Journal of Algebra E D I T O R - I N - C H I E F :

Graham Highman M a t h e m a t i c a l I n s t i t u t e 2 4 - 2 9 , S t . G i l e s O x f o r d , E n g l a n d

EDITORIAL B O A R D :

Richard Brauer D e p a r t m e n t of M a t h e m a t i c s H a r v a r d U n i v e r s i t y 2 D i v i n i t y A v e n u e C a m b r i d g e , M a s s a c h u s e t t s 0 2 1 3 8

R. H . Bruck D e p a r t m e n t of M a t h e m a t i c s V a n V l e c k H a l l U n i v e r s i t y of W i s c o n s i n M a d i s o n , W i s c o n s i n 53706

D . A . Buchsbaum D e p a r t m e n t of M a t h e m a t i c s B r a n d e i s U n i v e r s i t y W a l t h a m , M a s s a c h u s e t t s 0 2 1 5 4

P. M . Cohn D e p a r t m e n t of M a t h e m a t i c s B e d f o r d C o l l e g e Regents P a r k L o n d o n N . W . 1 , E n g l a n d

J. Dieudonn6 U n i v e r s i t e D y A i x - M a r s e i l l e

F a c u l t e D e s Sciences P a r c V a l r o s e N i c e , F r a n c e

Walter Feit D e p a r t m e n t of M a t h e m a t i c s Y a l e U n i v e r s i t y B o x 2155 Y a l e S t a t i o n N e w Häven, C o n n e c t i c u t 0 6 5 2 0

A. Fröhlich D e p a r t m e n t of M a t h e m a t i c s K i n g ' s C o l l e g e L o n d o n , W . C. 2 , E n g l a n d

A. W . Goldie M a t h e m a t i c s D e p a r t m e n t T h e U n i v e r s i t y of Leeds

Leeds, E n g l a n d J. A . Green

M a t h e m a t i c s I n s t i t u t e U n i v e r s i t y of W a r w i c k C o v e n t r y , E n g l a n d

Marshall Hall , Jr. S l o a n L a b o r a t o r y of M a t h e m a t i c s a n d P h y s i c s C a l i f o r n i a I n s t i t u t e of T e c h n o l o g y P a s a d e n a , C a l i f o r n i a 9 1 1 0 9

I. N . Herstein D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of C h i c a g o C h i c a g o , I l l i n o i s 6 0 6 3 7

B. Huppert M a t h e m a t i s c h e s I n s t i t u t der Universität Tübingen, G e r m a n y

Nathan Jacobson D e p a r t m e n t of M a t h e m a t i c s Y a l e U n i v e r s i t y

B o x 2155 Y a l e S t a t i o n N e w Häven, C o n n e c t i c u t 0 6 5 2 0

E . Kleinfeld D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of I o w a

I o w a C i t y , I o w a 5 2 2 4 0 Saunders MacLane

D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of C h i c a g o C h i c a g o , I l l i n o i s 6 0 6 3 7

G . B. Preston D e p a r t m e n t of M a t h e m a t i c s M o n a s h U n i v e r s i t y C l a y t o n V i c t o r i a 3168 A u s t r a l i a

H . J. Ryser D e p a r t m e n t of M a t h e m a t i c s C a l i f o r n i a I n s t i t u t e of T e c h n o l o g y P a s a d e n a , C a l i f o r n i a 9 1 1 0 9

J. Tits M a t h e m a t i s c h e s I n s t i t u t der Universität W e g e l e r s t r a s s e 10 B o n n , G e r m a n y

Guido Zappa I s t i t u t o Matemätico « U l i s s e Dini» Universitä d e g l i S t u d i i V i a l e M o r g a n i , 6 7 1 A

F i r e n z e , I t a l y

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© 1971 by Academic Press, Inc. P r i n t e d i n Bruges, B e l g i u m , by t h e S t C a t h e r i n e Press, L t d .

