Ascent and descent of Artinian module structures under flat base changes

TL;DR

This paper studies how Artinian module structures behave under flat local ring homomorphisms, providing descriptions of module structures after base change and clarifying conditions for Artinianess of local cohomology modules.

Contribution

It offers new insights into the ascent and descent of Artinian modules under flat base changes and characterizes when local cohomology modules remain Artinian.

Findings

Describes the structure of $A ensor_R S$ for Artinian $R$-modules.

Clarifies conditions for Artinianess of local cohomology modules.

Provides criteria for Artinian modules under flat base change.

Abstract

Let $\varphi: R\rightarrow S$ be a flat local homomorphism between commutative Noetherian local rings. In this paper, the ascent and descent of Artinian module structures between $R$ and $S$ are investigated. For an Artinian $R$-module $A$, the structure of $A\otimes_RS$ is described. As an application, the Artinianess of certain local cohomology modules is clarified.