Ž A, A m H . m A ª A m H H
B
j m a ¬ j Ž a.
HOPF ] GALOIS EXTENSIONS
271
is surjecti¨ e. Žd.
AaH *
A is a generator for the category A a H *-Mod. H
H
Žd9.
A AaH * is a generator for the category Mod-A a H *. H
H
Že.
A op is faithfully balanced and AaH * B
B A is finitely generated and
H
projecti¨ e. Že9.
B op
A AaH * is faithfully balanced and A B is finitely generated and H
projecti¨ e. Žf.
For e¨ ery M g A a H *-Mod, H
CM :
Hom AaH * Ž A, M . m A ª M H
B
j m a ¬ j Ž a. is an isomorphism. Žg.
For e¨ ery M g A a H *-Mod s M Ž H . AH H
FM :
M coH m A ª M B
m m a ¬ ma is an isomorphism. Moreo¨ er if these conditions are satisfied, A is a Frobenius extension of B. Proof. Ža. m Ža9. and Žb. m Žb9. follow by Remark 3.25. Žb. m Žc. follows by Proposition 3.20. Žd. « Ža. follows by Theorem 4.2Žf. « Ža.. Žd. m Že. m Žf. m Žg. follow by Theorem 4.2. Ža. « Žb. is trivial. Žb9. « Žd. Let us fix 0 / T g HHl * . Let ¨ s Žid A m w .( b 9, where w : H ª H * is the isomorphism defined in Section 3.34. Then ¨ : A m A ª B
A a H * is surjective. We prove that ¨ is a morphism of left A a H *-modH
H
ules, where A m A is endowed with its A a H *-module structure defined H
B
as in Section 3.16. Since A m A g GenŽ AaH * A. this will prove Žd.. B
H
272
MENINI AND ZUCCOLI
First of all note that for every a, b g A we have ¨ Ž a m b . s Ž baT .Ž aa« .. Let a, b g A, a ax g A a H *. Then H
¨ Ž Ž a ax . ? Ž a m b . . s ¨ Ž a m Ž a ax . ? b .
s¨Ža m
Ý b 0 a ² x , b1 : .
s Ž Ý b 0 a a² x , b1 :T . ? Ž aa« . s Ž Ý b 0 a a Ž b1 © x . T . ? Ž aa« . s Ž a ax . ? Ž baT . ? Ž aa« . s Ž a ax . ? ¨ Ž a m b . . The equivalences Ža9. m Žb9. m Žd9. m Že9. can be proved in an analogous way in view of Remark 4.6. Assume now that the foregoing equivalent conditions hold and let us prove that A is a Frobenius extension of B. From Že9. we get that A B is finitely generated and projective. Let us prove that B A A ( Ž . Ž . o p and let i: A ª R: a9 ¬ a9a« . B Hom B A B , BB A . Let R s A a H * H
Then, by Lemma 3.7, i is an injective ring morphism. Note that A is a left Žright. R-module, so that A can be considered as a left Žright. A-module via i. Now it is easy to prove that this left Žright. A-module structure coincides with the usual left Žright. A-module structure on A. Since B A R is faithfully balanced B BB ( Hom R ŽB A R , B A R ., so that
Ž A B , B BB . R (B Hom B Ž R A B , Hom R Ž B A R , B A R . .
B Hom B R
(B Hom R
ž
R
AB m B AR , B AR B
/
R
R
and, by the foregoing, we get that
Ž A B , B BB . A (B Hom R
B Hom B A
Let us fix a 0 / T g HHl * and let ¨ :
A
ž
A
AB m B AR , B AR B
/
. A
A B m B A R ªA R R be the map B
defined in the proof of Žb9. « Žd.. ¨ s Žid A m w .( b 9 so that, by Ža9., ¨ is bijective. Moreover ¨ is a morphism of right R-modules and for every a, b g A we know that ¨ Ž a m b . s Ž baT .Ž aa« .. Let us prove that ¨ is a morphism of left A-modules. Let a9 g A. Then ¨ Ž a9Ž a m b .. s ¨ Ž a9a m b . s Ž baT .Ž a9aa« . s Ž baT .Ž aa« .Ž a9a« . s ¨ Ž a m b .Ž a9a« . s a9¨ Ž a m b ..
HOPF ] GALOIS EXTENSIONS
273
Therefore we get
Ž A B , B BB . A (B Hom R Ž A R R , B A R . A (B A A ,
B Hom B A
where the last map is a morphism of right A-modules by the remarks made above on the structure of A A . 4.8. THEOREM. Let H a finite-dimensional Hopf algebra, A a right Hcomodule algebra, and B s AcoH . Then the following are equi¨ alent. Ža.
B ; A is a right Hopf]Galois extension and
B
A is a weak genera-
tor. Ža9. Žb.
B ; A is a left Hopf]Galois extension and A B is a weak generator. The functor map Hom AaH * Ž A, y . : H
A a H *-Mod ª Mod-B H
M ¬ Hom AaH * Ž A, M . H
is an equi¨ alence. Žc. AaH * A is a faithful quasiprogenerator and
B
A is finitely generated.
H
Žc9.
A AaH * is a faithful quasiprogenerator and A B is finitely generated. H
Žd.
AaH *
A is a progenerator.
H
Žd9.
A AaH * is a progenerator. H
Že.
B A is a progenerator and
AaH *
A B op is faithfully balanced.
H
Že9.
A B is a progenerator and
B op
A AaH * is faithfully balanced. H
Proof. Ža. « Ža9. By Theorem 4.7, B ; A is a left Hopf]Galois extension. Let B L g B-Mod, L / 0. Since B A is a progenerator and A A B ( Ž . A Hom B B A A , B BB B , by Theorem 4.7 we have 0 / Hom B Ž B A, B L . ( Hom B
ž
B
A, BB m B L B
/
( Hom B Ž B A, B BB . m B L ( A m B L. B
B
Ža9. « Ža. By Theorem 4.7, A B is a progenerator and B A A ( Ž . Ž . Ž . Hom B B A A B , B BB A . A proof similar to a « a9 applies. Ža. « Žb. By Theorem 4.7, B A is projective so that B A is faithfully flat
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since B A is a weak generator. Žb. follows by Ža. « Že. in Theorem 4.3. Žb. « Ža. By Že. « Ža. in Theorem 4.3. Žc. m Žd. m Že. follow by Theorem 4.3. Note added in proof. After the submission of this paper, the authors received the preprint ‘‘Galois Extensions for Co-Frobenius Hopf Algebras’’ by M. Beattie, S. Dascalescu, and ¸ S. Raianu where, using Theorem 3.27 above, it is proved that for every co-Frobenius Hopf algebra H, any H-Galois H-comodule algebra A is a flat left AcoH -module.
REFERENCES 1. F. W. Anderson and K. R. Fuller, ‘‘Rings and Categories of Modules,’’ Springer-Verlag, BerlinrNew York, 1992. 2. G. Azumaya, Some aspects of Fuller’s theorem, in Lecture Notes in Mathematics, Vol. 700, pp. 33]45, Springer-Verlag, BerlinrNew York, 1979. 3. S. Caeneppeel and S. Raianu, Induction functors for the Doi]Koppinen unified Hopf modules, in ‘‘Abelian Groups and Modules’’ ŽA. Facchini and C. Menini, Eds.., Kluwer Academic, DordrechtrBoston, 1995. 4. M. Cohen, D. Fischman, and S. Montgomery, Hopf]Galois extensions, smash products, and Morita equivalences, J. Algebra 133 Ž1990., 351]372. 5. S. Dal Pio and A. Orsatti, Representable equivalences for closed categories of modules, Rend. Sem. Mat. Uni¨ . Pado¨ a 92 Ž1994., 239]260. 6. Y. Doi, Unifying Hopf modules, J. Algebra 153 Ž1992., 373]385. 7. Y. Doi and M. Tacheuchi, Hopf]Galois extensions of algebras, the Miyashita]Ulbrich action, and Azumaya algebras, J. Algebra 121 Ž1989., 488]516. 8. K. R. Fuller, Density and equivalence, J. Algebra 29 Ž1974., 528]550. 9. F. Kasch, Projektive Frobenius-Erweiterungen, Sitaungsber. Heidelberger Akad. Wiss. Math.-Natur. K1 Ž1960r1961., 87]109. 10. H. F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Uni¨ . Math. J. 30Ž5. Ž1981., 675]692. 11. S. Montgomery, ‘‘Hopf Algebras and Their Actions on Rings,’’ CBMS, Vol. 82, Am. Math. Soc., Providence, 1993. 12. K. Morita, Duality of modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6 Ž1958., 85]142. 13. C. Nastasescu and B. Torrecillas, Graded coalgebras, Tsukuba J. Math. 17 Ž1993., ˘ ˘ 461]479. 14. H. J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72Ž1]2. Ž1990., 167]195. 15. M. E. Sweedler, ‘‘Hopf Algebras,’’ Benjamin, New York, 1969. 16. K. H. Ulbrich, Galois Erweiterungen von nicht-kommutativen Ringen, Comm. Algebra 10 Ž1982., 655]672. 17. R. Wisbauer, Foundations of Module and Ring Theory, Gordon & Breach, PhiladelphiarReading, 1991. 18. B. Zimmermann-Huisgen, Endomorphism rings of self-generators, Pacific J. Math. 61 Ž1975., 587]602.
