Cohen-Macaulay invariant subalgebras of Hopf dense Galois extensions

TL;DR

This paper investigates the Cohen-Macaulay properties of invariant subalgebras under Hopf algebra actions, establishing conditions under which these subalgebras inherit Cohen-Macaulayness and classifying Cohen-Macaulay modules in specific cases.

Contribution

It extends classical results by showing invariant subalgebras of certain Hopf Galois extensions are Cohen-Macaulay and classifies Cohen-Macaulay modules for these subalgebras.

Findings

Invariant subalgebras inherit AS-Cohen-Macaulay property under mild conditions.

When R is AS-regular of dimension 2, all indecomposable Cohen-Macaulay modules are direct summands of R.

R^H is Cohen-Macaulay-finite in the specified setting.

Abstract

Let $H$ be a semisimple Hopf algebra, and let $R$ be a noetherian left $H$-module algebra. If $R/R^H$ is a right $H^*$-dense Galois extension, then the invariant subalgebra $R^H$ will inherit the AS-Cohen-Macaulay property from $R$ under some mild conditions, and $R$, when viewed as a right $R^H$-module, is a Cohen-Macaulay module. In particular, we show that if $R$ is a noetherian complete semilocal algebra which is AS-regular of global dimension 2 and $H=\operatorname{\bf k} G$ for some finite subgroup $G\subseteq Aut(R)$, then all the indecomposable Cohen-Macaulay module of $R^H$ is a direct summand of $R_{R^H}$, and hence $R^H$ is Cohen-Macaulay-finite, which generalizes a classical result for commutative rings. The main tool used in the paper is the extension groups of objects in the corresponding quotient categories.