Abstract. Let H be a Hopf algebra over a field k, and A an Hcomodule algebra. The categories of comodules and relative Hopf modules are then Grothendieck categories with enough injectives. We study the derived functors of the associated Hom functors, and of the coinvariants functor, and discuss spectral sequences that connect them. We also discuss when the coinvariants functor preserves injectives.

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Abstract. Let k be a field, and H a Hopf algebra with bijective antipode. If H is commutative, noetherian, semisimple and cosemisimple, then the category H YD H of Yetter-Drinfeld modules is semisimple. We also prove a similar statement for the category of Long dimodules, without the assumption that H is commutative.

JE DEDIE CE TRAVAIL A ARSENE ASSOGBÁ`L et k be an algebraically closed field of characteristic zero, g a finite-dimen-Ž . sional Lie algebra over k, U g the enveloping algebra of g, R a Noetherian Ž . k-algebra on which g acts by derivations via a Lie algebra morphism, and R࠻U g Ž . the differential operator rings generated by R and U g . This paper is concerned Ž . with the homological algebra for R࠻U g -modules which are g-locally finite, especially with injective and projective modules, minimal injective resolutions, and cohomology.

Let k be a field and H a Hopf algebra over k with a bijective antipode. Suppose that H acts on an associative (left noetherian) k-algebra R such that R is an H -module algebra. We consider the categories of all H -modules, the subcategory of those which are H -locally finite, and the subcategories of each which are also R-modules in a compatible way. These categories are all abelian with enough injectives and we derive spectral sequences relating Ext * (−, −) in them. Now let (−) H denote taking H -invariants and set S = R H . We define a functor L S (R, −) from Mod S to Mod R(#H ) that has good behavior with respect to injective objects. We also show that the functor (−) H carries some injectives to injectives. When R is commutative, H is cocommutative, and k is projective in the category of finite-dimensional H -modules, we obtain more precise results, comparing, for example, the Picard groups Pic R (R, H ) and Pic(S).