JOURNAL OF ALGEBRA ARTICLE NO.
Equivalence Theorems and Hopf]Galois Extensions Claudia Menini* and Monica Zuccoli Dipartimento di Matematica, Uni¨ ersita ` di Ferrara, Via Machia¨ elli 35, 44100 Ferrara, Italy Communicated by Susan Montgomery Received August 2, 1996
1. INTRODUCTION In this article we apply equivalence theorems for categories of modules to the category M Ž H . AD of right Ž D y A.-Hopf modules to get a characterization of Hopf]Galois extensions. Indeed let P, Q g Mod-R and T s EndŽ PR ., and assume that P belongs to the category s Ž Q R . of modules subgenerated by Q R . In Theorem 2.3, by modifying a theorem formulated by Dal Pio and Orsatti w5, Theorem 2.6x, we study the situation where, for every M belonging to the category s Ž Q R ., the map CM :
Hom R Ž P , M . m P ª M T
j m x ¬ j Ž x. is an isomorphism in Mod-R. In particular we prove that this holds iff CM is an isomorphism for every M g GenŽ Q R . and T P is flat iff s Ž Q R . s s Ž PR . and PR generates all submodules of PRn for every n g N, n ) 0. Let now H be a Hopf algebra over a field k, let A be a right H-comodule algebra, let D be a right H-module coalgebra, and consider the category M Ž H . AD of right Ž D y A.-Hopf modules, i.e., of those D-comodules which are equipped with a suitable right A-module structure in such a way that *This article was written while the first author was a member of G.N.S.A.G.A. of C.N.R. with partial financial support from M.U.R.S.T. and H.C.M. project n.CHRX-CT93-0091. E-mail: [email protected]. 245 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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MENINI AND ZUCCOLI
these two structures are compatible Žsee Section 3.1.. The starting point for applying the previous result in this setting is that Žsee Lemma 3.9. M Ž H . AD s s Ž AaD* A m D . as remarked by Caeneppeel and Raianu in w3x. H
Moreover, using a grouplike element x g D, A can be endowed with a suitable D-comodule structure in such a way that A g M Ž H . AD Žsee Section 3.13.. Since End l AaD* Ž A. ( A x Žsee Theorem 3.15., we may apply H
the quoted result to the situation where R s A a D*, which is essentially H the smash product introduced by Doi in w6x, PR s AaD* A, Q R s AaD* A m D, H
H
and T s A x . Now if A xA is flat, CM is an isomorphism for every M g GenŽ AaD* A m D . iff CAmD is an isomorphism Žsee Lemma 3.22. and H
CAmD is an isomorphism iff
bx :
Am A ª A m D Ax
amb¬
Ý ab0 m x £ b1
is an isomorphism, i.e., iff A x ; A is a right Hopf]Galois extension. In this way we get Theorem 3.27, which characterizes right Hopf]Galois extensions A x ; A such that A xA is flat, using the structure of A as a left A a D*-module. In particular it is shown that these extensions are exactly H those for which the weak structure theorem in the sense of w7x holds. Also Theorem 3.27 enables us to apply the well-known Fuller theorem on equivalences of modules Žsee Theorem 2.5. in our setting. Thus we get Theorem 3.29, which characterizes right Hopf]Galois extensions A x ; A such that A xA is faithfully flat. In particular we prove that this holds iff A is a quasiprogenerator and s Ž AaD* A. s s Ž AaD* A m D . iff AaD* H
H
H
Žy. x : M Ž H . AD s s Ž AaD* A m D . ª Mod-A x is an equivalence, i.e., the H
strong structure theorem in the sense of w7x holds. In this case when D s H we thus get part of the famous Schneider’s theorem on Hopf]Galois extensions. Moreover, using the adjointness between the induction functor F and the functor G s y I H as described by Caeneppeel and Raianu D w3x, we easily get as a corollary ŽCorollary 3.32. Doi’s theorem on Hopf]Galois extensions w6, Theorem 2.3x. When dim k Ž D . - `, then s Ž AaD* A m D . s A a D*-Mod as remarked H
H
by Doi w6x. Therefore in this case A is a generator of s Ž AaD* A m D . if H
and only if A is a generator in A a D*-Mod. Hence this is equivalent, in H
HOPF ] GALOIS EXTENSIONS
247
the finite case, to A x ; A being a right Hopf]Galois extension and A xA being flat. Theorem 4.2 characterizes this situation. Also AaD* A is a H
quasiprogenerator and s Ž AaD* A. s s Ž AaD* A m D . in the finite case if H
and only if
AaD*
H
A is a progenerator. Theorem 4.3 characterizes this
H
situation. When D s H and dim k Ž H . - `, B s AcoH ; A is a right Hopf]Galois extension if and only if AaH * A is a generator for the category A a H *-Mod. H
H
Theorem 4.7 characterizes this situation. Here we prove that, in this case, A is a Frobenius extension of B. Theorem 4.7 contains essentially the famous result formulated by Cohen, Fischman, and Montgomery for Hopf]Galois extensions w4, Theorem 1.2 and Theorem 1.29x. Finally, using Theorem 4.7, we can prove that the assumption of faithfully flatness can be weakened in the finite case whenever D s H. In fact if B ; A is a right Hopf]Galois extension, then B A is faithfully flat if and only if B A is a weak generator. Theorem 4.8 characterizes such extensions. Part of this result appears in w4, Theorem 2.2x. We have made some efforts to get a paper that even someone not acquainted with Hopf algebra theory could read. Toward this end we have inserted some details that a Hopf-algebra expert would skip. The main results of this article were presented at the AMS-BENELUX meeting, held in May 1996 at Antwerp, Belgium.
2. PRELIMINARIES 2.1 Throughout this article, all rings have a nonzero identity and all modules are unital. Let R be a ring, PR a right R-module. We denote by GenŽ PR . the full subcategory of Mod-R generated by PR . The objects of GenŽ PR . are the right R-modules MR which are generated by PR , i.e., such that there exists a surjective morphism PRŽ X . ª M ª 0 for a suitable set X. GenŽ PR . is closed under taking epimorphic images and direct sums. Partially following w17x, we denote by s Ž PR . the full subcategory of Mod-R whose objects are the right R-modules subgenerated by PR , i.e., such that there exists an injective morphism M ¨ L for a suitable L g GenŽ PR .. s Ž PR . is closed under taking submodules, epimorphic images and direct sums. Clearly GenŽ PR . s s Ž PR . if and only if GenŽ PR . is closed under submodules.
