Flat commutative ring epimorphisms of almost Krull dimension zero

In this paper, we consider flat epimorphisms of commutative rings R -> U such that, for every ideal I subset of 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-&updatedExpOTTOM;0,U-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-1R of the ring R at a multiplicative subset S subset of R was considered.