Projective covers of flat contramodules
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 -strictly flat contramodules of projective dimension not exceeding , 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 -tilting cotorsion pair in an abelian category with exact direct limits.
Cite
@article{arxiv.1911.11720,
title = {Projective covers of flat contramodules},
author = {Silvana Bazzoni and Leonid Positselski and Jan Stovicek},
journal= {arXiv preprint arXiv:1911.11720},
year = {2022}
}
Comments
LaTeX 2e with pb-diagram and xy-pic, 31 pages, 3 commutative diagrams; v.2: a new author joined, the title changed, paper greatly expanded, main result generalized from countable to uncountable direct limits; v.4: Introduction expanded, many references added, several misprints corrected, the numbering of sections shifted to agree with the journal version, former Remark 7.4 is now Corollary 8.4