Morphisms of Relative Hopf Modules, Smash Products, and Duality

Journal of Algebra 219, 547᎐570 Ž1999. Article ID jabr.1998.7842, available online at http:rrwww.idealibrary.com on

Morphisms of Relative Hopf Modules, Smash Products, and Duality* Claudia Menini Uni¨ ersity of Ferrara, Department of Mathematics, Via Machia¨ elli 35, 44100 Ferrara, Italy E-mail: [email protected]

and

¸Serban Raianu Uni¨ ersity of Bucharest, Faculty of Mathematics, Str. Academiei 14, 70109 Bucharest 1, Romania E-mail: [email protected] Communicated by Susan Montgomery Received May 11, 1998

INTRODUCTION The aim of this paper is to extend a result of Schneider on endomorphism rings of relative Hopf modules w17, Theorem 3.2x to the case of projective Hopf algebras with bijective antipode, and to show the connection between this extension and several duality theorems for Hopf algebras. A consequence of this generalization is the following: if H is a coFrobenius Hopf algebra over the field k, ArB is a Hopf᎐Galois extension and M is a right Ž A, H .-Hopf module, then, considering the usual ring embeddings ENDA Ž M . 噛H * rat : ENDA Ž M . 噛H * : 噛 Ž H , ENDA Ž M . . , *This paper was written while the first author was a member of G.N.S.A.G.A. of the C.N.R. and while the second author was a visiting professor of the C.N.R. at the University of Ferrara. 547 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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we have the following description of the image of the smallest ring: there is an algebra isomorphism ENDA Ž M . 噛H * rat , ENDB Ž M . ⭈ H * rat . Also, as a consequence of our result, we get extensions of results of w2, 7, 8x as well as an easy proof for the duality theorems for co-Frobenius Hopf algebras from w20x. The point we want to make is that Schneider’s theorem may be found behind virtually all duality theorems for Hopf algebras. It was shown in w8x that two results of Ulbrich Žwhich combine to give a duality theorem for finite Hopf algebras. are particular cases of a theorem of H.-J. Schneider. In the same paper, a partial infinite dimensional version of Schneider’s result was given. After recalling in the first section some of the results we are going to use or improve, in the second section of this paper we give a full extension of Schneider’s theorem ŽTheorem 2.1.. We derive from it some new duality theorems ŽCorollary 2.2. involving endomorphism rings of a relative Hopf module, Doi’s smash product, and the usual smash product. Another application is Corollary 2.3, describing the situation for Hopf᎐Galois extensions, with emphasis on the case when H is co-Frobenius. From this we get extensions of the results in w8x and of some results obtained in w7x for the case of graded rings and modules, as well as an interpretation of the big smash product as an endomorphism ring, which extends a similar one given in w9x. We remark that we are able to prove w8, Proposition 3.7x without assuming that the Weak Structure Theorem holds. Recently, Van Daele and Zhang proved in w20x the duality theorems for actions and coactions of co-Frobenius Hopf algebras. The method of proof uses a Morita context associated with the Žco.action of a multiplier Hopf algebra. Our aim in the third section is to use the infinite dimensional version of Schneider’s result in order to give a very simple proof for the duality theorems for co-Frobenius Hopf algebras. This section may be read separately from the second section, as we give very simple alternate proofs for the results coming from Theorem 2.5. The important thing is, however, that the role played by Schneider’s theorem in these duality theorems is as effective as the one played in the duality theorem for finite Hopf algebras.

1. PRELIMINARIES A Hopf algebra H over the field k is called co-Frobenius if H has non-zero space of left Žright. integrals. Recall that H *rat is the unique maximal left Žright. rational submodule of the left Žright. H *-module H *,

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549

and H is co-Frobenius if and only if H *rat / 0. If H is co-Frobenius, the antipode S of H is bijective with inverse S w15, Proposition 2x. We let © and £ denote the usual left and right actions, respectively, of H on H *, i.e., Ž h © h*.Ž g . s h*Ž gh. and Ž h* £ h.Ž g . s h*Ž hg .. The subalgebra H * rat of H * is an H᎐H-bimodule algebra under these actions. If t is a left Žright. integral of H, then tS is a right Žleft. integral. Unless explicitly mentioned otherwise, H will be a Hopf algebra which is projective over the commutative ring k, A will denote a right H-comodule algebra with 1, with comodule structure map

␳ : A ª A m H,

␳ Ž a. s

Ý a0 m a1 .

We write Aco H for the coinvariant subalgebra of A, Aco H s  a g A: Ý a0 m a1 s a m 1 4 . All maps are assumed k-linear, m means mk , etc. We use Sweedler’s notation throughout. We denote by MAH the category of right Ž H, A.-Hopf modules. An object in MAH is a right A-module and right H-comodule M, such that

Ý Ž m ⭈ a. 0 m Ž m ⭈ a. 1 s Ý m 0 ⭈ a0 m m1 a1 for any m g M and a g A. For M a right H-comodule, M co H denotes the coinvariants of M, i.e., M co H s  m g M <Ým 0 m m1 s m m 14 . Maps in Ž . MAH are A-module and H-comodule maps. End H A M denotes the endoH morphism ring of M g MA . Recall that the right H-comodule algebra A is called a right H-Galois extension of its coinvariants Aco H if the map can: A mA co H A ª A m H ,

can Ž a m b . s

Ý ab0 m b1 ,

is bijective. Further details about Hopf algebras may be found in w1, 14, 18x. The reader is also referred to w2, 8x for additional background information. The first result we recall is the Schneider theorem. THEOREM 1.1 w17, Theorem 3.2x. Suppose that k is a field, the right H-comodule algebra A is an H-Galois extension of B [ Aco H , and H is finite dimensional. Then there exists an isomorphism of algebras End A Ž M . 噛H * , End B Ž M . , for any M g MAH .

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We recall now the infinite dimensional version of Theorem 1.1 given in w8x. If M g MAH and N g MA , then M is a left H *, right B-bimodule; therefore Hom B Ž M, N . is a right H *-module with the action defined by

Ž f ⭈ h* . Ž m . s f Ž h* ⭈ m . s Ý h* Ž m1 . f Ž m 0 . .

Ž 1.

