Relative derived dimensions for cotilting modules

TL;DR

This paper establishes a precise relationship between the derived dimension of a Noetherian ring relative to certain cotilting modules and the injective dimension of those modules, extending to Cohen-Macaulay modules in local rings.

Contribution

It proves that the derived dimension with respect to cotilting modules equals their injective dimension, generalizing known results for Cohen-Macaulay modules in local rings.

Findings

Derived dimension equals the injective dimension of cotilting modules.

In local rings with a canonical module, the derived dimension matches the ring's Krull dimension.

The results apply Auslander-Buchweitz theory and Ghost Lemma techniques.

Abstract

For a Noetherian ring $R$ and a cotilting $R$-module $T$ of injective dimension at least $1$, we prove that the derived dimension of $R$ with respect to the category $\mathcal{X}_T$ is precisely the injective dimension of $T$ by applying Auslander-Buchweitz theory and Ghost Lemma. In particular, when $R$ is a commutative Noetherian local ring with a canonical module $\omega_R$ and $\dim R\ge1$, the derived dimension of R with respect to the category of maximal Cohen-Macaulay modules is precisely $\dim R$.