Flat commutative ring epimorphisms of almost Krull dimension zero

Abstract

We consider flat epimorphisms of commutative rings R −→ U such that, for every ideal I ⊂ 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 ⊥0,1 in R-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 [6], where the case of the localization U = S −1 R of the ring R at a multiplicative subset S ⊂ R was considered.