PROPOSITION 1.2 w8, Proposition 3.7x. Suppose that S is bijecti¨ e and that the extension ArAco H satisfies the Weak Structure Theorem Ž this means that for any M g MAH , the canonical morphism ␾ M : A mB M co H ª M, gi¨ en by ␾ M Ž a m x . s ax, is an isomorphism Ž actually it is enough to assume that ␾ M is onto for any M ... Then the map ␾ : Hom AŽ M, N . m H * ª Hom B Ž M, N . defined by ␾ Ž f m h*. s f ⭈ h* is injecti¨ e. Let M, N g MAH. Then M, N g MBH , where B is a right H-comodule algebra with b ¬ b m 1. Then we can consider, as in w19x, HOM AŽ M, N . Žresp. HOM B Ž M, N .. consisting of those f g Hom AŽ M, N . Žresp. f g Hom B Ž M, N .. for which there exist f 0 g Hom AŽ M, N . Žresp. f 0 g Hom B Ž M, N .. and f 1 g H such that

Ý f Ž m 0 . 0 m f Ž m 0 . 1 S Ž m1 . s Ý f 0 Ž m . m f 1 .

Ž 2.

We remark that HOM AŽ M, N . Žresp. HOM B Ž M, N .. is the rational part of Hom AŽ M, N . Žresp. Hom B Ž M, N .. with respect to the left H *-module structure defined by

Ž h* ⭈ f . Ž m . s Ý h* Ž f Ž m 0 . 1 S Ž m1 . . f Ž m 0 . 0 .

Ž 3.

Since HOM AŽ M, N . is defined in the same way as HOM B Ž M, N ., the first is a subcomodule of the latter. Suppose now that M s N. Then recall from w19x that ENDA Ž M . s HOM A Ž M, M . is a right H-comodule algebra, with the comodule structure defined by f ¬ Ý f 0 m f 1. In the same way, ENDB Ž M . becomes also a right H-comodule algebra. We also recall that H * is a right H-module algebra with the action

Ž h* £ h . Ž g . s h* Ž hg . . Consequently, we can form the right smash product ENDAŽ M .噛H * with the multiplication

Ž f噛h* . Ž u噛g* . s Ý fu 0 噛 Ž h* £ u1 . g*.

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The infinite version of Theorem 1.1 given in w8x is the following: PROPOSITION 1.3 w8, Proposition 3.8x. With the hypotheses of Proposition 1.2, the map

␾ : ENDA Ž M . 噛H * ª End B Ž M . defined by ␾ Ž f m h*. s f ⭈ h*, is an injecti¨ e morphism of algebras. If A is a right H-comodule algebra, we shall denote by 噛Ž H, A. the ‘‘big’’ smash product defined on Hom k Ž H, A. by the multiplication

Ž f ⭈ g . Ž h . s Ý f Ž g Ž h 2 . 1 h1 . g Ž h 2 . 0 ,

f , g g 噛Ž H , A. ,

h g H.

Ž 4. With this multiplication, 噛Ž H, A. is an associative ring with multiplicative identity ␧ H , the counit of H w9, Sect. 4x. A is isomorphic to a subalgebra of 噛Ž H, A. by identifying a g A with the map h ¬ ␧ Ž h. a. Since the multiplication in Ž4. for maps from H to k is just convolution, we also have that H * s HomŽ H, k . is a subalgebra of 噛Ž H, A., and A噛H * embeds in 噛Ž H, A. via

Ž a噛h* . Ž h . s h* Ž h . a. Before presenting the connection between Theorem 1.1 and the duality results of w19x, as given in w8x, we recall the following DEFINITION 1.4 w23, 16x. We will denote by H MAH the category whose objects are right A-modules M which are also left H-modules and right H-comodules such that the following conditions hold: Ži. M is a left H, right A-bimodule; Žii. M is a right Ž A, H .-Hopf module; Žiii. M is a left᎐right H-Hopf module. The morphisms in this category are the left H-linear, right A-linear, and right H-colinear maps. Following w16x we will call H MAH the category of two-sided Ž A, H .-Hopf modules. EXAMPLE 1.5. Let M g MAH and consider H m M with the natural structures of a left H-module and right A-module, and with right H-comodule structure given by hmm¬

Ý h 1 m m 0 m h 2 m1 .

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Then H m M is a two-sided Ž A, H .-Hopf module. In particular, U s H m A is a two-sided Ž A, H .-Hopf module Žsee w19x.. Remark 1.6. Let N g MA . Then N m H gH MAH if we put

Ž n m h . b s Ý nb0 m hb1 , and

Ý Ž n m h . 0 m Ž n m h . 1 s Ý n m h1 m h 2 , the left H-module structure being the natural one. If M g MAH , then M m H , H m M Žas in Example 1.5. in H MAH , the isomorphism Žof right A-modules, left H-modules, and right H-comodules. being given by mmh¬

Ý hS Ž m1 . m m 0 and h m m ¬ Ý m 0 m hm1 .

In particular, U s H m A , A m H in THEOREM 1.7 Žsee w19x.. M gH MAH. Then

H

MAH Žsee also w6x..

Let A be a right H-comodule algebra, and

Ži. Žw19, Theorem 2.4x.: End H A Ž M . 噛H ª ENDA Ž M . ,

fmh¬f⭈h

is an isomorphism of right H-comodule algebras. Žii. Žw19, Theorem 1.3x.: If H is finite and U is as in Example 1.5, then Ž . the algebras A噛H * and End H A U are isomorphic. It was proved in w8, Corollary 2.5x that both i. Žin the finite case. and ii. of Theorem 1.7 are particular cases of Theorem 1.1. We shall use the following result LEMMA 1.8 Žw1, Lemma 3.3.7x.. If k is a field, H is a co-Frobenius Hopf algebra, t is a left integral of H, and a, b g H, then

Ý t Ž a2 S Ž b . . a1 s Ý t Ž aS Ž b1 . . b2 . We end by recalling from w2x the infinite dimensional analogue of Radford’s distinguished grouplike element.

MORPHISMS OF RELATIVE HOPF MODULES

553

PROPOSITION 1.9 w2, Proposition 1.3x. Let t be a nonzero integral of H Ž co-Frobenius o¨ er the field k .. Then there is a grouplike element g in H such that Ži. th* s h*Ž g . t for all h* g H *; Žii. tS s g © t, and tS s t £ g.

