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We consider flat epimorphisms of commutative rings $R\to U$ such that, forevery ideal $I\subset R$ for which $IU=U$, the quotient ring $R/I$ is semilocalof Krull dimension zero. Under these assumptions, we show that the projectivedimension of the $R$-module $U$ does not exceed $1$. We also describe theGeigle-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, weshow that all flat $R$-modules are $U$-strongly flat. Thus we obtain ageneralization of some results of the paper arXiv:1801.04820, where the case ofthe localization $U=S^{-1}R$ of the ring $R$ at a multiplicative subset$S\subset R$ was considered.

Flat commutative ring epimorphisms of almost Krull dimension zero