Flat commutative ring epimorphisms of almost Krull dimension zero

TL;DR

This paper studies flat epimorphisms of commutative rings with almost Krull dimension zero, showing bounds on projective dimension and characterizing flat modules, generalizing previous localization results.

Contribution

It introduces a broader class of flat epimorphisms with specific properties and extends known results on flat modules and localizations to this new setting.

Findings

Projective dimension of $U$ does not exceed 1.

Description of the Geigle-Lenzing perpendicular subcategory.

Under additional conditions, all flat $R$-modules are $U$-strongly flat.

Abstract

We consider flat epimorphisms of commutative rings $R\to U$ such that, for every ideal $I\subset R$ for which $IU=U$, the quotient ring $R/I$ is semilocal of Krull dimension zero. Under these assumptions, we show that the projective dimension of the $R$-module $U$ does not exceed $1$. We also describe the Geigle-Lenzing perpendicular subcategory $U^{\perp_{0,1}}$ in $R\mathsf{-Mod}$. Assuming additionally that the ring $U$ and all the rings $R/I$ are perfect, we show that all flat $R$-modules are $U$-strongly flat. Thus we obtain a generalization of some results of the paper arXiv:1801.04820, where the case of the localization $U=S^{-1}R$ of the ring $R$ at a multiplicative subset $S\subset R$ was considered.