On the dimension of cofinite modules

TL;DR

This paper proves a fundamental equality relating the dimension of certain modules over a complete local ring, specifically for I-cofinite modules, enhancing understanding of their structural properties.

Contribution

It establishes that for all I-cofinite modules over a complete local ring, the dimension of the quotient by the sum of the ideal and the annihilator equals the module's dimension.

Findings

Proves the equality $ ext{dim } R/(I+ ext{Ann}_R M) = ext{dim } M$ for I-cofinite modules.

Provides a key dimension relation in the theory of cofinite modules.

Enhances understanding of the structure of cofinite modules over complete local rings.

Abstract

Let $I$ be an ideal of a commutative Noetherian complete local ring $R$. In the present paper, we establish the equality $\dim R/(I+\Ann_R M)=\dim M$ for all $I$-cofinite $R$-modules $M$.