Derived $H$-module endomorphism rings

TL;DR

This paper investigates the relationship between derived endomorphism rings of modules over Hopf Galois extensions, establishing conditions under which these rings form Galois extensions and showing that the Koszul property is preserved.

Contribution

It provides new criteria for when derived endomorphism rings form Galois extensions in the context of Hopf algebra actions, and demonstrates the preservation of the Koszul property.

Findings

E_A(M) is a graded subalgebra of E_B(M) for semisimple Hopf algebras.

Necessary and sufficient conditions for E_B(M) to be an H^*-Galois extension of E_A(M).

Koszul property is preserved under Hopf Galois graded extensions.

Abstract

Let $H$ be a Hopf algebra, $A/B$ be an $H$-Galois extension. Let $D(A)$ and $D(B)$ be the derived categories of right $A$-modules and of right $B$-modules respectively. An object $M^\cdot\in D(A)$ may be regarded as an object in $D(B)$ via the restriction functor. We discuss the relations of the derived endomorphism rings $E_A(M^\cdot)=\op_{i\in\mathbb{Z}}\Hom_{D(A)}(M^\cdot,M^\cdot[i])$ and $E_B(M^\cdot)=\op_{i\in\mathbb{Z}}\Hom_{D(B)}(M^\cdot,M^\cdot[i])$. If $H$ is a finite dimensional semisimple Hopf algebra, then $E_A(M^\cdot)$ is a graded subalgebra of $E_B(M^\cdot)$. In particular, if $M$ is a usual $A$-module, a necessary and sufficient condition for $E_B(M)$ to be an $H^*$-Galois graded extension of $E_A(M)$ is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.