JOURNAL OF

Algebra

J E D I T O R - I N - C H I E F : Graham Higman

J E D I T O R I A L B O A R D :

R. H . Bruck D . A. Buchsbaum P. Cohn J. Dieudonne Walter Feit A. Fröhlich A. W. Goldie J. A. Green Marshall Hall, Jr.

I. N . Herstein B. Huppert Nathan Jacobson E. Kleinfeld Saunders MacLane G . B. Preston H . J. Ryser J. Tits Guido Zappa

Volume 18 • 1971

ACADEMIC PRESS New York and London

C O P Y R I G H T © 1971, B Y A C A D E M I C PRESS, INC.

A L L R I G H T S RESERVED

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Bay^rhche Staat.;. othek München

P r i n t e d by t h e S t C a t h e r i n e Press, L t d . , Bruges, B e l g i u m

CONTENTS OF VOLUME 18

NUMBER 1, M A Y 1971

JEAN CARCANAGUE. Ideaux bilateres d'un anneau de polynomes non commutatifs sur un corps 1

M . A N D R E . Hopf Algebras with Divided Powers 19 S. M . GERSTEN. On Mayer-Vietoris Functors and Algebraic i£-Theory 51 A. R. M A G I D . Galois Groupoids 89 K E V I N M C C R I M M O N . A Characterization of the Radical of a Jordan

Algebra 103 MITSUHIRO T A K E U C H I . A Simple Proof of Gabriel and Popesco's

Theorem 112 M I C H A E L ASCHBACHER. Doubly Transitive Groups in Which the

Stabilizer of Two Points is Abelian 114 ANDREAS DRESS. The Ring of Monomial Representations I. Structure

Theory 137

NUMBER 2, J U N E 1971

M . J. K A L L A H E R A N D T . G . OSTROM. Fixed Point Free Linear Groups, Rank Three Planes, and Bol Quasifields 159

M . SLATER. Alternative Rings with D . C . C . , III 179 STANLEY E . P A Y N E . Nonisomorphic Generalized Quadrangles . . . 201 NICOLAE POPESCU. Le spectre ä gauche d'un anneau 213 B. A. F. WEHRFRITZ. Remarks on Centrality and Cyclicity in Linear

Groups 229 C. B. T H O M A S . Frobenius Reciprocity of Hermitian Forms . . . . 237 ZVONIMIR JANKO. The Nonexistence of a Certain Type of Finite

Simple Group 245 J. T . ARNOLD A N D J. W . BREWER. On Fiat Overrings, Ideal Trans-

forms and Generalized Transforms of a Commutative Ring . . 254 PETER CRAWLEY AND A L F R E D W . H A L E S . The Structure of Abelian

^-Groups Given by Certain Presentations. II 264 L A N C E W . S M A L L . Localization in PI-Rings 269 SIGURD ELLIGER. Interdirekte Summen von Moduln 271 ERWIN KLEINFELD. Generalization of Alternative Rings, 1 304 ERWIN KLEINFELD. Generalization of Alternative Rings, II . . . . 326 GUNTER F. PILZ. Direct Sums of Ordered Near-Rings 340

N U M B E R 3, J U L Y 1971

D O R I N POPESCU. Categories de Faisceaux 343

J. A. GERHARD. The Number of Polynomials of Idempotent Semi-groups 366

JAMES M C C O O L AND A L F R E D PIETROWSKI. On Free Products with

Amalgamation of Two Infinite Cyclic Groups 377 M A R I O FIORENTINI. On Relative Regulär Sequences 384 GABRIEL SABBAGH. Embedding Problems for Modules and Rings

with Application to Model-Companions 390 W I L L I A M R. N I C O . Homological Dimension in Semigroup Algebras 4 0 4 R. H . D Y E . On the Conjugacy Classes of Involutions of the Simple

Orthogonal Groups over Perfect Fields of Characteristic Two . 414 G I O V A N N I ZACHER. Determination of Locally Finite Groups with

