On the cohomology of comodules over smash coproducts

TL;DR

This paper investigates the cohomology of comodules over smash coproduct coalgebras, establishing a spectral sequence that links various derived functors and connecting it to existing literature.

Contribution

It introduces a Grothendieck spectral sequence for comodules over smash coproduct coalgebras, providing new tools for understanding their cohomological properties.

Findings

Established a spectral sequence connecting different derived functors.

Connected new results to existing spectral sequences in literature.

Provided applications illustrating the spectral sequence's utility.

Abstract

We consider the category of comodules over a smash coproduct coalgebra $C\smashco H$. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from $C\smashco H$-colinear, $H$-colinear and rational $C$-colinear morphisms. We give several applications and connect our results to existing spectral sequences in the literature.