Generalized Benson-carlson Duality

1.1. Background. This paper deals with the landmark results of Benson and Carlson's Projective resolutions and Poincar e duality complexes [6]. Our goal is to provide some necessary background for that work, and to prove some of the results of [6] in a more general setting. In particular, we analyze what happens when we replace Benson and Carlson's complex C (in the notation of [6]) with an arbitrary Yoneda extension representing . Let be a nite-dimensional cocommutative Hopf algebra over an algebraically closed eld k. Because is cocommutative, H ( ; k) is a graded-commutative kalgebra (i.e., xy = ( 1)yx for homogeneous x; y). Throughout this paper we will also assume that has the following niteness property in cohomology: Ext (k; k) is a nitely generated k-algebra, and for any -modules M and N , Ext (M;N) is nitely generated as an Ext (k; k)-module. (By -module we always mean nitely generated left -module.) The second condition is equivalent to requiring H ( ;M) to be nitely generated over H ( ; k) for any -module M , because of the isomorphism Ext (M;N) = H ( ;Homk(M;N)). If is the group algebra of a nite group then certainly has the niteness property in cohomology, by a theorem of Evens (cf. [7, Theorem 7.4.1]). The niteness property also holds if is a nite-dimensional cocommutative connected Hopf algebra, e.g. the restricted enveloping algebra of a p-restricted Lie algebra (Bajer and Sadofsky [1, Lemma 6.2], Wilkerson [10]). In fact, at present we know of no example of a nite-dimensional Hopf algebra without this property. It is known that any nitedimensional Hopf algebra is a Frobenius algebra (Larson and Sweedler [9]), which implies that a -module is projective if and only if it is injective, and this property is fundamental for the proofs given here.