On the H-finite 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=RH. We define a functor from ModS to ModR(#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 PicR(R,H) and Pic(S).