The cotorsion pair generated by the class of flat Mittag-Leffler modules

TL;DR

This paper extends the understanding of cotorsion pairs generated by flat Mittag-Leffler modules to a broader class of rings, showing that over these rings, flat modules have finite projective dimension.

Contribution

It generalizes previous results to rings where flat modules are filtered by limits of projective modules, including countable, perfect, and valuation domains.

Findings

Orthogonal class of flat Mittag-Leffler modules consists of cotorsion modules for these rings.

Flat modules over these rings have finite projective dimension under certain set-theoretic assumptions.

The class of rings includes all countable, perfect, and valuation domains.

Abstract

Let $R$ be a ring and denote by $\mathcal{FM}$ the class of all flat and Mittag-Leffler left $R$-modules. In \cite{BazzoniStovicek2} it is proved that, if $R$ is countable, the orthogonal class of $\mathcal{FM}$ consists of all cotorsion modules. In this note we extend this result to the class of all rings $R$ satisfying that each flat left $R$-module is filtered by totally ordered limits of projective modules. This class of rings contains all countable, left perfect and discrete valuation domains. Moreover, assuming that there do not exist inaccessible cardinals, we obtain that, over these rings, all flat left $R$-modules have finite projective dimension.