Covers and direct limits: a contramodule-based approach (Q2231124)

The authors continue their earlier work (with their coauthors). Of interest are applications of techniques of contramodules and categorical tilting theory to what is called here the Enochs conjecture: In a module category over an associative ring, a covering class in that category is closed under direct limits. The first third of the paper is devoted to terminology (not always attributed to the original sources or original names) and previous results. The following topics are treated: Covers and those reduced to projective covers, telescoping Hom exactness condition, perfect decompositions, functor purity in abelian categories, self-pure projective and direct limit pure rigid objects, covers in hereditary cotorsion pairs, the tilting-cotilting correspondence, conditions for left tilting to be class covering, injective ring epimorphisms of projective dimension 1, covers and direct limits for injective ring epimorphisms. The paper wisely develops results from simpler cases to more general one's. Thus assumptions of countability is imposed initially, and so is a restriction to the class of (self-)small modules... One of the cases the authors establish is as follows: If \(\mathcal{A}\) is an additive category with (co)kernels and a precovering class \(\mathcal{L}\subset\mathcal{A}\) is closed under summands, an object \(N\in\mathcal{A}\) has an \(\mathcal{L}\)-cover if and only if a certain object \(\Psi(N)\) in an abelian category \(\mathcal{B}\) with enough projectives has a projective cover.