2. SCHNEIDER’S RESULT REVISITED In this section we prove the main result of this paper. It is a complete infinite analogue of Schneider’s Theorem 1.1 Žrecall that the Hopf algebras are projective throughout.. From this result we deduce immediately Propositions 1.2 and 1.3, as well as several new results, some of them extending results from w7x on graded modules and their endomorphism rings. THEOREM 2.1. Let H be a Hopf algebra with bijecti¨ e antipode S, A a right H-comodule algebra, M g MAH , N g MA . Then the following assertions hold: Ži. Hom k Ž H, Hom AŽ M, N .. , Hom HA Ž M m H, N m H . as right H *modules. The right H *-module structure on Hom k Ž H, Hom AŽ M, N .. is gi¨ en by

Ž f ⭈ h* . Ž h . s f Ž h* ⭈ h . s Ý h* Ž h 2 . f Ž h1 . ,

Ž 5.

and the right H *-module structure on Hom HA Ž M m H, N m H . is defined as

Ž u ⭈ h* . Ž m m h . s Ý h* Ž m1 S Ž h1 . . u Ž m 0 m h 2 . . Žii.

The isomorphism from Ži. induces an algebra embedding 噛 Ž H , ENDA Ž M . . ª End H AŽ MmH..

Žiii. If k is a field and H is co-Frobenius, then Hom A Ž M, N . m H * rat , Hom HA Ž M m H , N m H . ⭈ H *rat . Proof. Ži. Define

␤ : Hom k Ž H , Hom A Ž M, N . . ª Hom HA Ž M m H , N m H . by

␤ Ž f . Ž m m h. s

Ý f Ž m1 S Ž h 1 . . Ž m 0 . m h 2 .

Ž 6.

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We have that ␤ Ž f . is right A-linear, because

␤ Ž f . Ž Ž m m h . a . s ␤ Ž f . Ž Ý ma0 m ha1 . s

Ý f Ž m1 a1 S Ž h1 a2 . . Ž m 0 a0 . m h 2 a3 s Ý f Ž m1 a1 S Ž a2 . S Ž h1 . . Ž m 0 a0 . m h 2 a3 s Ý f Ž m1 S Ž h1 . . Ž m 0 a0 . m h 2 a1 s Ý f Ž m1 S Ž h1 . . Ž m 0 . a0 m h 2 a1 s Ž Ý f Ž m1 S Ž h1 . . Ž m 0 . m h 2 . a s Ž ␤ Ž f . Ž m m h . . a, and it is also right H-colinear, since

Ý ␤ Ž f . Ž m m h 1 . m h 2 s Ý f Ž m1 S Ž h 1 . . Ž m 0 . m h 2 m h 3 s Ý ␤ Ž f . Ž m m h. 0 m ␤ Ž f . Ž m m h. 1 . Define now

␣ : Hom HA Ž M m H , N m H . ª Hom k Ž H , Hom A Ž M, N . . by

␣ Ž u. Ž h. Ž m. s

Ý Ž I m ␧ . u Ž m 0 m S Ž h . m1 . .

For all u g Hom HA Ž M m H, N m H . and h g H, we have

␣ Ž u . Ž h . Ž ma. s s

Ý Ž I m ␧ . u Ž m 0 a0 m S Ž h . m1 a1 . Ý Ž I m ␧ . Ž u Ž m 0 m S Ž h . m1 . a . s ␣ Ž u . Ž h . Ž m . a,

since Ž I m ␧ .ŽŽÝn i m h i . a. s Ýn i ␧ Ž h i . a s ŽŽ I m ␧ .ŽÝn i m h i .. a, and so ␣ Ž u.Ž h. is right A-linear. We now compute

␤ Ž ␣ Ž u. . Ž m m h. s

Ý ␣ Ž u . Ž m1 S Ž h 1 . . Ž m 0 . m h 2

Ý Ž I m ␧ . u Ž m 0 m h 1 S Ž m 2 . m1 . m h 2 s Ý Ž I m ␧ . u Ž m m h1 . m h 2 s Ý Ž I m ␧ . u Ž m m h . 0 m u Ž m m h . 1 Ž u is colinear . s

s uŽ m m h. ,

MORPHISMS OF RELATIVE HOPF MODULES

555

since

Ý Ž I m ␧ . Ž ni m hi . m hi 1

2

s

Ý ni m hi ,

Ž 7.

and

␣ Ž ␤ Ž f . . Ž h. Ž m. s

Ý Ž I m ␧ . Ž ␤ Ž f . Ž m 0 m S Ž h . m1 . .

s

Ý Ž I m ␧ . Ž f Ž m1 S Ž m 2 . h 2 . Ž m 0 . m S Ž h 1 . m 3 .

s

Ý Ž I m ␧ . Ž f Ž h 2 . Ž m 0 . m S Ž h 1 . m1 . s f Ž h . Ž m . ,

thus ␣ and ␤ are inverse one to each other. It remains to prove that ␣ is right H *-linear. We have

␤ Ž ␣ Ž u . ⭈ h* . Ž m m h .

Ý Ž ␣ Ž u . ⭈ h*. Ž m1 S Ž h1 . . Ž m 0 . m h 2 s Ý h* Ž m 2 S Ž h1 . . ␣ Ž u . Ž m1 S Ž h 2 . . Ž m 0 . m h 3 s

Ý h* Ž m 3 S Ž h1 . . Ž I m ␧ . u Ž m 0 m h 2 S Ž m 2 . m1 . m h 3 s Ý h* Ž m1 S Ž h1 . . Ž I m ␧ . u Ž m 0 m h 2 . m h 3 s Ý h* Ž m1 S Ž h1 . . u Ž m 0 m h 2 . Ž again from Ž 7 . . , s

so we proved both that Hom HA Ž M m H, N m H . is an H *-module and that ␣ and ␤ are H *-linear. Žii. We show that the restriction of ␤ is an algebra map. Let f, f ⬘ g 噛Ž H, ENDAŽ M ... We have

␤ Ž f ⭈ f ⬘. Ž m m h . s

Ý Ž f ⭈ f ⬘ . Ž m1 S Ž h 1 . . Ž m 0 . m h 2 s Ý f Ž f ⬘ Ž m 2 S Ž h 1 . . 1 m1 S Ž h 2 . . f ⬘ Ž m 2 S Ž h 1 . . 0 Ž m 0 . m h 3 s Ý f Ž f ⬘ Ž m 3 S Ž h 1 . . Ž m 0 . 1 S Ž m1 . m 2 S Ž h 2 . . = Ž f ⬘ Ž m 3 S Ž h1 . . Ž m 0 . 0 . mh 3 Ž since f ⬘ Ž m 2 S Ž h1 . . g ENDA Ž M . . s ␤ Ž f . Ž Ý f ⬘ Ž m1 S Ž h 1 . . Ž m 0 . m h 2 . s ␤ Ž f . Ž ␤ Ž f ⬘ . Ž m m h . . . Žiii. Let

␮ : Hom A Ž M, N . m H * ª Hom k Ž H , Hom A Ž M, N . .