248
MENINI AND ZUCCOLI
2.2. THEOREM. equi¨ alent.
Let PR g Mod-R, T s EndŽ PR .. Then the following are
Ža. For e¨ ery n g N, n ) 0, PR generates all submodules of PRn. Žb. GenŽ PR . s s Ž PR .. Žc. For e¨ ery M g s Ž PR . the map CM :
Hom R Ž P , M . m P ª M T
j m x ¬ j Ž x. is an isomorphism in Mod-R. Žd. For e¨ ery M g s Ž PR ., CM is a surjecti¨ e morphism in Mod-R. Že. T P is flat and the functor Hom R Ž T PR , y . : Gen Ž PR . ª Mod-T is full and faithful. Moreo¨ er if these conditions are fulfilled H s Hom R ŽT PR , y. induces an equi¨ alence between GenŽ PR . s s Ž PR . and ImŽ H .. Proof. Ža. « Žb. follows from Ž1. « Ž2. of Lemma 1.4 in w18x. Žb. « Žc. follows from Ž1. of Lemma 1.3 in w18x. Žc. « Žb., Žc. « Žd., and Žd. « Žb. are trivial. Žb. « Že. follows from Ž2. « Ž3. of Lemma 1.4 in w18x. Že. « Ža. follows from Ž3. « Ž1. of Lemma 1.4 in w18x. Note that the equivalences Ža. m Žb. m Že. and the last assertion appear in w5, Theorem 2.6x. 2.3. THEOREM. Let P, Q g Mod-R, let T s EndŽ PR ., and assume that P g s Ž Q R .. Then the following are equi¨ alent. Ža.
For e¨ ery M g GenŽ Q R . CM :
Hom R Ž P , M . m P ª M T
j m x ¬ j Ž x. is an isomorphism in Mod-R and T P is flat. Žb. For e¨ ery M g s Ž Q R ., CM is an isomorphism in Mod-R.
HOPF ] GALOIS EXTENSIONS
249
Žc. s Ž Q R . s GenŽ PR .. Žd. T P is flat, s Ž Q R . s s Ž PR . and the functor Hom R Ž P , y . : Gen Ž PR . ª Mod-T is full and faithful. Že. s Ž Q R . s s Ž PR . and PR generates all submodules of PRn, for e¨ ery n g N, n ) 0. Proof. Ža. « Žb. Let M g s Ž Q R .. Then there exist a right R-module N g GenŽ Q R . and an injective morphism i: M ª N. By applying the functor H s Hom R Ž P, y. to the exact sequence NrM
6
0,
6
6
p
N
6
i
M
0
where p denotes the canonical projection, we get the following exact sequence: 6
6
HŽ N .
H Žp .
H Ž NrM . .
6
H Ži.
HŽ M .
0
By the flatness of T P, we obtain the following exact sequence: T
HŽ N . m P
h
T
H Ž NrM . m P ,
6
f
6
HŽ M . m P
6
0
T
where f s H Ž i . m id P and h s H Žp . m id P . Consider the following comT
T
HŽ N . m P T
T
CNr M X
p
NrM
0,
6
N
6
i
6
6
6
M
H Ž NrM . m P
CN X
6
CM
0
h
6
f
6
HŽ M . m P
6
0
6
T
mutative diagram:
where CN and CN r M are isomorphisms as both N and NrM g GenŽ Q R .. It follows that CM is an isomorphism too. Žb. « Žc. Since P g s Ž Q R ., GenŽ PR . : s Ž Q R .. As CM is an isomorphism for every M g s Ž Q R ., s Ž Q R . : GenŽ PR .. Žc. « Žd. Since P g s Ž Q R ., we get GenŽ PR . : s Ž PR . : s Ž Q R . s GenŽ PR . : s Ž PR .. Hence GenŽ PR . s s Ž PR . and we can apply Žb. « Že. of Theorem 2.2. Žd. m Že. follows from Že. m Ža. in Theorem 2.2. Žd. « Ža. By Že. « Žc. in Theorem 2.2, CM is an isomorphism for every M g s Ž PR . s s Ž Q R . = GenŽ Q R ..
250
MENINI AND ZUCCOLI
2.4. DEFINITION. Let R be a ring, M a left R-module. R M is called a weak generator if Y m M s 0 for every right R-module Y implies Y s 0. R
Recall that a module PR is called a quasiprogenerator if PR is finitely generated, is quasiprojective, and generates each of its submodules. 2.5. THEOREM w2, 8x. equi¨ alent.
Let
T
PR be a bimodule. Then the following are
Ža. Hom R Ž P, y.: s Ž PR . ª Mod-T and y
m P: T
Mod-T ª s Ž PR .
are in¨ erse category equi¨ alences. Žb. Hom R Ž P, y.: s Ž PR . ª Mod-T is a category equi¨ alence. Žc. y m P: Mod-T ª s Ž PR . is a category equi¨ alence. T
Žd.
The map CM :
Hom R Ž P , M . m P ª M T
j m x ¬ j Ž x. is an isomorphism for e¨ ery M g s Ž PR . and CNX :
N ª Hom R P , N m P
ž
n¬
ž
CNX Ž n . :
T
/
P ª NmP T
x¬nmx
/
is an isomorphism for e¨ ery N g Mod-T. Že. CM is an isomorphism for e¨ ery M g s Ž PR . and T P is faithfully flat. Žf. CM and CTX r I are isomorphisms for e¨ ery M F PR and e¨ ery I F TT . Žg. PR is quasiprojecti¨ e and generates each of its submodules, T P is a weak generator, and T ( EndŽ PR . canonically. Žh. PR is a quasiprogenerator and T ( EndŽ PR . canonically. Ži. T P is a weak generator, CM is an isomorphism for e¨ ery M g Ž Gen PR ., and T ( EndŽ PR . canonically. Proof. Ža. m Žd. m Žf. m Žh. are in Theorem 2.6 in w8x. Ža. m Žb. m Žc. follow by adjoint properties of Hom R Ž P, y. and y m P. T
HOPF ] GALOIS EXTENSIONS
251
Žh. « Ži. Since we already know that Žh. « Žd., we get that GenŽ PR . s s Ž PR . so that Ž4. « Ž3. of Theorem 10 in w2x applies. Ži. « Žh. By Ž3. « Ž4. of Theorem 10 in w2x. Žf. « Žg. By Žc. « Žd. of Theorem 2.6 in w8x. Žg. « Že. and Žg. « Žf. By Lemma 2.2 in w8x we get s Ž PR . s GenŽ PR . so that, by Lemma 2.1 in w8x, T P is flat and hence faithfully flat, being a weak generator. Now apply Žd. « Žc. and Žd. « Žb. of Theorem 2.6 in w8x. Že. « Ži. Since CP is an isomorphism and T P is faithfully flat it is easy and straightforward to prove that the canonical morphism T ª EndŽ PR . is bijective. We recall now the celebrated Morita theorem. 2.6. THEOREM w12x. are equi¨ alent.
Let PR g Mod-R, T s EndŽ PR .. Then the following
Ža. PR is a faithful quasiprogenerator; T P is finitely generated. Žb. PR is a progenerator. Žc. T P is a progenerator and T PR is faithfully balanced. Žd. The functor map Hom R Ž P , y . :
Mod-R ª Mod-T M ¬ Hom R Ž P , M .
is an equi¨ alence. Proof. See w12, Theorems 3.2 and 3.4x.
