Finiteness of homological dimensions of Ext modules

TL;DR

This paper explores how the finiteness of the homological dimensions of Ext modules influences the finiteness of the dimensions of the modules themselves over a Noetherian local ring, revealing conditions under which these properties are equivalent.

Contribution

It establishes new conditions linking the finiteness of Ext modules' homological dimensions to the finiteness of the modules' own dimensions, especially involving the restricted flat dimension.

Findings

Finite projective dimension of Ext modules implies finiteness of module dimensions under certain conditions.

Equivalence of finite projective or injective dimension of modules based on Ext modules' properties.

Provides criteria connecting Ext module dimensions with module homological dimensions.

Abstract

Let $R$ be a commutative Noetherian local ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. In this paper, we investigate how the finiteness of the homological dimension of Ext modules between $M$ and $N$ affects that of $M$ and $N$. One of our main result states that if $\operatorname{Ext}^i_R(M,N)$ has finite projective dimension for any $0\le i\le \operatorname{Rfd}_R M$, where $\operatorname{Rfd}_R M$ is the (large) restricted flat dimension of $M$, then $M$ has finite projective or injective dimension if and only if $N$ does.