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be the usual embedding:

␮ Ž u m h* . Ž h . Ž m . s h* Ž h . u Ž m . . Composing this embedding with the isomorphism ␤ from i., we get an embedding

␤ ( ␮ : Hom A Ž M, N . m H * ª Hom HA Ž M m H , N m H . , which sends u m h* to Ž ␤ ( ␮ .Ž u m h*., where

Ž ␤ ( ␮ . Ž u m h* . Ž m m h . s Ý h* Ž m1 S Ž h1 . . u Ž m 0 . m h 2 s Ž Ž Ž ␤ ( ␮ . Ž u m ␧ . . ⭈ h* . Ž m m h . . Consequently,

Ž ␤ ( ␮ . Ž u m h* . s Ž Ž ␤ ( ␮ . Ž u m ␧ . . ⭈ h* and we find that ␤ ( ␮ induces an embedding Hom A Ž M, N . m H * rat ª Hom HA Ž M m H , N m H . ⭈ H * rat , which we are going to show is also surjective. Let f g Hom HA Ž M m H, N m H ., h* g H * rat . We would like to find some f i g Hom AŽ M, N ., hUi g H * rat such that

Ž ␤ ( ␮ . Ž Ý f i m hUi . s f ⭈ h*. This will happen if and only if

␣ Ž f ⭈ h* . s ␮ Ž Ý f i m hUi . , i.e.,

␣ Ž f ⭈ h* . Ž h . Ž m . s ␮ Ž Ý f i m hUi . Ž h . Ž m . s

Ý hUi Ž h . f i Ž m . ,

for all h g H, m g M. Since h* g H *rat , we have that the space  Ýh*Ž h 2 . S Ž h1 .< h g H 4 is finite dimensional. If  e1 , e2 , . . . , e n4 is a basis in this space, then we can write

Ý h* Ž h 2 . S Ž h1 . s Ý hUi Ž h . ei ,

MORPHISMS OF RELATIVE HOPF MODULES

557

for some hUi g H * rat . Then we have

␣ Ž f ⭈ h* . Ž h . Ž m . s s

Ý Ž I m ␧ . Ž f ⭈ h*. Ž m 0 m S Ž h . m1 . Ý h* Ž m1 S Ž m 2 . h 2 . Ž I m ␧ . f Ž m 0 m S Ž h1 . m 3 .

Ý h* Ž h 2 . Ž I m ␧ . f Ž m 0 m S Ž h1 . m1 . s Ý hUi Ž h . Ž I m ␧ . f Ž m 0 m e i m1 . s Ý hUi Ž h . f i Ž m . , s

where f i Ž m. s ÝŽ I m ␧ . f Ž m 0 m e i m1 ., which is right A-linear since

Ž I m ␧ . ž Ž Ý n i m ai . a / s Ž I m ␧ . Ž Ý n i a0 m ai a1 . s

Ý n i ␧ Ž ai . a s Ž I m ␧ . Ž Ý n i m ai . a,

and the proof is complete. We now give some consequences of Theorem 2.1. The first one is a new duality theorem for infinite Hopf algebras. COROLLARY 2.2. Let H be a Hopf algebra with bijecti¨ e antipode, A a right H-comodule algebra, and M g MAH which is finitely generated as an A-module. Then the following assertions hold: Ži. We ha¨ e algebra isomorphisms 噛 Ž H , End A Ž M . . , End H AŽ MmH., and 噛 Ž H , End A Ž M . . 噛H , ENDA Ž M m H . . Žii. Assume, moreo¨ er, that M gH MAH . Then we ha¨ e an algebra isomorphism H 噛 Ž H , End H A Ž M . 噛H . , End A Ž M m H . .

Proof. Ži. Since M is finitely generated, ENDAŽ M . s End AŽ M ., and the first assertion follows directly from Theorem 2.1. The second isomorphism follows from the first one, using Example 1.5, Remark 1.6, and Theorem 1.7 i.. Žii. follows from the first isomorphism of i. and Theorem 1.7 i.. We show now that Propositions 1.2 and 1.3 may also be obtained Žand extended. using Theorem 2.1. Moreover, the following result will provide a

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characterization of the image of ENDAŽ M .噛H * rat through ␾ from Proposition 1.3. COROLLARY 2.3. Let H be a Hopf algebra with bijecti¨ e antipode, A a right H-comodule algebra, B s Aco H , such that ArB is an H᎐Galois extension. Let M g MAH and N g MA . Then the following assertions hold: Ži. We ha¨ e Hom k Ž H , Hom A Ž M, N . . , Hom B Ž M, N . , f¬m¬

Ý f Ž m1 . Ž m 0 .

as right H *-modules, where Hom k Ž H, Hom AŽ M, N .. is a right H *-module as in Ž5. and the right H *-module structure on Hom B Ž M, N . is gi¨ en by Ž1.. Žii. The isomorphism from Ži. induces an algebra embedding 噛 Ž H , ENDA Ž M . . ª End B Ž M . . Žiii. Ž See w9, 4.5x. 噛 Ž H , A . , End B Ž A . . Živ.

If k is a field and H is co-Frobenius, then Hom A Ž M, N . m H *rat , Hom B Ž M, N . ⭈ H * rat , u m h* ¬ m ¬

Ý h* Ž m1 . u Ž m 0 . .

Proof. Ži. We have that the map ␯

Hom HA Ž M m H , N m H . ª Hom B Ž M, N . ,

Ž 8.

sending a map to its restriction to Ž M m H . co H s M m 1 , M is an isomorphism of right H *-modules, as may be easily seen from the definition of the H *-module structures. We note, for further use, that the inverse of ␯ , since ArB is a Galois extension, sends f g Hom B Ž M, N . to f g Hom HA Ž M m H, N m H ., where f Ž m m h. s

Ý f Ž mri Ž h . . l i Ž h . 0 m l i Ž h . 1 ,

Žhere canŽÝri Ž h. m l i Ž h.. s Ýri Ž h. l i Ž h. 0 m l i Ž h.1 s 1 m h.. Žii. If M s N, then Ž8. yields an algebra isomorphism End H A Ž M m H . , End B Ž M . . Žiii. follows from Žii. for M s A.