3. MAIN RESULTS 3.1 Let H be a Hopf algebra over the field k with comultiplication D, counit « , and antipode S. Let A be a right H-comodule algebra with structure map r , and let B s AcoH s a g A < r Ž a. s a m 1 A 4 Žsee w11x for an explanation and definition.. Recall that a right H-module coalgebra is a coalgebra D together with a right H-module structure m D : D m H ª D: d m h ¬ d £ h such that m D is a coalgebra map, that is, D D Ž d £ h. s
Ý d1 £ h 1 m d 2 £ h 2 ,
«D Ž d £ h. s «D Ž d . « Ž h. . Following Doi we introduce the category of right Ž D y A.-Hopf modules denoted by M Ž H . AD Žor simply MAD . as follows. An object in M Ž H . AD is a
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MENINI AND ZUCCOLI
right D-comodule with structure map r M , endowed with a right A-module structure such that, for every m g M and a g A, r M Ž ma. s Ým 0 a0 m m1 £ a1. Clearly a morphism in M Ž H . AD is defined to be both a morphism of right D-comodules and of right A-modules. The definition of left H-module coalgebra is given in a similar way. Let D be a left H-module coalgebra, A a right H-comodule algebra. The category of right]left Ž D y A.-Hopf modules denoted by A M Ž H . D Žor simply A M D . is defined as follows. An object in A M Ž H . D is a right D-comodule endowed with a left A-module structure such that, for every m g M and a g A, r M Ž am. s Ýa0 m 0 m a1 © m1. 3.2. EXAMPLE. For every M g Mod-A, M m D g M Ž H . AD via r MmD s id M m D D and m m d ? a s Ýma0 m d £ a1 , m g M, d g D, a g A. Moreover if M g M Ž H . AD it is easy to prove that r M : M ª M m D is a morphism in M Ž H . AD . 3.3. DEFINITION. Let A be a right H-comodule algebra, D a right H-module coalgebra. Let A a D* s A m D* as a k-vector space and let H
aag s a m g for every a g A, g g D*. Given a, b g A, g , x g D*, set aag ? bax s
Ý b0 aa Ž b1 © g . x ,
where ² h © g , d : s ²g , d £ h: for every h g H, d g D, g g D*. By linearity this defines a multiplication in A a D* which becomes a ring with H
identity 1 Aa« D . The ring A a D* will be called the smash product of A H and D* Žover H .. 3.4. DEFINITION. Let A be a right H-comodule algebra, D a left H-module coalgebra. Similarly A a9 D* is defined by setting H
aag ? bax s
Ý ab0 a Ž g £ b1 . x ,
where ²g £ b1 , d : s ²g , b1 © d : for every d g D. Note that this is exactly the smash product introduced in w6x. 3.5. Remark. Assume that the antipode Žof H . S is bijective. In this case the opposite ring H op of H is a Hopf algebra with comultiplication D, counit « , and antipode S, the composition inverse of S. In this case, given a left H-module coalgebra D, D can be regarded as a right H op -module coalgebra and Aop as a right H op -comodule algebra. Then it is easy to check that Aop aop D* s A a9 D*. H
H
HOPF ] GALOIS EXTENSIONS
3.6. PROPOSITION. L:
253
Assume that S is bijecti¨ e. Then the map
ž A a H */
op
H
ª Aop aop H * H
a ag ¬
Ý a 0aŽ S Ž a1 . © g .( S
is a ring isomorphism. Proof. Using the following equalities S Ž h1 g . © Ž Ž h 2 © x . g . s Ž S Ž g 2 . © x .Ž S Ž hg 1 . © g . ,
Ž S Ž h . © g . ( S £ g s S Ž hg . © g ( S, Ž xg . ( S s Ž g ( S . Ž x ( S . , which hold for every h, g g H and g , x g H *, as it is easily checked, it is straightforward to prove that L is a ring morphism. Now it is easy to prove that h© Ž x ( S . s Ž x £ S Ž h . . ( S,
Ž xg . ( S s Ž g ( S .Ž x ( S . . Using these equalities it can be proved that Ly1 :
Aop aop H * ª A a H *
ž
H
a ag ¬
H
op
/
Ý a 0 a Ž g £ S Ž a 1 . . ( S.
3.7. LEMMA. Let A a right H-comodule algebra, D a right H-module coalgebra. The maps f:
Aop ª A a D* H
a ¬ aa« D and g:
D* ª A a D* H
g ¬ 1 Aag are injecti¨ e ring morphisms. In particular e¨ ery left A a D*-module is a right A-module ¨ ia f.
H
254
MENINI AND ZUCCOLI
3.8 Let M g M Ž H . AD . M becomes a left A a D*-module by setting H
Ž a ag . ? m s
Ý m 0 a ² g , m1 : .
We denote this module by GŽ M .. Moreover, given M, N g M Ž H . AD and an abelian group morphism f : M ª N, f is a morphism in M Ž H . AD if and only if f is a morphism of left A a D*-modules. In this case we denote by H GŽ f .: GŽ M . ª GŽ N . the map f regarded as a morphism in A a D*-Mod. H
The assignments M ¬ GŽ M ., f ¬ GŽ f . define a covariant functor D
G: M Ž H . A ª A a D*-Mod. H
In particular, by Example 3.2, for every M g Mod-A, M m D is a left A a D*-module via H
Ž aag . ? Ž m m d . s
Ý ma0 m d1 £ a1²g , d2 : .
3.9. LEMMA w3, 13x. Im Ž G . s s
ž
AaD*
AmD .
/
H
Therefore G induces an equi¨ alence D
MŽ H.A ª s
ž
AaD*
AmD .
H
/
Proof. Let M g M Ž H . AD . Then r M : M ª M m D is an injective morphism in M Ž H . AD . Let pM :
AŽ M . ª M
Ž am . mgM ¬
Ý mgM
mam .
HOPF ] GALOIS EXTENSIONS
255
Then pM is a surjective morphism in Mod-A and it is easy to prove that pM m id D : AŽ M . m D ª M m D is a surjective morphism in M Ž H . AD . As AŽ M . m D ( Ž A m D .Ž M . in M Ž H . AD , we get that GŽ M . g s Ž AaD* A m D .. H
Conversely let M g s Ž AaD* A m D .. Then there exist a set X, a left H
A a D*-module L and a surjective morphism of left A a D*-modules p H
H
such that Ž A m D .Ž X . p
6
i
M¨
L
As remarked in Lemma 3.7, M is a right A-module via m ? a s Ž aa« D . m. Let us prove that M is a right D-comodule. In fact Ž A m D .Ž X . is a right D-comodule and hence a rational left D*-module. By Lemma 3.7, p is a surjective morphism of left D*-modules so that L is a rational D*-module too. Therefore L is a right D-comodule so that both i and p are morphisms of right D-comodules. Now, by Lemma 3.7, both p and i are right A-module morphisms. As p is surjective and Ž A m D .Ž X . g M Ž H . AD , we get that L itself satisfies the requirement of being a right Ž D y A.-Hopf module, i.e., L g M Ž H . AD . Since i is injective we get that also M g M Ž H . AD . By means of the foregoing theorem, in the following we will often identify M Ž H . AD with s Ž AaD* A m D .. H
3.10 w6x Let D be a right H-module coalgebra and x g D a grouplike element. Let px : H ª D h ¬ x £ h.
px is a morphism of right H-module coalgebras.
3.11. PROPOSITION.
3.12. Remark. Regarding H as a right H-module coalgebra we may consider M Ž H . AH , which will be simply denoted by MAH . Note that A g MAH . In fact for every a, b g A we have r Ž ab. s Ýa0 b 0 m a1 b1. 3.13 Let M g MAH and set
r˜M s Ž id M m px . ( r M :
MªMmD m¬
Ý m 0 m x £ m1 .
256
MENINI AND ZUCCOLI
M can be regarded as an object of M Ž H . AD via r˜M . We will denote this object by Mp x. Given a morphism f : M ª N in MAH , it is easy to prove that f is a morphism Mp x ª Np x in M Ž H . AD . In this way we get a covariant functor D
MAH ª M Ž H . A M ¬ Mp x .
In the following for a given M g MAH we will simply write M instead of Mp x, whenever no confusion will arise. 3.14. DEFINITION.
For every M g M Ž H . AD set M x s m g M < rM Ž m . s m m x 4 .
In particular A x s a g A <Ýa0 m x £ a1 s a m x 4 . In the particular case where D s H and x s 1 H , M x will be denoted by M coH . 3.15. THEOREM. Let A a right H-comodule algebra, D a right H-module coalgebra, and x g D a grouplike element. Then Ž1.
For e¨ ery M g M Ž H . AD ,
n M : Hom AaD* Ž A, M . ª M x H
f ¬ f Ž 1A . is an isomorphism and its in¨ erse is VM :
M x ª Hom AaD* Ž A, M . H
m¬ Ž2.
