Very flat, locally very flat, and contraadjusted modules

TL;DR

This paper explores the structure and approximation properties of very flat, locally very flat, and contraadjusted modules over noetherian rings, establishing their relation to the finiteness of the spectrum and providing criteria over Dedekind domains.

Contribution

It proves the equivalence of certain module class properties with the finiteness of the spectrum of a noetherian domain and offers an analog of Pontryagin's criterion for Dedekind domains.

Findings

Very flat modules form a covering class if the spectrum is finite.

Locally very flat modules are precovering under spectrum finiteness.

Contraadjusted modules form an enveloping class in this context.

Abstract

Very flat and contradjusted modules naturally arise in algebraic geometry in the study of contraherent cosheaves over schemes. Here, we investigate the structure and approximation properties of these modules over commutative noetherian rings. Using an analogy between projective and flat Mittag-Leffler modules on one hand, and very flat and locally very flat modules on the other, we prove that each of the following statements are equivalent to the finiteness of the Zariski spectrum Spec(R) of a noetherian domain R: (i) the class of all very flat modules is covering, (ii) the class of all locally very flat modules is precovering, and (iii) the class of all contraadjusted modules is enveloping. We also prove an analog of Pontryagin's criterion for locally very flat modules over Dedekind domains.