Ž 9.

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Živ. is clear, but, since we will need it soon, we write down explicitly the preimage of some f ⭈ h*, f g Hom B Ž M, N ., h* g H * rat . By the proof of Theorem 2.1 iii., we have f ⭈ h* s Ý f i ⭈ hUi , where f i has the form fi Ž m. s

Ý Ž I m ␧ . f Ž m o m em1 . ,

Ž 10 .

for some e g H. Since the Weak Structure Theorem holds by w2, Theorem 3.1x, we can write an m g M as m s Ým j a j , m j g M co H , and using Ž9., we get fi Ž m. s s

Ý fi Ž m j a j . Ý f ž Ž m j a j . 0 ri Ž e Ž m j a j . 1 . / l i Ž e Ž m j a j . 1 . Ž by Ž 9. , Ž 10. .

s

Ý f Ž m j a j ri Ž a j . r k Ž e . . l k Ž e . l i Ž a j . s Ý f Ž m j rk Ž e . . l k Ž e . a j , 0

1

1

since Ýa0 ri Ž a1 . m l i Ž a1 . s 1 m a Žthis is checked by applying can.. Remarks 2.4. 1. Taking M s N s A in Theorem 2.1 i., and taking into account Remark 1.6, we obtain the isomorphism given by Ulbrich in w19, Lemma 1.2x, composed with Hom k Ž S, A.. This is a result of the type of w12, Proposition 4.1x: to see this, consider first H m A as an object in A M H via aŽ h m b . s

Ý a1 h m a0 b

and

hmb¬

Ý h1 m b m h 2 .

Consequently, by w12, Ž3.2., p. 66x, H m A is a left 噛Ž H, A.-module by f ⭈ Ž h m a. s

Ý f Ž h 2 . 1 h1 m f Ž h 2 . 0 b.

Ž 11 .

Now A is a right H op cop-comodule algebra via a¬

Ý a0 m S Ž a1 . ;

and we denote it by A⬘. In this way, 噛 Ž H , A . ª 噛 Ž H op cop , A⬘ . ,

f¬fS

is a ring isomorphism, and its composition with Ulbrich’s isomorphism, regarded as 噛 Ž H op cop , A⬘ . ª Hom HA Ž H m A . . is given exactly by the left module structure as in Ž11..

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2. Theorem 2.1 i. extends Theorem 4.3 of w7x. 3. From Corollary 2.3 i. we get Proposition 1.2, under the weaker hypothesis that the extension is Galois. Note that we also obtain, when H is co-Frobenius, the description of the image of Hom AŽ M, N .噛H * rat in Corollary 2.3 iii.. 4. If M s N, then we get Proposition 1.3 from Corollary 2.3 ii.. This extends Theorem 4.8 of w7x. 5. Suppose that k is a field, the antipode of H is bijective, and let A⬘噛␴ H be a crossed product with invertible cocycle ␴ Žsee w5x or w10x.. Then we have an algebra isomorphism 噛 Ž H , A⬘噛␴ H . , RF MH Ž A⬘ . , where RF MH Ž A⬘. are the row finite matrices, with rows and columns indexed by a basis of H, and with entries in A⬘. Indeed, we know from w4x that A⬘噛␴ HrA⬘ is a Galois extension. Then we find from Corollary 2.3 iii. that 噛 Ž H , A⬘噛␴ H . , End A⬘ Ž A⬘噛␴ H . , and since the antipode is bijective, A⬘噛␴ H is free as a right A⬘-module Žsee w4x., and so the result follows. This is, in fact, a version of w11, Proposition 4.1x, which, as remarked in w11x, can also be deduced from w9, 4.5x and w3, 1.18x. What it shows is that Schneider’s theorem, in the form of Theorem 2.1, is also present in the proof of the duality theorems for infinite Hopf algebras. In fact, this may also be seen in w3x, where the proof of the duality theorem used embeddings in an endomorphism ring. The following result improves Proposition 1.3 in the co-Frobenius case by describing the image of ENDAŽ M .噛H * rat in End B Ž M .. THEOREM 2.5. If k is a field, ArB is Galois, H is co-Frobenius, M g MAH , and N g MA , then HOM A Ž M, N . m H *rat , HOM B Ž M, N . ⭈ H * rat . We also ha¨ e a ring isomorphism ENDA Ž M . 噛H * rat , ENDB Ž M . ⭈ H * rat . Proof. We know from Corollary 2.3 iv. that the map Hom A Ž M, N . m H * rat ª Hom B Ž M, N . ⭈ H * rat ,

f m h* ¬ f ⭈ h*

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561

is a bijection, and the restriction to HOM AŽ M, N . m H * rat is clearly sent to HOM B Ž M, N . ⭈ H * rat . We have to show only that for any f g HOM B Ž M, N . and h* g H *rat , f ⭈ h* is in the image. We know from the proof of Corollary 2.3 iv. that f ⭈ h* s Ý f i ⭈ hUi , where the f i g Hom AŽ M, N . have the form f˜Ž m . s

Ý f Ž m j ri Ž e . . l i Ž e . a j .

We know that there are f 0 g Hom B Ž M, N . and f 1 g H satisfying Ž2., and we want to prove that any map of the form f˜ has the same property, i.e., it belongs to HOM AŽ M, N .. We have

Ý f˜Ž m 0 . 0 m f˜Ž m 0 . 1 S Ž m1 . s Ý f Ž m j ri Ž e . . 0 l i Ž e . 0 a j

0

m f Ž m j r i Ž e . . 1 l i Ž e . 1 a j1 S Ž a j 2 .

s

Ý f Ž m j ri Ž e . . 0 l i Ž e . 0 a j m f Ž m j ri Ž e . . 1 l i Ž e . 1 s Ý f Ž m j ri Ž e . 0 . 0 l i Ž e . 0 a j m f Ž m j ri Ž e . 0 . 1 S Ž ri Ž e . 1 . ri Ž e . 2 l i Ž e . 1 s

Ý f 0 Ž m j ri Ž e . 0 . l i Ž e . 0 a j m f 1 ri Ž e . 1 l i Ž e . 1 s Ý f˜0 Ž m . m f˜1 ,

and since f˜0 is A-linear the proof of the first assertion is complete. The other assertion is now clear. Remark 2.6. When H is cocommutative it is easy to see that Hom AŽ M, N . and Hom B Ž M, N . are H *-bimodules with the left module structure given by Ž3. and the right module structure given by Ž1.. In this case we obtain that, if H is co-Frobenius, then the image of HOM AŽ M, N . m H * contains HOM B Ž M, N ., similar to the graded case Žsee w7, Theorem 4.3x..