Duals 426 P A U L C A M I O N , L . S. L E V Y , AND H . B. M A N N . Linear Equations over

a Commutative Ring 432 C E M A L K O C . On a Generalization of ClifTord Algebras 474 J O H N R. D U R B I N AND M E R R Y M C D O N A L D . Groups with a Character­

istic Cyclic Series 453 Earl J. T A F T . On the Splitting of Hopf Algebra Modules 461 A R O N SIMIS. Projective Modules of Certain Rings and the Existence

of Cyclic Basis 468 M A R T H A S M I T H . Group Algebras 477

N U M B E R 4, A U G U S T 1971

M . OJANGUREN AND R. SRIDHARAN. Cancellation of Azumaya Algebras 501 P A U L C H A B O T . Some Sylow 2-Groups W^hich Cannot Occur in Simple

Groups 506 K A R L E G I L AUBERT. Additive Ideal Systems 511

A. D O N E D D U . Sur les extensions quadratiques des corps non com-mutatifs 529

J. C. L E N N O X . On a Centrality Property of Finitely Generated Torsion Free Soluble Groups 541

D. B. C O L E M A N AND JOEL C U N N I N G H A M . Harrison's Witt Ring of a

Commutative Ring 549 K E V I N M C C R I M M O N . A Characterization of the Jacobson-Smiley

Radical 565 C. J. G R A D D O N . J^-Reducers in Finite Soluble Groups 574 BODO PAREIGIS. When Hopf Algebras Are Frobenius Algebras . . 588 B. L . OSOFSKY. On Twisted Polynomial Rings 597 S. B. C O N L O N . A Short Communication 608 A U T H O R INDEX 609

JOURNAL OF ALGEBRA 18, 588~596 (1971)

When Hopf Algebras Are Frobenius Algebras

BODO PAREIGIS

M a t h e m a t i s c h e s I n s t i t u t der Universität München 8 München J3, Schellingstrasse 2-8, G e r m a n y

C o m m u n i c a t e d by A . Fröhlich

Received August 19, 1970

R. Larson and M . Sweedler recently proved that for free finitely generated Hopf algebras H over a principal ideal domain R the following are equivalent: (a) H has an antipode and (b) H has a nonsingular left integral. In this paper 1 give a generalization of this result which needs only a minor restriction, which, for example, always holds if p i c ( R ) — 0 for the base ring R . A finitely generated projective Hopf algebra H over R has an antipode if and only if H is a Frobenius algebra with a Frobenius homomorphism ifi such that 2 h ( i ) t/j(h(2)) = tp(h) - 1 for all h e H . We also show that the antipode is bijective and that the ideal of left integrals is a free rank 1, i?-direct summand of H .

L In Ref. [4], Larson and Sweedler proved the equivalence for a finite-dimensional Hopf algebra over a principal ideal domain to have a (necessarily unique) antipode and to have a nonsingular left integral. It is easy to see that this result implies that a finite-dimensional Hopf algebra over a principal ideal domain is a Frobenius algebra, which generalizes the well-known fact that a group ring of a finite group is Frobenius as well as the result of Berkson [1], that the restricted universal enveloping algebra of a finite-dimensional restricted Lie algebra is Frobenius. This result has consequences with respect to a cohomology theory of Hopf algebras which will be exhibited in a subsequent paper.

In this paper we want to generalize the main result of [4] to arbitrary commutative rings R and finitely generated projective Hopf algebras H . We need only a slight restriction on H or on R y viz., pic(i?) = 0 to get the equivalence between the existence of an antipode and the fact that H is a Frobenius algebra with a Frobenius homomorphism I/J, such that Z(Ä) Ä(I)0(A<2>) = ' 1 f o r a 1 1 he H, where £<Ä) Ä ( i ) (X) Ä< 2 ) = A(h) is the

Sweedler notation. We do not know whether the imposed restrictions on R or H are necessary for the above result.1 In this context we also prove that the

1 See footnote at the end.

588

WHEN HOPF ALGEBRAS ARE FROBENIUS ALGEBRAS 589

antipode of a finitely generated projective Hopf algebra is bijective. This holds without any further restrictions.

