Cosupport of artinian modules

TL;DR

This paper proves the equality of two notions of cosupport for artinian modules over certain local rings, using properties of linearly compact modules.

Contribution

It establishes the equality of cosupport and small cosupport for artinian modules over semi-discrete linearly compact local rings.

Findings

Proves cosupp_R M equals Cosupp_R M for artinian modules.

Uses semi-discrete linearly compactness of Hom modules in the proof.

Provides a new understanding of support concepts in module theory.

Abstract

Let $(R,\mathfrak{m})$ be a commutative local noetherian ring. For an artinian $R$-module $M$, we show the equality $$\mathrm{cosupp}_RM=\mathrm{Cosupp}_RM$$ using the semi-discrete linearly compactness of $R$-module $\mathrm{Hom}_R(R_\mathfrak{p},M)$ where $\mathfrak{p}$ is a prime ideal of $R$.