3. APPLICATION TO THE DUALITY THEOREMS FOR CO-FROBENIUS HOPF ALGEBRAS Throughout this section, H will be a co-Frobenius Hopf algebra over the field k. In this section we show that it is possible to derive the duality theorems for co-Frobenius Hopf algebras from Theorem 2.5. We remark first that if A is a right H-comodule algebra, then ;

ENDA Ž A . ,

6

A

a ¬ fa ,

is an isomorphism of H-comodule algebras.

f a Ž b . s ab

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Indeed, if h* g H * and a, b g A, then

Ž h* ⭈ f a . Ž b . s Ý f a Ž b 0 . 0 h* Ž f a Ž b 0 . 1 S Ž b1 . . s

Ý a0 b0 h* Ž a1 b1 S Ž b2 . . s Ý a0 bh* Ž a1 . s Ý h* Ž a1 . f a Ž b . 0

which proves the desired isomorphism. Identifying A with ENDAŽ A. via this isomorphism, the map ␾ from Proposition 1.3 becomes

␲ : A噛H *rat ª End Ž A A co H . ␲ Ž a噛h* . Ž b . s

rat

,

Ý ab0 h* Ž b1 . ,

which is exactly the map ␲ 1 from Corollary 3.4 of w2x. With this notation we have the following immediate consequence of Corollary 2.3 iv.. PROPOSITION 3.1. phism.

If ArB is Galois, then the map ␲ is a ring isomor-

Remark 3.2. It is also possible to derive Proposition 3.1 directly from Proposition 1.3. We know from the above remarks and from Proposition 1.3 that ␲ is an injective ring morphism. ŽNote also that this was proved directly in w2, Corollary 3.4x.. Thus it remains to show that ␲ is surjective. Let f g EndŽ A A co H . rat , and let h* g H *rat such that f s f ⭈ h*. Let t⬘ be a right integral and Ý i a i m bi g A mA co H A such that 1 m h*S s

Ý ai bi

0

m S Ž bi1 . © t⬘.

i

Then, for any b g A we have f Ž b . s Ž f ⭈ h* . Ž b . s

Ý h* Ž b1 . f Ž b0 .

s

Ý h*S Ž S Ž b1 . . f Ž b0 .

s

Ý f Ž ai bi

s

Ý f Ž ai . bi

s

Ý f Ž ai . bi

0

b 0 . t⬘ Ž S Ž b1 . S Ž bi1 . . 0

b 0 t⬘S Ž bi1 b1 .

ž Ž S Ž b . © t⬘. S / Ž b . s ␲ ž Ý f Ž a . b 噛 Ž S Ž b © t⬘ . S . / Ž b . i

and the proof is complete.

0

b0

i0

i1

1

i1

MORPHISMS OF RELATIVE HOPF MODULES

563

From Proposition 3.1 we obtain, immediately, the duality theorem for actions of co-Frobenius Hopf algebras. COROLLARY 3.3 w20, Theorem 5.3x. If A⬘噛␴ H is a crossed product with in¨ ertible ␴ , then we ha¨ e an isomorphism of algebras,

Ž A⬘噛␴ H . 噛H *rat , MHf Ž A⬘ . , where MHf Ž A⬘. denotes the ring of matrices with rows and columns indexed by a basis of H, and with only finitely many non-zero entries in A⬘. Proof. The right H *-module structure on EndŽ A⬘噛␴ HA⬘ . is given by the left H *-module structure on A⬘噛␴ H: h* ⭈ Ž a噛h. s Ýa噛h1 h*Ž h 2 .. It is well known from w4, Theorem 1.18x that A⬘噛␴ H , H m A⬘ as right A⬘modules and left H *-modules via a噛h ¬

Ý h 4 m Ž S Ž h1 . ⭈ a . ␴ Ž S Ž h 2 . , h1 . ,

and so A⬘噛␴ HA⬘ is free. Then EndŽ A⬘噛␴ HA⬘ . is the ring of row-finite matrices with entries in A⬘ and rows and columns indexed by a basis of H. In order to see who is EndŽ A⬘噛␴ HA⬘ . rat we choose the following basis of H: as in w13x, write H s [EŽ S␭ ., where the S␭’s are simple left H-comodules and EŽ S␭ . denotes the injective envelope of S␭ . Then we denote by  h i 4i the basis of H obtained by putting together bases in the EŽ S␭ .’s, and use the basis  h i m 14i in H m A⬘ , A⬘噛␴ H to write EndŽ A⬘噛␴ HA⬘ . as a matrix ring. We denote by p␭ the map in H *rat equal to ␧ on EŽ S␭ . and 0 elsewhere. In order to prove the assertion in the statement it is enough to prove that any f with f Ž h i m 1. / 0 and f Ž h j m 1. s 0 for j / i is in the rational part; say h i g EŽ S␭ .. Then it is easy to see that f s f ⭈ p␭ . The converse inclusion is clear, since for any f and any p␭, f ⭈ p␭ is 0 outside EŽ S␭ .. We have seen how the first duality theorem for co-Frobenius Hopf algebras may be derived from Proposition 3.1. There is yet another, even easier way to prove this duality theorem. We sketch it since it provides a hint on how to attack the second duality theorem. It is based on the following simple LEMMA 3.4. Let R be a ring without unit and MR a right R-module with the property that there exists a common unit Ž in R . for any finite number of elements in R and M. Then MR , Hom R Ž R R , MR . ⭈ R, where, for f g Hom R Ž R R , MR . and r, s g R, Ž f ⭈ r .Ž s . s f Ž rs ..

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Proof. The isomorphism sends m g M to ␳mŽ r . s mr, for all r g R. The inverse sends Ý f i ⭈ ri to Ý f i Ž ri .. COROLLARY 3.5.

