Product of flat modules and global dimension relative to $\mathcal F$-Mittag-Leffler modules

TL;DR

This paper characterizes when all direct products of flat modules have finite flat dimension in terms of relative projective dimensions of finitely generated ideals, extending classical global dimension results.

Contribution

It establishes a new equivalence relating the flat dimension of products of flat modules to the relative projective dimension of ideals, generalizing known global dimension theorems.

Findings

Finite flat dimension of products of flat modules characterized by relative projective dimensions.

General result on global relative dimension for classes of modules closed under filtrations.

Extension of classical global dimension results to relative and Gorenstein contexts.

Abstract

Let $R$ be any ring. We prove that all direct products of flat right $R$-modules have finite flat dimension if and only if each finitely generated left ideal of $R$ has finite projective dimension relative to the class of all $\mathcal F$-Mittag-Leffler left $R$-modules, where $\mathcal F$ is the class of all flat right $R$-modules. In order to prove this theorem, we obtain a general result concerning global relative dimension. Namely, if $\mathcal X$ is any class of left $R$-modules closed under filtrations that contains all projective modules, then $R$ has finite left global projective dimension relative to $\mathcal X$ if and only if each left ideal of $R$ has finite projective dimension relative to $\mathcal X$. This result contains, as particular cases, the well known results concerning the classical left global, weak and Gorenstein global dimensions.