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COHOMOLOGY OF ALGEBRAS OVER HOPF ALGEBRASBY

MOSS EISENBERG SWEEDLER

We present a cohomology theory for algebras which are modules over a givenHopf algebra. The algebras are commutative, the Hopf algebra cocommutative andunder the module action the underlying coalgebra of the Hopf algebra " respects "the multiplication and unit in the algebras.

The cohomology is defined by means of an explicit complex. Whenever C is acoalgebra and A an algebra Horn (C, A) has a certain natural algebra structure.The groups in our complex consist of the multiplicative group of invertible ele-ments in Horn (C, A) where C is the underlying coalgebra of the Hopf algebratensored with itself a number of times. The complex arises as the chain complexassociated with a semi-cosimplicial complex whose face operators are inducedby maps of the form (g)'l+1 H-+ (g)n H, h0 - - h0 - ii + l hn. Under Horn (*, A) these maps become coface operators.

Familiar examples of Hopf algebras are the group algebra kG of the group Gand the universal enveloping algebra UL of the Lie algebra L. If the commutativealgebra A is an admissible &G-module then the Hopf algebra cohomology H'(kG, A)is canonically isomorphic to H'(G, A'), the group cohomology of G in the multi-plicative group of invertible elements of A. If A is an admissible UL-moduie, thenfor /> 1 the Hopf cohomology H\UL, A) is canonically isomorphic to H*(L, A +),the Lie cohomology of L in the underlying vector space of A. If A has enoughnilpotent elements then H1(UL, A)^H1(L, A+). All the preceding isomorphismsarise from isomorphisms on the complex level.

We consider relative cohomology and show how an injective (surjective) algebramorphism A -* can give rise to a long exact cohomology sequence relating//*(//, A), H*(H, ) and the relative cohomology groups. Using relative co-homology one can completely recover the usual group (Lie) cohomology theoryfor modules. One considers the module as a trivial algebra and adjoins a unit toform A. The injection of the ground field into A gives rise to relative cohomologygroups which are precisely the classical cohomology groups of the group (Liealgebra) with coefficients in the module.

The last comparison is that of H*(H, A) with the Amitsur cohomology of A.First we show that there always is a natural transformation from the Amitsurcohomology of A to //*(//, A). We then specialize to the case that A is a finitefield extension and give conditions on A and H which imply the natural trans-formation is an isomorphism. We also show that many field extensions A can

Received by the editors April 11, 1967.

205

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206 M. E. SWEEDLER [August

satisfy these conditions ; such as, separable normal extensions, purely inseparableextensions which are the tensor product of primitively generated extensions, etc.

The last half of the paper is devoted to studying extensions. An extension of analgebra by a Hopf algebra is itself an algebra and has further properties. Wedescribe equivalence and product of extensions and arrive at the usual result thatH2(H, A) is isomorphic to the group of equivalence classes of extensions. Part ofthe theory involves the definition of certain algebras which we call crossed products.They generalize existing instances of crossed products.

The extension theory is particularly interesting for field extensions A, where Aand 77 satisfy certain conditions. (These conditions imply that the Amitsur co-homology is isomorphic to 77*(77, A).) When the conditions are satisfied anycrossed product of A by 77 is a central simple algebra (over the ground field) withsplitting field A. This leads to an isomorphism between 772(77, A) and the subgroupof the Brauer group over A consisting of classes split by A. One of the key resultsneeded to give the isomorphism is the existence of "inner" coalgebra actions. Thisresult generalizes known results about inner automorphisms and derivations.

1. Preliminaries. All vector spaces are over the ground field A, which hascharacteristic/;. A coalgebra is a vector space C equipped with maps A : C -* C Cand e: C^ k satisfying (7 $ A)A=(A 7)A and (e 7)A = 7=(7 *)A. The firstidentity is called coassociativity, A is called the diagonal map and e is called thecounit or augmentation. If C is a coalgebra the structure morphisms pertaining toC may be denoted Ac and ec, to avoid confusion. Similarly, if A is an algebra thestructure morphisms may be denoted mA: A A -> A and pA: A -* A, (X-+ XI).When no confusion can arise we omit the subscripts C and A.

For ceCwe write 2(o c(1) c(2) to denote A(c), 2(o c V; let Lo/fon, , c(B)) denote /(2(c) c c(B)). In thisnotation the identity relating e and A becomes 2(o e(cw)c(2) = c='2.ic) ca)e(ci2)).

If V and W are vector spaces we use f to denote the twist map t : V W->W V, v w-w v. A coalgebra C is called cocommutative if fA=A or forall ceC, 2(o C(n).

If C and D are coalgebras then C D is a coalgebra where AC(8D = (7(g) t 7)(ACAD) and ecD = ec D. Thus AcglD(c ) = 2(o,i> foi da)) (C(2) (2)).