If R and M are as in Lemma 3.4 and MR , R R , then R , End Ž MR . ⭈ R

Ž as rings.. LEMMA 3.6. Then

Let H m A be the two-sided Hopf module from Example 1.5. H m A , A噛H * rat

as right A噛H *rat-modules, ¨ ia hma¬

Ý a0 噛t £ ha1 ,

where t is a right integral of H. Proof. Recall that H m A becomes a right A噛H *rat-module via

Ž h m a. Ž b噛h* . s Ý h1 m a0 b 0 h* Ž S Ž h 2 a1 b1 . . . If we denote by ␪ the map in the statement, which is clearly a bijection, then we have to show that

␪ Ž Ž h m a . Ž b噛h* . . s Ž ␪ Ž h m a . . Ž b噛h* . , i.e., that

Ý a0 b0 噛t £ h1 a1 b1 h* Ž S Ž h 2 a2 b2 . . s Ý a0 b0 噛 Ž t £ ha1 b1 . h*. If we apply the second parts to an element x, and denote y s ha1 b1 and z s SŽ y ., then this gets down to

Ý t Ž S Ž z 2 . x . z1 s Ý t Ž S Ž z . x 1 . x 2 , which is exactly Lemma 1.8 applied to H op cop. As a consequence of the above we obtain the infinite dimensional version of Theorem 1.7 ii.: COROLLARY 3.7. algebra, then

If H is co-Frobenius and A is a right H-comodule

A噛H * rat , End A噛H * ra t Ž H m A A噛H * ra t . ⭈ A噛H *rat rat s End H A Ž H m A. ⭈ H * .

MORPHISMS OF RELATIVE HOPF MODULES

565

Let us now describe, explicitly, the left H *-module structure on H m A obtained via the isomorphism ␪ of Lemma 3.6. This is given by h* ⭈ Ž h m a . s

Ý h* Ž S Ž h1 . gy1 . h 2 m a,

Ž 12 .

where h* g H *, h g H, a g A, and g is the distinguished grouplike element from Proposition 1.9. Indeed, we have

Ž 1噛h* . ␪ Ž h m a. s Ž 1噛h* . Ž Ý a0 噛t £ ha1 . s

Ý a0 噛 Ž h* £ a1 . Ž t £ ha2 . s Ý a0 噛h* Ž S Ž h1 . gy1 . t £ h 2 a1 s ␪ Ž Ý h* Ž S Ž h1 . gy1 . h 2 m a . , since we have, for all h, x g H

Ý h* Ž S Ž h1 . gy1 . t £ h 2 x s Ý Ž h* £ x 1 . Ž t £ hx 2 . . The last equality may be checked as follows. Apply both sides to an element w g H and denote y s xw to get

Ý h* Ž S Ž h1 . gy1 . t Ž h 2 y . s Ý h* Ž y1 . t Ž hy 2 . . Now denote u s SŽ gh. and t⬘ s t £ gy1 to obtain

Ý t⬘ Ž S Ž u1 . y . u 2 s Ý t⬘ Ž S Ž u . y 2 . y1 , which is exactly Lemma 1.8 applied to H op , for which t⬘ is still a left integral. This proves Ž12.. Remark 3.8. Corollary 3.7 should be compared to Theorem 2.1 iii., Ž .Ž noting that the right H *-modules End H A H m A the structure being given Ž . Ž Ž .. by Ž12.. and End H A m H as in 6 are isomorphic. A Indeed, if we denote by ␶ : A m H ª H m A the isomorphism from Remark 1.6, and if we define

␰ : End HA Ž A m H . ª End HA Ž H m A . ,

␰ Ž f . s ␶ f␶y1 ,

then, after a short computation, we find that

␰ Ž f ⭈ h* . s ␰ Ž f . ⭈ Ž g © h* . .

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MENINI AND RAIANU

Ž Since h* ¬ g © h* is an automorphism of H *, it follows that End H A Am Ž . H . , End H H m A as right H *-modules. A As we said, the above corollary provides a new proof for Proposition 3.1. COROLLARY 3.9.

If H is co-Frobenius and ArAco H is Galois, then A噛H * rat , End Ž A A co H . ⭈ H *rat .

Proof. Since the Weak Structure Theorem holds, we have that co H

End H A Ž H m A . , End Ž Ž H m A . A co H . . On the other hand, H m A , A m H via h m a ¬ Ýa0 m ha1 as right A-modules and right H-comodules Žas in Remark 1.6., and thus we have that Ž H m A. co H , Ž A m H . co H , A. The isomorphism in the statement then follows from Corollary 3.7. It remains to show that this gives another proof for Proposition 3.1. Indeed, the above isomorphism sends a噛h* to ␳ a噛h* , left multiplication by a噛h* restricted to the coinvariants of A噛H *rat . But Ž A噛H *rat . co H s A噛t, where t is a right integral of H. We compute thus Ž a噛h*.Ž b噛t . s Ýab0 噛Ž h* £ b1 . t s Ýab0 h*Ž b1 gy1 .噛t, where g is the distinguished group-like element from Proposition 1.9. Since this isomorphism is nothing but ␲ from Proposition 3.1, composed with the automorphism of A噛H * rat sending a噛h* to a噛gy1 © h*, it follows that ␲ is an isomorphism too. We move next to the second duality theorem for co-Frobenius Hopf algebra. The sketch of the proof is the following: Step I. Write Theorem 1.7 i. for the two-sided Hopf module H m A from Example 1.5: End H A Ž H m A . 噛H , ENDA Ž H m A . . Step II. Define the right H *-module structure on ENDAŽ H m A. as Ž . follows: If f g ENDAŽ H m A. and f s Ý f i e i for some f i g End H A HmA and e i g H, then f ⭈ h* s

Ý Ž f i ⭈ h*. ei .

Ž 13 .

Step III. Take the rational part on both sides of the isomorphism in Step I with respect to the H *-module structure in Step II, and use Corollary 3.7 to get

Ž A噛H *rat . 噛H , ENDA Ž H m A . ⭈ H *rat .

MORPHISMS OF RELATIVE HOPF MODULES

567

Step IV. Describe ENDAŽ H m A. ⭈ H * rat as a matrix ring. We are ready to prove the second duality theorem for co-Frobenius Hopf algebras. THEOREM 3.10 Žw20, Theorem 5.5x.. Let H be a co-Frobenius Hopf algebra and A a right H-comodule algebra. Then

Ž A噛H *rat . 噛H , MHf Ž A . , where MHf Ž A. denotes the ring of matrices with rows and columns indexed by a basis of H, and with only finitely many non-zero entries in A. Proof. It is clearly enough to prove the assertion in Step IV. above, i.e., that ENDA Ž H m A . ⭈ H * rat , MHf Ž A . , where the H *-module structure is the one given by Ž13.. We define another H *-module structure on ENDAŽ H m A. and show that the rational parts with respect to this module structure and the one given by Ž13. are the same. Define, for h* g H *, f g ENDAŽ H m A.