ž
V M Ž m. :
AªM a ¬ Ž aa« D . ? m
/
The map
nA : End lAaD* Ž A . ª A x H
is a ring isomorphism. Therefore A is an A a D* y Aop x -bimodule. H
Ž3.
For e¨ ery M g M Ž H . AD , Hom AaD* Ž AaD* A A opx , H
AaD*
H
A x-module and n M is an isomorphism of right A x-modules.
H
M . is a right
HOPF ] GALOIS EXTENSIONS
257
Proof. Ž1. Let f g Hom AaD* Ž A, M .. Then H
r M Ž f Ž 1 A . . s Ž Ž f m id D . ( r˜A . Ž 1 A . s Ž f m id D . Ž 1 A m x . s f Ž 1 A . m x. Note that the first equality holds as f : A ª M is a morphism of left D*-modules and hence a right D-comodule morphism. Therefore we get f Ž1 A . g M x . Let m g M x . We have
Ž n M ( V M . Ž m . s Ž V M Ž m . . Ž 1 A . s Ž 1 Aa« D . ? m s m « D Ž x . s m,
Ž Ž V M ( n M . Ž f . . Ž a. s Ž V M Ž f Ž 1 A . . . Ž a. s Ž aa« D . ? f Ž 1 A . s f Ž Ž aa« D . 1 A . s f Ž a . . Ž2. We prove that V A is a ring morphism. Let b, b9 g A x , a g A. We have
Ž V A Ž bb9. . Ž a. s Ž aa« D . ? bb9 s Ž bb9. a, Ž V A Ž b . ( V A Ž b9. . Ž a. s V A Ž b . Ž Ž aa« D . ? b9 . s V A Ž b . Ž b9a. s Ž b9aa« D . ? b s b Ž b9a . . Ž3. Let b g A x . We will prove that n M Ž fb . s n M Ž f . b. We have
n M Ž fb . s f Ž b . , by Section 3.8 and
n M Ž f . b s f Ž 1 A . b s Ž ba« D . f Ž 1 A . s f Ž Ž ba« D . 1 A . s f Ž b . . 3.16 Let V g Mod-A x . Since A is an A x y Ž A a D*. op -bimodule, V m A is a H
left A a D*-module, where for a given a ag g A a D* we have H
H
Ž a ag . ? Ž ¨ m a. s ¨ m Ž Ž a ag . ? a. s
Ý ¨ m a0 a ²g , x £ a1 :
Ax
258
MENINI AND ZUCCOLI
for every ¨ g V and a g A. Clearly V m A g GenŽ AaD* A. : s Ž AaD* A Ax
H
H
m D . s M Ž H . AD . It is easy to prove that r VmA s id V m r˜ and Ž ¨ m a. ? b Ax
s ¨ m ab for every ¨ g V, a, b g A. Let M g M Ž H . AD s s Ž AaD* A m D .. H
We set
Mx m A ª M
FM :
Ax
m m a ¬ ma. FM is a morphism in A a D*-Mod. H
Moreover let
Hom AaD* Ž A, M . m A ª M
CM :
H
Ax
j m a ¬ j Ž a. . Note that this is exactly the map introduced in Theorem 2.2 for R s A a D*, T s A x , and P s A. H
For e¨ ery M g M Ž H . AD the diagram
3.17. PROPOSITION.
H
Ax
n Mmid A
FM
cM
66
Hom AaD* Ž A, M . m A
M
6
Mx m A Ax
is commutati¨ e. In particular CM is an isomorphism if and only if FM is an isomorphism. Proof. Let a g A and f g Hom AaD* Ž A, M .. We have H
Ž FM ( Ž n M m id A . . Ž f m a. s FM Ž f Ž 1 A . m a. s f Ž 1 A . a s f Ž a . s CM Ž f m a . , as f is a morphism of right A-modules by Section 3.8. 3.18. LEMMA.
For e¨ ery M g Mod-A the map LM :
M ª Ž M m D. x
m¬mmx is an isomorphism of right A x-modules and its in¨ erse is the map GM :
Ž M m D. x ª M n
n
Ý mi m di ¬ Ý mi «D Ž di . . is1
is1
HOPF ] GALOIS EXTENSIONS
259
Therefore QM sGM ( n MmD :
Hom AaD* Ž A, M m D . ª M H
n
f¬
Ý yi « D Ž d i . , is1
where f Ž1 A . s Ý nis1 yi m d i , is an isomorphism. 3.19 Let M g Mod-A and let à M s QM m id A . Then
à M : Hom AaD* Ž A, M m D . m A ª M m A H
Ax
Ax
n
Ý yi « D Ž d i . m a,
fma¬
is1
where f Ž1 A . s Ý nis1 yi m d i , is an isomorphism. Let b xM s Ž FMmD .(Ž L M m id A ., we have
b xM :
Mm A ª M m D Ax
m m a ¬ Ž m m x. a s
Ý ma0 m x £ a1 .
For e¨ ery M g Mod-A the diagram
3.20. PROPOSITION. H
Ax
ÃM
cMm D b xM
MmD
6 6
Hom AaD* Ž A, M m D . m A 6
Mm A Ax
is commutati¨ e. In particular CMmD is injecti¨ e Ž resp. surjecti¨ e . if and only if b xM is injecti¨ e Ž resp. surjecti¨ e .. In the following b xA will be simply denoted by b x so that
bx :
Am A ª A m D Ax
amb¬
Ý ab0 m x £ b1 .
3.21. LEMMA. The map
m:
A a D* ª A H
aag ¬ aag ? 1 A
260
MENINI AND ZUCCOLI
is a surjecti¨ e morphism of left A a D*-modules. In particular A is a cyclic left H
A a D*-module. H
Proof. aa« D ? 1 A s a « D Ž x . s a. Assume that
3.22. LEMMA.
CAmD :
A xA
is flat and that
Hom AaD* Ž A, A m D . m A ª A m D H
Ax
j m a ¬ j Ž a. is an isomorphism. Then for e¨ ery M g s Ž AaD* A m D ., CM is an injecti¨ e H
morphism. In particular CM is an isomorphism for e¨ ery M g GenŽ AaD* A. H
= GenŽ AaD* A m D .. H
Proof. Note that, as CAmD is an isomorphism, A m D is generated by A in A a D*-Mod so that H
Gen
ž
AaD* H
A m D : Gen
/
ž
AaD* H
/
A .
Let M g s Ž AaD* A m D . s M Ž H . AD . As M g Mod-A, there exists an exact H
sequence h
AŽY . ª AŽ X . ª M ª 0, where X and Y are suitable sets. By applying the functor ym D we get that the sequence AŽY . m D ª AŽ X . m D ª M m D ª 0
Ž ).
is exact in A a D*-Mod. Recalling now that, by Lemma 3.21,
AaD*
H
A is
H
cyclic, we get the following isomorphisms of right A x-modules: Hom AaD* Ž A, AŽ X . m D . ( Hom AaD* Ž A, Ž A m D . H
ŽX.
H
( Hom AaD* Ž A, A m D .
ž
H
/
.
ŽX.
.
Now it is easy to prove that, given a nonempty set X, CAŽ X .mD is an isomorphism, CAmD being an isomorphism. Let H s Hom AaD* Ž A, y.. H
HOPF ] GALOIS EXTENSIONS
261
From the exact sequence Ž). we get the following diagram: f
H Ž A, M m D .
6
H Ž A, AŽ X . m D .
6
H Ž A, AŽY . m D . QAŽY . X
QAŽ X . X
QM X
AŽY .
AŽ X .
M
6
6
6
6
6
6
0.