Since integrals are also of interest in this general Situation, we shall prove that in a Hopf algebra H which is Frobenius with a Frobenius homomorphism I/J such that Ä(i)0(Ä(2)) = ${h) ' 1 for all he H the two-sided H ideal of left integrals is an iWree rank 1 i?-direct summand of H .

2. Let R be a commutative ring (associative with unit). All modules are assumed to be unitary R modules. All algebras are assumed to be associative R algebras with unit.

A c o a l g e b r a C is a module C together with homomorphisms A : C —> C ® C (the tensor product is taken over R ) y the diagonal, and e : C —> R y the counit or augmentation, such that

c — ^ — * c ® c

c®c^c®c®c commute where we identify C ® R , C, and R ® C. We adopt the Sweedler notation A ( c ) = Y , ( c ) c ( i ) ® c ( 2 ) a s explained in [7, p. 10].

A H o p f a l g e b r a H is an algebra H with structure maps [x : H ® H —• H and r) : R - + H , which is also a coalgebra with structure maps A : H ^ H ® H and € : H - + R such that A and e are algebra homomorphisms. As in [7, Proposition 3.1.1] one shows that fx and rj are coalgebra homomorphisms.

Let C be a coalgebra. A C r i g h t comodule is a module M together with a homomorphism ^ : M —• M ® C such that

M * >M®C M^M®C

M ® C ̂ > M ® C ® C M

commute, where we identify M and M (x) R . Here again we use the Sweedler notation x i m ) = m ( o ) ® m ( i ) • Observe that the ^( '̂s for i ^ 1 are elements in C, whereas the W(0)'s are in M .

Let // be a Hopf algebra. An H r i g h t H o p f m o d u l e is an H right module M which is also an H right comodule such that

x ( m h ) = £ m i 0 ) h { l ) ® m ( l ) h { 2 ) . ( m ) , ( h )

590 PAREIGIS

Let H be a Hopf algebra. Then h o m R ( H , H ) is an associative R algebra with unit 77c, if we define the multiplication by f * g :.== t x ( f ( x ) g ) A , i.e.,

f * g { h ) = Z(A)/ (^(I))#(2)) 17 > P- ^O, exercise 1]. The a n t i p o d e S of H is (if it exists) the two-sided inverse of the identity \ H of H in homÄ(//,

3* L E M M A 1. L e t A , B , a n d C be R m o d u l e s . I f C is finitely

generated a n d p r o j e c t i v e , t h e n X : hom(^4, B ® C ) —> hom(C* ® A , B ) w i t h

Kf)ic* ® a ) 0 ® c * ) f ( a ) " a n i s o m o r p h i s m .

P r o o f . X defines a natural transformation

hom(^, B ® — ) - + h o m ( - * ® A , B ) ,

which is an isomorphism for C — R . By [6, 4.11. Lemma 2] the above lemma holds.

PROPOSITION 1. L e t C be a f i n i t e l y generated p r o j e c t i v e R c o a l g e b r a a n d M be a n R m o d u l e . T h e n x M —>• M ® C defines a C r i g h t comodule if a n d o n l y if X ( x ) : C * ® M —• M defines a C* left m o d u l e , w h e r e X is defined as i n L e m m a 1.

P r o o f . C * is an R algebra by c*d*(c) = X(c) c*( cd)) ^*(^(2))- Assume X : M —> M ® C defines a C right comodule. Then

(Cl*c2*)m = £ ™<o)(*i*c2*0«<i)))

i m )

i m )

= Z C1*W(0)(^2*(^(1)))

= c^{c^m)

and

^ = Z m ( o A m d ) ) = m > (m)

so M is a C * left module. Now let M be a C * left module. The natural transformation

p : M ( g ) C ® D - > hom(C* ® Z>* M )

defined by /)(/« ® c ® d)(c* ® d*) = c*(c) d * { d ) m is an isomorphism for finitely generated projective modules C and £>, since it is an isomorphism for C = D = R [6, 4.11. Lemma 2]. So M ® C ® C hom(C* ® C * M ) for our finitely generated projective coalgebra C.