Ž f 䉺h* . Ž h m a. s Ý f Ž h* Ž S Ž h1 . gy1 . h 2 m a . . We check that ENDAŽ H m A. ⭈ H * rat s ENDAŽ H m A. 䉺H * rat . Let f g Ž . ENDAŽ H m A., f s Ý f j e j , f j g End H A H m A , e j g H. Then

Ž f ⭈ h* . Ž h m a. s Ý Ž Ž f j ⭈ h* . e j . Ž h m a. s

Ý Ž f j ⭈ h*.Ž e j h m a.

s

Ý h* Ž S Ž e j h1 . gy1 . f j Ž e j h 2 m a. Ž we used Ž 12. .

s

Ý ž S Ž ge j

s

Ý Ž f j e j . 䉺 ž S Ž ge j

1

2

1

2

gy1 . © h* Ž S Ž h1 . gy1 . Ž f j e j 2 . Ž h 2 m a .

/

1

gy1 . © h* Ž h m a . ,

/

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MENINI AND RAIANU

which shows that ENDAŽ H m A. ⭈ H *rat : ENDAŽ H m A. 䉺H *rat . ConŽ . versely, if f g ENDAŽ H m A., f s Ý f j e j , f j g End H A H m A , e j g H, then

Ž f 䉺h* . Ž h m a. s ž Ž Ý f j e j . 䉺h* / Ž h m a. s

Ý h*S Ž gh1 . f j Ž e j h 2 m a. s Ý h*S Ž gS Ž e j . e j h1 . f j Ž e j h 2 m a . 1

2

3

gy1 . ge j 2 h1 f j Ž e j3 h 2 m a .

s

Ý h*S ž S Ž ge j

s

Ý h* ž S Ž ge j h1 . S 2 Ž ge j

s

Ý ž S 2 Ž ge j

s

Ý ž f j ⭈ ž S 2 Ž ge j

s

ž Ý ž f ⭈ ž S Ž ge

1

/

2

1

gy1 . f j Ž e j3 h 2 m a .

/

gy1 . © h* Ž S Ž ge j 2 h1 . . f j Ž e j3 h 2 m a .

/

2

j

1

1

gy1 . © h* y1

j1 g

/ / Ž e h m a. . © h* / / e / Ž h m a. , j2

j2

showing that ENDAŽ H m A. 䉺H *rat : ENDAŽ H m A. ⭈ H * rat . Now write H s [EŽ S␭ ., where the S␭’s are simple right H-comodules, and take  h i 4 a basis in each EŽ S␭ .. Put them together and take  gy1 h i m 14 , which is an A-basis for H m A. Use this basis to view the elements of ENDAŽ H m A. as row finite matrices with entries in A. We show now that, under this identification, the elements of ENDAŽ H m A. 䉺H rat are represented by finite matrices. As in the proof of Corollary 3.3, we denote by p␭ the linear form on H equal to ␧ on EŽ S␭ . and 0 elsewhere. It is easy to see that for every f g End AŽ H m A., f 䉺 p␭ S is represented by a finite matrix for all ␭. Conversely, if f Ž gy1 h i m 1. / 0 and f Ž gy1 h j m 1. s 0 for all j / i, then clearly f g ENDAŽ H m A.. Moreover, if h i g EŽ S␭ ., then if h j g EŽ S␭ ., we have

Ž f 䉺 p␭ S . Ž gy1 h j m 1 . s Ý p␭ S ž S Ž ggy1 h j . / f Ž gy1 h j 1

s

Ý ␧ Ž h j . f Ž gy1 h j 1

2

2

m 1.

m 1.

s f Ž gy1 h j m 1 . . Since for h j f EŽ S␭ ., Ž f 䉺 p␭ S .Ž gy1 h j m 1. s f Ž gy1 h j m 1. s 0, we find that f s f 䉺 p␭ S g ENDAŽ H m A. 䉺H * rat , and the proof is complete. We end by giving some comments on the isomorphisms and the structures involved in the proof of the duality theorems.

MORPHISMS OF RELATIVE HOPF MODULES

569

Ž . Ž . Remark 3.11. w19, Lemma 12x shows that End H A H m A , Hom k H, A as vector spaces. Taking the embedding of A噛H * to 噛Ž H, A., and then mapping through Ulbrich’s isomorphism, we obtain the map

␲ ⬘: A噛H * ª End HA Ž H m A . , ␲ ⬘ Ž b噛h* . Ž h m a . s

Ý h 2 S Ž b1 . m h* Ž S Ž h1 . . b0 a.

It is not hard to check that ␲ ⬘ is actually a ring embedding. In fact, ␲ ⬘ is the embedding of A噛H * composed with the isomorphism ␤ from Theorem 2.1 i. Žsee Remark 2.4 1.. If ArAco H is Galois and H is co-Frobenius, then composing the restriction of ␲ ⬘ to A噛H *rat with the isomorphism Ž . Ž . End H A H m A , End A A co H , we get a¬am1 ¬

Ý S Ž a1 . m a0

¬

Ý S Ž a1 . 2 S Ž b1 . m h* Ž S Ž S Ž a1 . 1 . . b0 a0

s

Ý S Ž b1 a1 . m h* Ž a2 . b0 a0

¬

Ý h* Ž a2 . b0 a0 m S Ž b2 a2 . b1 a1 s Ý h* Ž a1 . ba0 m 1 ¬ Ý h* Ž a1 . ba0 s ␲ Ž b噛h* . Ž a . . This shows that A噛H * rat , 噛Ž H, A. ⭈ H * rat . Compare this to the remark, made in w2, p. 169x, that even if ArAco H is Galois, A噛H *rat may be properly contained in H * rat ⭈ Ž噛Ž H, A... Finally, we remark that if we replace, in the definition of ␪ from Lemma 3.6, the right integral with a left integral, and use this bijection to identify H m A with A噛H * rat , then the map ␲ ⬘ above is nothing else but right multiplication with elements from A噛H *. This explains the presence of the distinguished grouplike element in the definition of the module structure given by Ž12..

REFERENCES 1. E. Abe, ‘‘Hopf Algebras,’’ Cambridge Univ. Press, Cambridge, UK, 1977. 2. M. Beattie, S. Dascalescu, and ¸ S. Raianu, Galois Extensions for Co-Frobenius Hopf ˘ ˘ Algebras, J. Algebra 198 Ž1997., 164᎐183.

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