It is easy to prove that this diagram is commutative; moreover its vertical arrows are isomorphisms by Lemma 3.18. It follows that the sequence H Ž A, AŽY . m D . ª H Ž A, AŽ X . m D . ª H Ž A, M m D . ª 0 Ž )) . is exact. Let us prove that CMmD is an isomorphism. By applying the functor ym A to the exact sequence Ž)). we get the diagram
CAŽ Y .m D X
H Ž A, M m D . m A CMm D
6
t
6
AŽ X . m D
MmD
0,
6
l
0
Ax
CA Ž X .m D X
6
6
AŽY . m D
h
Ax
6
Ax
6
H Ž A, AŽ X . m D . m A
6
g
6
Ax
H Ž A, AŽY . m D . m A
whose rows are exact. It is straightforward to prove that this diagram is commutative. Since CAŽY .mD and CAŽ X .mD are isomorphisms we get that CMmD is an isomorphism. By Example 3.2 and Section 3.8, r M : M ª M m D is an injective morphism in M Ž H . AD s s Ž AaD* A m D .. Therefore, since H
A xA
is flat, we get the exact sequence 0 ª H Ž A, M . m A ª H Ž A, M m D . m A Ax
Ax
so that from the commutative diagram Ax
H Ž A, M m D . m A Ax
X CMm D
CM
6
6
M
6
H Ž A, M . m A
6
6
0
MmD
we get that CM is injective. 3.23. DEFINITION. Let A a right H-comodule algebra, D a right Hmodule coalgebra, x g D a grouplike element. We say that A x ; A is a right Hopf]Galois extension if the map
bx :
Am A ª A m D Ax
amb¬ is bijective.
Ý ab0 m x £ b1
262
MENINI AND ZUCCOLI
3.24. DEFINITION. Let A a right H-comodule algebra, D a left Hmodule coalgebra, x g D a grouplike element. We say that A x ; A is a left Hopf]Galois extension if the map
b xX :
Am A ª A m D Ax
amb¬
Ý a0 b m a1 © x
is bijective. 3.25. Remark. Taking D s H, we have that H can be considered both as a right and as a left H-module coalgebra with respect to the natural H-module structure of H. Moreover x s 1 H is a grouplike element of H. In this case we simply set b s b 1 H and b 9 s b 1X H . Whenever the antipode S of H is bijective with composition inverse S, the map
u:
AmHªAmH amh¬
Ý a0 m a1 S Ž h .
is bijective and its inverse is the map
u 9:
AmHªAmH amh¬
Ý a0 m S Ž h . a1 .
Moreover it is straightforward to prove that b 9 s u ( b Žsee w10, Proposition 1.2x.. 3.26. Remark. If A s D s H, then H is a right Žleft. Hopf]Galois extension. In fact it is easy to prove that H coH s k ? 1 H and that the map
a:
HmHªHmH hmg¬
Ý hS Ž g 1 . m g 2
is the two-sided inverse of the map
b:
HmHªHmH hmg¬
Ý hg 1 m g 2 .
3.27. THEOREM. Let k be a field, H a Hopf algebra, A a right H-comodule algebra, D a right H-module coalgebra, and x g D a grouplike element.
HOPF ] GALOIS EXTENSIONS
263
Then the following are equi¨ alent. A x ; A is a right Hopf]Galois extension and For e¨ ery M g M Ž H . AD s s Ž AaD* A m D .,
Ža. Žb.
A xA
is flat.
H
Mx m A ª M
FM :
Ax
m m a ¬ ma is an isomorphism of left A a D*-modules, i.e., the weak structure theorem in H the sense of w7x holds. Žc. The map CAmD :
Hom AaD* Ž A, A m D . m A ª A m D H
Ax
j m a ¬ j Ž a. is an isomorphism of left A a D*-modules and H
A xA
is flat.
For e¨ ery M g GenŽ AaD* A m D .,
Žd.
H
CM :
Hom AaD* Ž A, M . m A ª M H
Ax
j m a ¬ j Ž a. is an isomorphism of left A a D*-modules and H
A xA
is flat.
For e¨ ery M g s Ž AaD* A m D .,
Že.
H
CM :
Hom AaD* Ž A, M . m A ª M H
Ax
j m a ¬ j Ž a. is an isomorphism of left A a D*-modules. H
Žf.
s Ž AaD* A m D . s GenŽ AaD* A..
Žg.
Ž . Ž . A xA is flat, s AaD* A s s AaD* A m D , and
H
H
H
H
Hom AaD* Ž A, y . :
Gen
H
ž
AaD* H
A ª Mod-A x
/
M ¬ Hom AaD* Ž A, M . H
is full and faithful. Žh. s Ž AaD* A. s s Ž AaD* A m D . and H
H
ules of A , for e¨ ery n g N, n ) 0. n
AaD* H
A generates all submod-
264
MENINI AND ZUCCOLI
Proof. Ža. m Žc. follows from Proposition 3.20. Žb. m Že. follows from Proposition 3.17. Žc. « Žd. follows from Lemma 3.22. Žd. « Žc. is trivial. Žd. m Že. m Žf. m Žg. m Žh. follow from Theorem 2.3 applied to R s A a D*, T s A x , P s A, and Q s A m D, in view of Theorem 3.15, ReH
mark 3.12, and Section 3.13. 3.28. Remark. Clearly analogous results hold in the category A M Ž H . D , where A a9 D* and b 9 play the role of b and A a D*, respectively. H
H
3.29. THEOREM. Let k be a field, H a Hopf algebra, A a right H-comodule algebra, D a right H-module coalgebra, and x g D a grouplike element. Then the following are equi¨ alent. Ža.
A x ; A is a right Hopf]Galois extension and
Žb.
The map
Ž y. x :
D
M Ž H . A ss
ž
AaD*
A xA
is faithfully flat.
A m D ª Mod-A x
/
H
M ¬ Mx is an equi¨ alence, i.e., the strong structure theorem in the sense of w7x holds. Žc.
The map CAmD :
Hom AaD* Ž A, A m D . m A ª A m D H
Ax
j m a ¬ j Ž a. is an isomorphism and Žd.
A xA
is faithfully flat.
The map CM :
Hom AaD* Ž A, M . m A ª M H
Ax
j m a ¬ j Ž a.
HOPF ] GALOIS EXTENSIONS
265
is an isomorphism for e¨ ery M g Ž AaD* A m D . and H
N ª Hom AaD* A, N m A
CNX :
H
n¬
ž
Ax
A ª Nm A Ax
a¬nma
/
0
is an isomorphism for e¨ ery N g Mod-A x . Že. The map Hom AaD* Ž A, y . : H
s
ž
AaD*
A m D ª Mod-A x
/
H
M ¬ Hom AaD* Ž A, M . H
is an equi¨ alence. Žf. The map y m A: Ax
Mod-A x ª s
ž
AaD*
AmD
H
/
N ¬ Nm A Ax
is an equi¨ alence. Žg. AaD* A is quasiprojecti¨ e and generates each of its submodules,
A xA
H
is a weak generator, and s Ž AaD* A m D . s s Ž AaD* A.. H
H
A is a quasiprogenerator and s Ž AaD* A m D . s s Ž AaD* A.. AaD*
Žh.
H
H
H
A xA is a weak generator, CM is an isomorphism for e¨ ery M g GenŽ AaD* A., and s Ž AaD* A m D . s s Ž AaD* A..
Ži.
H
H
H
Proof. Recall now, by Theorem 3.15, that End l AaD* Ž A. ( A x . H
Ža. m Žc. follows from Proposition 3.20. Žb. m Že. follows from Theorem 3.15. Žc. « Žd. By Žc. « Že. in Theorem 3.27, CM is an isomorphism for every M g s Ž AaD* A m D .. Moreover, by Žc. « Žh. in the same theorem, H
s Ž AaD* A m D . s s Ž AaD* A.. So we may apply Že. « Žd. of Theorem 2.5. H
H
H
H
Žd. « Žc. and Žd. « Že.. By Že. « Žg. in Theorem 3.27 we get that s Ž AaD* A m D . s s Ž AaD* A.. Now Žd. « Žc. follows from Žd. « Že. in
266
MENINI AND ZUCCOLI
Theorem 2.5 and Žd. « Že. follows from Žd. « Žb. in the same theorem. Že. « Žf. Note that for every N g Mod-A x we have N m A g Ax
GenŽ AaD* A. : s Ž AaD* A mD .. Therefore ImŽym A. : s Ž AaD* A mD .. H
Since y
Ax
H
m A is a left adjoint of Hom AaD* Ž A, y. we get Ž f .. Ax
H
H
Žf. « Žg. We have
s
ž
AaD* H
A m D s Im y m A : Gen
ž
/
:s
ž
/
Ax
AaD*
ž
AaD* H
A :s
/
ž
AaD* H
A
/
AmD .
