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.

Differential graded Koszul duality: An introductory survey

“…In the meantime, what came to be known as ‘Ext‐$$p$$‐complete’ or ‘weakly‐$$l$$‐complete’ abelian groups [12, 36] or (in a later terminology) ‘cohomologically $$I$$‐adically complete’ or ‘derived $$I$$‐adically complete modules’ (for a finitely generated ideal $$I$$ in a commutative ring) [60] were defined and studied by authors who remained apparently completely unaware of the connection with Eilenberg and Moore's contramodules. We refer to the presentation [73] for a detailed discussion of this part of the history.…”