A Hopf algebra 77 is an algebra and a coalgebra where the coalgebra structuremorphisms AH, eH are algebra homomorphisms. Thus for example, AH(g/t) =2(i).() ITdAn g(2)(2), g, 77. If F is a left 77-module then F F is naturallya left 77 77-module. By pull back along A:77->7777, F V becomes a left77-module. Specifically, h iv v) = 2i) u> w(2> The augmentatione: H->k gives A the structure of a left 77-module.

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1968] COHOMOLOGY OF ALGEBRAS OVER HOPF ALGEBRAS 207

Definition. An algebra A which is a left //-module is called a (left) H-modulealgebra if mA and pA are //-module morphisms. A coalgebra C which is a left//-module is called an H-module coalgebra if Ac and ec are //-module morphisms.(A

208 M. E. SWEEDLER [August

Definition. If A is an algebra A" denotes the multiplicative group of invertible-regular-elements of A. If C is a coalgebra and A an algebra Reg (C, A) denotesHorn (C, A)r.

Observe Reg (C, A) is a multiplicative abelian group when C is cocommutativeand A is commutative.

Suppose C is an //-module coalgebra and A an //-module algebra. HomH (C, A)denotes the //-module morphisms from C to A. Clearly p.Aec e HomH (C, A).Suppose fige HomH (C, A) then

h [f* g(c)] = 2h~ L/X%>)s(c)] = 2 (*0)/(ca>)XA /(*))(c) (ft).(c)

= 2 /(a)-C(1))g(A(2)-c(a))=/*g(A-c),(W.(c)

for he H, ce C. Thus HomH (C, A) is a subalgebra of Horn (C, y4) and we defineRegH (C, /f) to be Homw (C, /i)r. This is the subgroup of Reg (C, A)D consistingof all //-module morphisms.

2. Definition of the cohomology. Throughout the paper H will denote a co-commutative Hopf algebra; i.e., where the underlying coalgebra is cocommutativeand A will denote a commutative algebra.

We form a semisimplicial complex [7, p. 55], [8, p. 56], whose objects are the//-module coalgebras {(g)5 + 1 //}0g0 of Example 1.2. The object of ^-degree is(g)a + 1 H for q=0, 1,.... The face operators are given by d, : (g)+ x H -* (g) H,(*o xq) - (*o *f*i+i x) for /=0,..., q-\ and0,: (g)'+1 //-> (g)5 //, (* xq.x)e(xg).

The degeneracy operators are given by St'. (g" + 1 H-> (g)',+2 //, (x0 x)-> (x0 *i 1 xt + ! x,) for /=0,..., q. All the face and degeneracyoperators are //-module coalgebra morphisms; i.e., //-module morphisms andcoalgebra morphisms. We omit the calculations verifying the face-degeneracyoperator identities.

Suppose A is an //-module algebra. We have the contravariant functorRegH (*, A) from cocommutative //-module coalgebras to abelian groups. Weapply this functor to the above semisimplicial complex to obtain a semi-cosimplicialcomplex whose objects are {RegH ((g)"+ 1 H, /4)},6o. We denote the coface op-erators, RegH (0 A): Reg^ttg" H, A) -> Reg ((g)'+ 1 H, A), by & for /=0, ...,

1968] COHOMOLOGY OF ALGEBRAS OVER HOPF ALGEBRAS 209

(In [17, p. 235] the homology of a semisimplicial complex is defined. Our complexis obtained from a contravariant functor applied to a semisimplicial complex;hence, is a semi-cosimplicial complex. By dualizing the theory in [17, p. 235] oneobtains the above homology of {Reg (0+1 77, A)}.)

Remark. Since (g1 77=77 and we have e: 77-> A it seems as if

)+177

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210 M. E. SWEEDLER [August

the simplicial complex {Reg ((g" H, A), />"}9a0 is isomorphic to the simplicialcomplex {RegH ((g"""1 H, A), *}0owill be denoted {Reg* (//, /4), D"},^, and referred to as the standard complex tocompute H"(H, A).

We briefly look at Hl(H, A) for /=0, 1. Reg0 (H, A) ~ Ar and if a g H(H, A)then (A a)a~x = e(h) for all heH. Thus A a = e(A)a for all A g //. We denote by Hthe set {aeA\ha = e(h)a for all heH}. This is a subalgebra of ,4 since A is an//-module algebra. Suppose aeA'nA". For all heH, e(h)=hl=h(aa~x)= 2() (A(D a)(A(2) a'x) = 2m e(hm)a(h{2) a~x)=a(h a~x) which implies a~x e A"n Ar. Thus //(//, /i) = ^Hr. Note, A" is just the "invariants" with respect to theHochschild theory, [6, p. 170].

Iff: H^Aisa 1-cocycle then p.(e e) = Dx(f) = [

1968] COHOMOLOGY OF ALGEBRAS OVER HOPF ALGEBRAS 211

theory may be useful in the area of affine algebraic groups where the commutativeHopf algebra is taken to be the coordinate ring of the algebraic group. Rationalmodules for the group correspond to the comodules for the Hopf algebra.

3. Comparison with group cohomology. Suppose G is a group and