Covers and direct limits: a contramodule-based approach

Abstract

We present applications of contramodule techniques to the Enochs conjecture about covers and direct limits, both in the categorical tilting context and beyond. In the n-tilting–cotilting correspondence situation, if $\mathsf A$ is a Grothendieck abelian category and the related abelian category $\mathsf B$ is equivalent to the category of contramodules over a topological ring $\mathfrak R$ belonging to one of certain four classes of topological rings (e. g., $\mathfrak R$ is commutative), then the left tilting class is covering in $\mathsf A$ if and only if it is closed under direct limits in $\mathsf A$, and if and only if all the discrete quotient rings of the topological ring $\mathfrak R$ are perfect. More generally, if M is a module satisfying a certain telescope Hom exactness condition (e. g., M is $\Sigma$-pure-${{\,\mathrm{Ext}\,}}^1$-self-orthogonal) and the topological ring $\mathfrak R$ of endomorphisms of M belongs to one of certain seven classes of topological rings, then the class $\mathsf {Add}(M)$ is closed under direct limits if and only if every countable direct limit of copies of M has an $\mathsf {Add}(M)$-cover, and if and only if M has perfect decomposition. In full generality, for an additive category $\mathsf A$ with (co)kernels and a precovering class $\mathsf L\subset \mathsf A$ closed under summands, an object $N\in \mathsf A$ has an $\mathsf L$-cover if and only if a certain object $\Psi (N)$ in an abelian category $\mathsf B$ with enough projectives has a projective cover. The 1-tilting modules and objects arising from injective ring epimorphisms of projective dimension 1 form a class of examples which we discuss.