W H E N HOPF ALGEBRAS ARE FROBENIUS ALGEBRAS 591

Let x = ((x ® l) x - (1 ® J)x)(m) e M ® C ® C. Then ® ' 2*) =

c i * ( c 2 * m ) ~ ( c i*^2*) m = 0 for all C!*, c 2* e C * , so x = 0, i.e., x is coasso-

ciative. Furthermore, we have (1 ® <0x(m) = S(m) m ( o ) e ( m ( i ) ) = em = m , since € is the unit element in C * . So M is a C right comodule.

PROPOSITION 2. Le£ H be a finitely generated p r o j e c t i v e H o p f a l g e b r a w i t h a n t i p o d e S. Then H * is a n H r i g h t H o p f m o d u l e .

Proof. H * is an H * left module, so it is an H right comodule. We have

for g*9 h* e H * , and h e H and the comodule map x • H * —> H * ® H with

X(**) =Z<»> *<*o)® *(*!,-

£*A* = X (1)

and

(A) ( h * )

H * is also an H right module by A* • h = S{h) o A*, where (A o A*)(a) = h * ( a h ) for all fle£ For g*, A* e / / * , and a, 6 6 ff and x : ff* - * #* ® H hom(//*, //*), we have

X(A* • «)(**)(*) = ( S * ( h * • a ) ) { b )

(b)

= I |-*(*<D«(«ö)))A*(*ö)S(«<i>)) (a)(&)

= I («(3)^*)(*(i)5(«tt)))A*(*(«)S(«ü))) (a ) (b)

= Z ( ( % ) » l ^ * ) ( 4 % i ) ) ) (a)

= 1 ( ( ( « ( 2 ) »**)**) '«(DK*) (a)

= Z ((A(o)(ö(2) °^*)(A(*i))) ' *<!>)(*) ( a ) ( Ä * )

= Z ((*(*) ' fld))(fl(2) °<§r*)(A(*i)))(^) ( a ) ( A * )

= Z ((Ä*o) ' fl(i)k*(A*i)«(2)))W-( 0 ) ( Ä * )

592 PAREIGIS

This implies x(A* ' )̂ = £<a)U*) hfo) ' a d ) ® h * i ) a ( 2 ) which proves the proposition.

Let M and AT be ff right Hopf modules over a Hopf algebra ff with antipode S. We define an R module P ( M ) = { m e M \ x { m ) = m ® 1}. L e t / : M - > N be a module and comodule homomorphism. We define P ( f ) as restriction of/

to P ( M ) . Then # ( / ) ( « ) ) = X(/(m)) = ( / ® \ ) X { m ) = / ( m ) ® 1 € P(iV). Obviously P is a functor from the category of ff right Hopf modules to the category of R modules.

L E M M A 2. L e t M be a H o p f m o d u l e o v e r a H o p f a l g e b r a H with a n t i p o d e S. T h e n M ^ P ( M ) ® H as r i g h t H o p f m o d u l e s . F u r t h e r m o r e , P ( M ) is a n R d i r e c t s u m m a n d of M .

P r o o f . The natural injection P ( M ) —> M has a retraction M S W I H -

Now OL : P ( M ) ® ff-> Af and ß : P ( M ) ® H y defined by oc(m ® h ) = m h and ß(m) = Y*(m) m ( o ) S ( m d ) ) ® m ( 2 ) » a r e inverse R homomorphisms of each other. OL being an H module homomorphism, ß is an H module homo­morphism. Furthermore, xß(m) = (ß ® A)x(w) implies that ß and, conse-quently, also a is a comodule homomorphism.

PROPOSITION 3. L e t H be a finitely g e n e r a t e d p r o j e c t i v e H o p f a l g e b r a zvith a n t i p o d e S. T h e n P ( H * ) is a finitely g e n e r a t e d p r o j e c t i v e r a n k 1 R m o d u l e .

