Flat commutative ring epimorphisms of almost Krull dimension zero

Abstract

In this paper, we consider flat epimorphisms of commutative rings [Formula: see text] such that, for every ideal [Formula: see text] for which [Formula: see text], the quotient ring [Formula: see text] is semilocal of Krull dimension zero. Under these assumptions, we show that the projective dimension of the [Formula: see text]-module [Formula: see text] does not exceed [Formula: see text]. We also describe the Geigle–Lenzing perpendicular subcategory [Formula: see text] in [Formula: see text]. Assuming additionally that the ring [Formula: see text] and all the rings [Formula: see text] are perfect, we show that all flat [Formula: see text]-modules are [Formula: see text]-strongly flat. Thus, we obtain a generalization of some results of the paper [6], where the case of the localization [Formula: see text] of the ring [Formula: see text] at a multiplicative subset [Formula: see text] was considered.