Cohomology of Modules Over -categories and Co--categories

Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category ${\mathcal{C}}$ as modules over the smash extension ${\mathcal{C}}\#H$. We construct Grothendieck spectral sequences for the cohomologies as well as the $H$-locally finite cohomologies of these objects. We also introduce relative $({\mathcal{D}},H)$-Hopf modules over a Hopf comodule category ${\mathcal{D}}$. These generalize relative $(A,H)$-Hopf modules over an $H$-comodule algebra $A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational $\text{Hom}$ objects and higher derived functors of coinvariants.

The Factorization for a Class of Hom-Coalgebras

Let (H,α) be a Hom-Hopf algebra and (C,β) a Hom-coalgebra. In this paper, we first introduce the notions of Hom-crossed coproduct C ⋊λ H and cleft coextension and then discuss the equivalence between them. Furthermore, we discuss the relation between cleft coextension and Hom-module coalgebra with the Hom-Hopf module structure and obtain a Hom-coalgebra factorization for Hom-module coalgebra with the Hom-Hopf module structure.

ON GORENSTEINNESS OF HOPF MODULE ALGEBRAS

Let H be a Hopf algebra with a bijective antipode, A an H-simple H-module algebra finitely generated as an algebra over the ground field and module-finite over its centre. The main result states that A has finite injective dimension and is, moreover, Artin–Schelter Gorenstein under the additional assumption that each H-orbit in the space of maximal ideals of A is dense with respect to the Zariski topology. Further conclusions are derived in the cases when the maximal spectrum of A is a single H-orbit or contains an open dense H-orbit.

Total integrals of Doi Hom-Hopf modules

Let [Formula: see text] be a monoidal Hom-Hopf algebra, [Formula: see text] a right [Formula: see text]-Hom-comodule algebra and [Formula: see text] a right [Formula: see text]-Hom-module coalgebra. We first investigate the criterion for the existence of a total integral of [Formula: see text] in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral [Formula: see text] if and only if any representation of the pair [Formula: see text] is injective in a functorial way, which generalizes Menini and Militaru’s result. Finally, we extend to the category of [Formula: see text]-Doi Hom-Hopf modules a result of Doi on projectivity of every relative [Formula: see text]-Hopf module as an [Formula: see text]-module.

The space of framed chord diagrams as a Hopf module

This paper is devoted to the study of Hopf module structures on the spaces of framed chord diagrams and framed graphs. We also introduce a framed version of the chromatic polynomial and propose two methods to construct framed weight systems.