Homological flat dimensions

TL;DR

This paper introduces new homological invariants called complete intersection flat and Cohen-Macaulay flat dimensions for modules over local rings, extending classical notions to infinitely generated modules.

Contribution

It defines and studies these new invariants, generalizing existing homological dimensions to a broader class of modules.

Findings

New invariants coincide with classical ones for finitely generated modules

Extension of homological dimensions to infinitely generated modules

Provides a framework for analyzing modules beyond traditional limits

Abstract

For finitely generated module $M$ over a local ring $R$, the conventional notions of complete intersection dimension $\cid_R M$ and Cohen-Macaulay dimension $\cmdim_R M$ do not extend to cover the case of infinitely generated modules. In this paper we introduce similar invariants for not necessarily finitely generated modules, (namely, complete intersection flat and Cohen-Macaulay flat dimensions) which for finitely generated modules, coincide with the corresponding classical ones.