/
H
Now apply Žc. « Žg. of Theorem 2.5. Žg. « Žc. Since A m D g s Ž AaD* A m D . s s Ž AaD* A., by Žg. « Že. H
H
in Theorem 2.5, we get that CAmD is an isomorphism and A xA is faithfully flat. Žg. m Žh. m Ži. follow from Žg. m Žh. m Ži. in Theorem 2.5. 3.30. Remark. Clearly analogous results hold in the category A M Ž H . D , where A a9 D* and b 9 play the role of b and A a D*, respectively. H
H
3.31. Remark. The equivalence Ža. m Žf. for D s H is part of the famous Schneider’s theorem w14, Theorem 1x. 3.32. COROLLARY w6, Theorem 2.3x. Let H be a Hopf algebra, A a right H-comodule algebra, D a right H-module coalgebra, and x g D a grouplike element. Assume that H is a faithfully coflat left D-comodule ¨ ia px , i.e., h ¬ ÝŽ x £ h1 . m h 2 . Then A x ; A is a right Hopf]Galois extension and A xA is flat Ž resp. faithfully flat . if B ; A is a right Hopf]Galois extension and B A is flat Ž resp. faithfully flat .. In this case Theorem 3.27 Ž resp. Theorem 3.29. applies. Proof. Consider
px :
HªD h ¬ x £ h.
px is a coalgebra morphism. Therefore by Theorems 1.1 and 1.3 in w3x the induction functor F: M Ž H . AH ª M Ž H . AD defined by F Ž M . s M endowed with the right D-comodule structure r F Ž M .Ž m. s Ým 0 m px Ž m1 . has, as a
HOPF ] GALOIS EXTENSIONS
267
right adjoint, the functor G: M Ž H . AD ª M Ž H . AH defined by GŽ M9. s M9 I H, with structure map rGŽ M 9.ŽÝmXi m h i . s ÝmXi m h i, 1 m h i, 2 and D ŽÝmXi m h i . ? a s ÝmXi a0 m h i a1. Since H is a faithfully coflat left D-comodule, G is a faithful and exact functor. Assume now that B ; A is a right Hopf]Galois extension and B A is flat. Then, by Theorem 3.27, A is a generator of M Ž H . AH . Let g: M1 ª M2 be a nonzero morphism in M Ž H . AD . Then GŽ g .: GŽ M1 . ª GŽ M2 . is a nonzero morphism in M Ž H . AH and hence HomŽ A, GŽ g ..: HomŽ A, GŽ M1 .. ª HomŽ A, GŽ M2 .. is a nonzero morphism, A being a generator of M Ž H . AH . Thus we get that Ap x s F Ž A. is a generator in M Ž H . AD so that GenŽ AaD* A. s s Ž AaD* A m D . and A x ; A is right H-Galois and A xA is H
H
flat. Assume now that B A is faithfully flat. Then, by theorem 3.29, Hom M Ž H . AH Ž A, y. s Žy. coH : M Ž H . AH ª Mod-B is an equivalence. It follows that the functor Hom M Ž H . AD Ž F Ž A., y. is exact, so that Ap x s F Ž A. is a projective object of s Ž AaD* A m D . s GenŽ AaD* A.. In particular AaD* A H
H
H
is quasiprojective. Since A is a cyclic left A a D*-module we conclude that AaD*
H
A is a quasiprogenerator.
H
3.33. Remark. We insert the following well-known facts for the generic reader’s sake. By applying Theorem 3.29 for D s A s H Žsee Remark 3.26. we get ‘‘the fundamental theorem of Hopf-modules.’’ The functor
Ž y.
coH
MHH ª k-Mod
:
M ¬ M coH is an equivalence. Its inverse is the functor ym H :
k-Mod ª MHH V ¬ V m H.
I
We recall now that H s H * g with respect to the right H-module structure defined by setting ²g h, x : s ²g , xSŽ h.: for every g g H * rat , h, x g H Žsee w15, Theorem 5.1.2x. and Ž H I . coH s x g H * < g ? x s ²g , 1 H :x for every g g H *4 s HHl * s the space of left integrals in H * Žor left integrals on H *.. Therefore we get that FH I :
MHH
rat
l
HH * m H ª H gmh¬g
is an isomorphism.
I
h
268
MENINI AND ZUCCOLI
If dim k Ž H . - `, H I s H * so that we get that FH * :
l
HH * m H ª H * gmh¬g
h
is an isomorphism. Since dim k Ž H . s dim k Ž H *. we get, in particular, dim k Ž HHl * . s 1. Assume now that h g kerŽ S . and let 0 / T g HHl * . Then for every x g H we have ²T h, x : s ²T, xSŽ h.: s 0. Hence FH * ŽT m h. s T h s 0, by injectivity of FH * , we get T m h s 0 and hence h s 0. Therefore S: H ª H is injective and hence also bijective. 3.34 Let now T / 0 be a fixed element of map
w:
l
HH *. Then, by the foregoing, the
H ª H* h¬T
S Ž h.
is an isomorphism. Note that, for every x g H, we have ²T
S Ž h . , x : s ²T , xS Ž S Ž h . . : s ²T , xh: s ² h © T , x:.
Therefore, for every h g H, T g H *, we have
w Ž h. s h © T . 4. THE FINITE CASE 4.1. LEMMA.
Assume that dim k Ž D . - `. Then
s
ž
AaD* H
A m D s A a D*-Mod.
/
H
Proof. See w6, page 375x. Using the above result, from Theorems 3.27 and 3.29 we get the following results. 4.2. THEOREM. Let A be a right H-comodule algebra, D a finite-dimensional right H-module coalgebra, and x g D a grouplike element. Then the
HOPF ] GALOIS EXTENSIONS
269
following are equi¨ alent. A x ; A is a right Hopf]Galois extension and A is a generator in A a D*-Mod. AaD*
Ža. Žf.
is flat.
H
H
Ži.
A xA
AaD*
A A opx is faithfully balanced and A A opx is finitely generated and
H
projecti¨ e. Moreo¨ er each of these conditions is equi¨ alent to each of the conditions Žb., Žc., Žd., Že., Žg., and Žh. of Theorem 3.27 with s Ž AaD* A m D . s A a D*H
H
Mod.
Proof. Ža. m Žb. m Žc. m Žd. m Že. m Žf. m Žg. m Žh. follow from Theorem 3.27 and Lemma 4.1. Žf. m Ži. follows from Theorem 17.8 in w1x. 4.3. THEOREM. Let A be a right H-comodule algebra, D a finite-dimensional right H-module coalgebra, and x g D a grouplike element. Then the following are equi¨ alent. Ža. Žf.
A x ; A is a right Hopf]Galois extension and A xA is faithfully flat. A is a faithful quasiprogenerator, A xA is finitely generated. AaD* H
Žg.
AaD*
A is a progenerator.
H
Žh.
A xA
is a progenerator and
AaD*
A A opx is faithfully balanced.
H
Moreo¨ er each of these conditions is equi¨ alent to each of the conditions Žb., Žc., Žd., and Že. of Theorem 3.29 with s Ž AaD* A m D . s A a D*-Mod. H
H
Proof. Apply Lemma 4.1, Theorem 3.29, and Theorem 2.6. 4.4. DEFINITION Žsee w9x.. Let B be a subring of a ring A with identity 1 A Žsuch that 1 A g B .. A is a Frobenius extension of B if A B is finitely generated and projective and moreover
B
A A (B Hom B Ž A B , BB . A .
4.5. Remark Ž see w9, Bemerkung 1x.. A is a Frobenius extension of B if and only if B A is finitely generated and projective and A
A B (A Hom B Ž A B , BB . B .