P r o o f . For each prime ideal p in R the isomorphism ff* ^ P(ff*) ® ff implies ffp* ^ P(ff*) p ® Ä ffp . Now ffp* ^ ffp as free finite-dimensional P p modules. So dim(P(ff*)p) = 1 for all prime ideals p in R . Thus, P(ff*) has rank 1. Furthermore, it is finitely generated projective as a direct summand

4* An R algebra ff is a F r o b e n i u s a l g e b r a if ff is a finitely generated projective R module and if there is an isomorphism 0 : H H ^ H H * , where we consider ff* as an ff left module via h o h * ( a ) = h * ( a h ) for all A e ff, A* G ff* [2]. 3> is called a F r o b e n i u s i s o m o r p h i s m . & ( \ ) = : 0 is a free generator of ff* as an ff left module called F r o b e n i u s h o m o m o r p h i s m . By [3, p. 220, (4)], 0 is also a free generator of ff* as an ff right module, where A* o : == h * { h a ) for all A* e ff*, and h y A G ff. This is a consequence of the proof that the

Z(™) m ( 0 ) 4 S ( ™ ( l ) ) e P ( M ) f o r

= Z w(o)S("*(i>) ® 1.

of ff*.

WHEN HOPF ALGEBRAS ARE FROBENIUS ALGEBRAS 593

conditions for a Frobenius algebra are independent of the choice of the sides, i.e., H H ^ H H * implies H H ^ H H * (if ff is finitely generated and projective). \jj is unique up to multiplication with an invertible element of ff [5, Satz 1].

If a Frobenius algebra ff has an augmentation e, then the element TV with A~ o i/j — e is called a left n o r m of ff. A left norm N is also unique up to multiplication with an invertible element of ff from the right side. We have { ( h N ) o i/j)(a) = ifj{ahN) = ( N ° i/j)(ah) = € ( a h ) = e ( a ) e ( h ) = ( N o </0(a)e(A) = ( e ( h ) N o $ ) { a ) for all a, A e ff. This implies

AAf = e ( h ) N for all A e ff.

A n element # e ff of an augmented algebra ff with A<z = e(A)a for all A e ff is called a /e// i n t e g r a l of the augmented algebra H . So a left norm is in particular a left integral.

PROPOSITION 4. L e t H be a finitely g e n e r a t e d p r o j e c t i v e H o p f a l g e b r a with a n t i p o d e S. T h e n S is b i j e c t i v e .

P r o o f . S is injective: Let 6 : H * ^ P ( H * ) ® H be the isomorphism of H right modules defined by Proposition 2 and Lemma 2. Let £(A) — 0 and p®aeP(H*) ® H y then

d~\p ® a h ) = 6-\p ® a ) • A = 5(A) o ® a) = 0.

So p ® a h = 0 for all >̂ ® a e P(//*) ® ff and the P endomorphism of P(H*) ® ff, defined by the multiplication with A, is the zero endomorphism.

Let m be a maximal ideal of R . If we localize with respect to m, we get an R m Hopf algebra ffm with antipode Sm . Furthermore, (ff*) m ^ (#m)*> where the second * means dualization with respect to R m . Also, P { M ) m ^ P ( M m ) for an ff Hopf module M since P ( M ) is the kernel of x — I M ® V and localization is an exact functor. So (P(ff *) ® ff)m ^ P((ffm)*) ® / ? m -^ m . As in Proposition 3, P((ffm)*) is a free P m module on one generator so P((ffm)*) ® R ffm ^ ffm . The multiplication of ffm with h on the right is a zero morphism for all maximal ideals m of R . So multiplication of ff with h on the right must be zero which implies A — 0. So S is injective.

S is surjective: Let 0 —>- ff —> ff —> £) — > 0 be an P exact sequence. Then for all maximal ideals m Q P, we have 0 —> ffm -> ffm -> Q m — > 0 is P m exact. So ffm/mffni ffm/mffTU —• ö m / m ö m 0 is P m / m P m exact, where T is the antipode of the P m / m P m Hopf algebra ffm/mffm . Since this Hopf algebra is finitely generated, T is injective so that T is bijective and ö m / m ö m = 0* Since g and ^ m are finitely generated, £ ) m = 0 for all maximal ideals m C R . So Q = 0 and 5 is surjective.

