Projective covers of flat contramodules

TL;DR

This paper investigates conditions under which direct limits of certain contramodules are projective, introduces results for flat contramodules using the topological Jacobson radical, and discusses covers in relation to the Enochs conjecture.

Contribution

It provides new criteria for projectivity of direct limits of contramodules and offers an elementary proof of the Enochs conjecture in specific settings.

Findings

Direct limits of projective contramodules are projective if they have a projective cover.

Results for $ abla$-strictly flat contramodules with projective dimension ≤ 1.

An elementary proof of the Enochs conjecture for n-tilting cotorsion pairs.

Abstract

We show that a direct limit of projective contramodules (over a right linear topological ring) is projective if it has a projective cover. A similar result is obtained for $\infty$-strictly flat contramodules of projective dimension not exceeding $1$, using an argument based on the notion of the topological Jacobson radical. Covers and precovers of direct limits of more general classes of objects, both in abelian categories with exact and with nonexact direct limits, are also discussed, with an eye towards the Enochs conjecture about covers and direct limits, using locally split (mono)morphisms as the main technique. In particular, we offer a simple elementary proof of the Enochs conjecture for the left class of an $n$-tilting cotorsion pair in an abelian category with exact direct limits.