On the Cohomology of Relative Hopf Modules

Abstract

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.

Coactions on Spaces of Morphisms

“…If in Proposition 2.1 we take N = k with the standard H-comodule structure, we see that M * has a right H-comodule structure given by (1) ).…”

“…Conversely, if H has non-zero integrals, then the injective envelope of any simple right H-comodule is finite dimensional by [8,Theorem 3]. (1) ). In particular if S 2 = Id, then φ is an isomorphism of H-comodules.…”

“…If M is a right C-comodule, with comodule structure map ρ : M → M ⊗ C, ρ(m) = m (0) ⊗ m (1) , then on the underlying space M we have a right C-comodule structure M (F) defined by ρ (F) (1) ). In general the comodules M and M (F) are not isomorphic, even when F is a coalgebra automorphism of C. However, we have the following.…”

“…If moreover M is finite dimensional, then HOM(M, N) = Hom(M, N). Variants of this construction have been considered, see for example [1]. We study this comodule structure on a space of morphisms and discuss its behavior when we take dual spaces, tensor products and double dual spaces.…”

“…We work with Hopf algebras H over a field k. The antipode of H is denoted by S. (1) the comodule structure. M is called absolutely irreducible if its endomorphisms (as an H-comodule) are the scalar multiples of the identity map.…”