T H E O R E M 1. L e t ff be a finitely g e n e r a t e d p r o j e c t i v e H o p f a l g e b r a with

594 PAREIGIS

a n t i p o d e S. L e t P(ff *) _ i R as R modules. T h e n ff is a F r o b e n i u s a l g e b r a w i t h a F r o b e n i u s homomorphism $ such that

Z*(i#ti))=r<*)l M all h e H . ( h )

P r o o f . By Proposition 2 and Lemma 2 there exists an isomorphism ff* P(ff*) ® ff of ff right modules where (A* • h ) ( a ) = h * ( a S ( h ) ) or h * - h = S ( h ) o A* is the definition of the module structure on ff*. Since P(ff *) ^ P, let 0 : ff* _^ ff be an isomorphism of ff right modules. Define 0 = 0-i<>-i : ff ^ ff*, where we use Proposition 4. Then 0 ( h a ) = e-*S-\ha) = fl-^S-^S-^A)) = fl-i(^a)) • S-X(A) = *(a) • *S_1(A) = A o 0 ( a ) > so ff is a Frobenius algebra. Before we prove the formula on the Frobenius homomorphism we prove the following:

L E M M A 3. L e t ff be a finitely g e n e r a t e d p r o j e c t i v e H o p f a l g e b r a with a n t i p o d e S with P(ff*) P. L e t 0 : ff _< ff* be t h e F r o b e n i u s isomorphism c o n s t r u c t e d above a n d let ip = 0 ( 1 ) be a F r o b e n i u s h o m o m o r p h i s m . T h e n ift e P(ff*) a n d ift is a left i n t e g r a l i n ff*.

Proof. 0 ( \ ) = ifj implies S 6 ( i p ) = 1 and also 0(</>) = 1. 0 is a comodule homomorphism so E 0(0<o>) ® 0<D = (0 ® l)xW0 = ^WO) = ^0) = 1 ® 1 = Q{xjj) ® 1. 0 ® 1 being an isomorphism this implies x(<A) = <A ® 1 soi/reP(ff*). Now

A* (ZA ( 1 ) ^(A ( 2 ) ) ) =EA*(A ( l ) )0 (A ( 2 ) )

= ( * * # ( * )

= Itoo»(*) **(*&>) (by(i)) (<A)

= 0(A)A*(1)

= A*(0(A)1) for all A G ff and A* G ff*. This implies

Z A ( 1 ) ^(A ( 2 ) ) = </<A)l for all A G ff. ( h )

This also means (A*</r)(A) = A*(l)</f(A), which proves that 0 is a left integral in ff*.

COROLLARY 1. L e t R be a c o m m u t a t i v e r i n g with pic(P) = 0. T h e n e a c h finitely g e n e r a t e d p r o j e c t i v e H o p f a l g e b r a with a n t i p o d e is a F r o b e n i u s a l g e b r a .

W H E N HOPF ALGEBRAS ARE FROBENIUS ALGEBRAS 595

5* L E M M A 4. L e t P be a finitely generated p r o j e c t i v e R m o d u l e and l e t f : P —• P be a n e p i m o r p h i s m . Then f is a n i s o m o r p h i s m .

Proof. The sequences

0 - + A - > P - U P - > 0 ,

0 —> A m —> P m — > P m —• 0,

0 -> A m J Y x A m - > P m / m P m -4 P m / m P m -> 0

are all split exact. But / is an isomorphism by reasons of dimension so A m l m A m = 0. A m is finitely generated so that A m = 0 for all maximal ideals m C P, so that A — 0 and/is an isomorphism.

T H E O R E M 2. Le£ H be a H o p f a l g e b r a and a Frobenius a l g e b r a w i t h a Frobenius h o m o m o r p h i s m ift such t h a t £ (A) A(1)^(A(2)) = ^(A)l /or #// h e H . Then H has a n a n t i p o d e S.