270
MENINI AND ZUCCOLI
4.6. Remark. Let H be a finite-dimensional Hopf algebra and A a right H-comodule algebra. Regarding H as a left H-module coalgebra, we have that A gA M Ž H . H s A a9 H *-Mod. Now, by Remark 3.5, A a9 H * s H
H
Aop aop H * and, by Proposition 3.6, H
L:
ž A a H */
op
H
ª Aop aop H * H
aag ¬
Ý a 0aŽ S Ž a1 . © g .( S
is a ring isomorphism. Therefore A becomes a right A a H *-module with H
respect to a ? Ž a ax . s L Ž a ax . ? a s Ž x ( S . ? Ž a a. ,
a g A, a ax g A a H *. This is the right A a H *-module structure of A H
considered in the following theorem.
H
4.7. THEOREM Žw4, Theorems 1.2 and 1.29x, w10, 1.7x, w16, 1.1x.. Let H be a finite-dimensional Hopf algebra and A a right H-comodule algebra, B s AcoH . Then the following are equi¨ alent. Ža. Ža9. Žb.
B ; A is a right Hopf]Galois extension. B ; A is a left Hopf]Galois extension. The map
b:
Am A ª A m H B
amb¬
Ý ab0 m b1
is surjecti¨ e. Žb9. The map
b 9:
Am A ª A m H B
amb¬
Ý a0 b m a1
is surjecti¨ e. Žc. The map CAmH :
Hom AaH * Ž A, A m H . m A ª A m H H
B
j m a ¬ j Ž a.
HOPF ] GALOIS EXTENSIONS
271
is surjecti¨ e. Žd.
AaH *
A is a generator for the category A a H *-Mod. H
H
Žd9.
A AaH * is a generator for the category Mod-A a H *. H
H
Že.
A op is faithfully balanced and AaH * B
B A is finitely generated and
H
projecti¨ e. Že9.
B op
A AaH * is faithfully balanced and A B is finitely generated and H
projecti¨ e. Žf.
For e¨ ery M g A a H *-Mod, H
CM :
Hom AaH * Ž A, M . m A ª M H
B
j m a ¬ j Ž a. is an isomorphism. Žg.
For e¨ ery M g A a H *-Mod s M Ž H . AH H
FM :
M coH m A ª M B
m m a ¬ ma is an isomorphism. Moreo¨ er if these conditions are satisfied, A is a Frobenius extension of B. Proof. Ža. m Ža9. and Žb. m Žb9. follow by Remark 3.25. Žb. m Žc. follows by Proposition 3.20. Žd. « Ža. follows by Theorem 4.2Žf. « Ža.. Žd. m Že. m Žf. m Žg. follow by Theorem 4.2. Ža. « Žb. is trivial. Žb9. « Žd. Let us fix 0 / T g HHl * . Let ¨ s Žid A m w .( b 9, where w : H ª H * is the isomorphism defined in Section 3.34. Then ¨ : A m A ª B
A a H * is surjective. We prove that ¨ is a morphism of left A a H *-modH
H
ules, where A m A is endowed with its A a H *-module structure defined H
B
as in Section 3.16. Since A m A g GenŽ AaH * A. this will prove Žd.. B
H
272
MENINI AND ZUCCOLI
First of all note that for every a, b g A we have ¨ Ž a m b . s Ž baT .Ž aa« .. Let a, b g A, a ax g A a H *. Then H
¨ Ž Ž a ax . ? Ž a m b . . s ¨ Ž a m Ž a ax . ? b .
s¨Ža m
Ý b 0 a ² x , b1 : .
s Ž Ý b 0 a a² x , b1 :T . ? Ž aa« . s Ž Ý b 0 a a Ž b1 © x . T . ? Ž aa« . s Ž a ax . ? Ž baT . ? Ž aa« . s Ž a ax . ? ¨ Ž a m b . . The equivalences Ža9. m Žb9. m Žd9. m Že9. can be proved in an analogous way in view of Remark 4.6. Assume now that the foregoing equivalent conditions hold and let us prove that A is a Frobenius extension of B. From Že9. we get that A B is finitely generated and projective. Let us prove that B A A ( Ž . Ž . o p and let i: A ª R: a9 ¬ a9a« . B Hom B A B , BB A . Let R s A a H * H
Then, by Lemma 3.7, i is an injective ring morphism. Note that A is a left Žright. R-module, so that A can be considered as a left Žright. A-module via i. Now it is easy to prove that this left Žright. A-module structure coincides with the usual left Žright. A-module structure on A. Since B A R is faithfully balanced B BB ( Hom R ŽB A R , B A R ., so that
Ž A B , B BB . R (B Hom B Ž R A B , Hom R Ž B A R , B A R . .
B Hom B R
(B Hom R
ž
R
AB m B AR , B AR B
/
R
R
and, by the foregoing, we get that
Ž A B , B BB . A (B Hom R
B Hom B A
Let us fix a 0 / T g HHl * and let ¨ :
A
ž
A
AB m B AR , B AR B
/
. A
A B m B A R ªA R R be the map B
defined in the proof of Žb9. « Žd.. ¨ s Žid A m w .( b 9 so that, by Ža9., ¨ is bijective. Moreover ¨ is a morphism of right R-modules and for every a, b g A we know that ¨ Ž a m b . s Ž baT .Ž aa« .. Let us prove that ¨ is a morphism of left A-modules. Let a9 g A. Then ¨ Ž a9Ž a m b .. s ¨ Ž a9a m b . s Ž baT .Ž a9aa« . s Ž baT .Ž aa« .Ž a9a« . s ¨ Ž a m b .Ž a9a« . s a9¨ Ž a m b ..
HOPF ] GALOIS EXTENSIONS
273
Therefore we get
Ž A B , B BB . A (B Hom R Ž A R R , B A R . A (B A A ,
B Hom B A
where the last map is a morphism of right A-modules by the remarks made above on the structure of A A . 4.8. THEOREM. Let H a finite-dimensional Hopf algebra, A a right Hcomodule algebra, and B s AcoH . Then the following are equi¨ alent. Ža.
B ; A is a right Hopf]Galois extension and
B
A is a weak genera-
tor. Ža9. Žb.
B ; A is a left Hopf]Galois extension and A B is a weak generator. The functor map Hom AaH * Ž A, y . : H
A a H *-Mod ª Mod-B H
M ¬ Hom AaH * Ž A, M . H
is an equi¨ alence. Žc. AaH * A is a faithful quasiprogenerator and
B
A is finitely generated.
H
Žc9.
A AaH * is a faithful quasiprogenerator and A B is finitely generated. H
Žd.
AaH *
A is a progenerator.
H
Žd9.
A AaH * is a progenerator. H
Že.
B A is a progenerator and
AaH *
A B op is faithfully balanced.
H
Že9.
A B is a progenerator and
B op
A AaH * is faithfully balanced. H
Proof. Ža. « Ža9. By Theorem 4.7, B ; A is a left Hopf]Galois extension. Let B L g B-Mod, L / 0. Since B A is a progenerator and A A B ( Ž . A Hom B B A A , B BB B , by Theorem 4.7 we have 0 / Hom B Ž B A, B L . ( Hom B
ž
B
A, BB m B L B
/
( Hom B Ž B A, B BB . m B L ( A m B L. B
B
Ža9. « Ža. By Theorem 4.7, A B is a progenerator and B A A ( Ž . Ž . Ž . Hom B B A A B , B BB A . A proof similar to a « a9 applies. Ža. « Žb. By Theorem 4.7, B A is projective so that B A is faithfully flat
274
MENINI AND ZUCCOLI
since B A is a weak generator. Žb. follows by Ža. « Že. in Theorem 4.3. Žb. « Ža. By Že. « Ža. in Theorem 4.3. Žc. m Žd. m Že. follow by Theorem 4.3. Note added in proof. After the submission of this paper, the authors received the preprint ‘‘Galois Extensions for Co-Frobenius Hopf Algebras’’ by M. Beattie, S. Dascalescu, and ¸ S. Raianu where, using Theorem 3.27 above, it is proved that for every co-Frobenius Hopf algebra H, any H-Galois H-comodule algebra A is a flat left AcoH -module.
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