Proof. We define S : H - > H by S(A) = Z ( N ) N^hN®), where iVis a left norm, i.e., ATo $ = € . Then

Z h ( D S ( h ( 2 ) ) = I h(i)N(i)*l>(h(2)N{2)) <A) ( h ) { N )

= < h ) l

= V < ( h )

for all he F l b y definition of N . So 1 * S = 77c

Now homÄ(ff, ff) is an associative P algebra with multiplication f * g = H>(f®g)A. HomÄ(ff, ff) is also finitely generated and projective since ff is. The map

hom^ff, ff)9/H>/*Se hom^ff, ff)

is an P epimorphism for (/* 1) * S = / * (1 * S) = /* rje = / . By the preceeding lemma, — * 5 is an isomorphism with inverse map — * 1. So S * 1 = * $ * 1 = T)€, i.e., *S is an antipode.

T H E O R E M 3. L e t ff be a H o p f a l g e b r a and a Frobenius a l g e b r a . Then t h e t w o - s i d e d i d e a l of left i n t e g r a l s he H ( w i t h a h = e ( a ) h f o r a l l a e H ) is a free R d i r e c t summand of H of r a n k 1 w i t h basis { N } y a left norm of ff.

Proof. Let h be a left integral, then i/j(ah) = i/j(e(a)h) = ifj(h)€(a) = W W a N ) = <l>(aij>(h)N) for all a e H , so that h o 0 = 0(Ä)AT o $ or A = flA)iV.

596 PAREIGIS

Furthermore, € is an epimorphism since er) = l R . So e* : R —> ff* is a

monomorphism. Also p : R—> ff* ^ ff is a monomorphism. But e*(r)(A) —

re*(l)(A) = re(h) = i / j ( h r N ) , so that p(r) = r N . Since p is injective,

P 9 r e ff is injective. Thus, RN is free of rank 1.

Finally, ff 9 A h -> i f j ( h ) N e RN is a retraction for the inclusion i W C ff,

in which case i W splits off as a direct summand.

By Theorem 1, I/J is a left integral and H 3 h\-> h o ifj e ff* as well as

H3h\->i/johe ff* are isomorphisms, so that 0 is a nonsingular left integral

in ff*. Now let pic(JR) = 0. If ff is a finitely generated projective Hopf

algebra with an antipode, then so is ff*. Furthermore, ff** —< ff as Hopf

algebras with antipode and also ff has a nonsingular left integral. This

implies the main result of (see remark on page 588) [4] for the case that R

is a principal ideal domain, for then pic(P) = 0 holds.

N o t e added i n proof. P . Gabriel gave an example of a finitely generated projective Hopf algebra with antipode which is not a Frobenius algebra, showing that pic(i?) = 0 is a necessary condition for Corollary 1. We have however the following

T h e o r e m . Let H b e a finitely generated projective Hopf algebra over a commutative ring R . H has an antipode if and only if H is a quasi Frobenius algebra and ff* has a finite set of generators ipi i/rn as an H left module ( h ° h * ( a ) = h * ( a h ) ) such that for all k = 1,..., n

£ A(i>lMA<.>) = U h ) \ for all h e ff.

Here a quasi Frobenius algebra is taken in the sense of B. Müller, Quasi-Frobenius-Erweiterungen, M a t h . Z e i t s c h r . 85, 345-368 (1964). This theorem guarantees that the cohomology theory of Hopf algebras can be developed without the restrictive condition pic(i?) = 0.

REFERENCES

1. A . BERKSON, Proc. Amer. Math. Soc. 15 (1964), 14, 15.

2. F. K A S C H , Sitzungsber. Heidelberg. Akad. Wiss. (1960/61), 89-109 .

3. F. K A S C H , Math. Z. 77 (1961), 219-227.

4. R . G . L A R S O N AND M . E. SWEEDLER, Amer. J. Math. 91 (1969), 75-94.

5. B. PAREIGIS, Ann. Math. 153 (1964), 1-13.

6. B. PAREIGIS, "Kategorien und Funktoren," Teubner, Stuttgart, 1969. 7. M . E. SWEEDLER, "Hopf Algebras," W. A. Benjamin Inc., New York, 1969.