Covers and direct limits: a contramodule-based approach

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 A is a Grothendieck abelian category and the related abelian category B is equivalent to the category of contramodules over a topological ring R belonging to one of certain four classes of topological rings (e. g., R is commutative), then the left tilting class is covering in A if and only if it is closed under direct limits in A, and if and only if all the discrete quotient rings of the topological ring R are perfect. More generally, if M is a module satisfying a certain telescope Hom exactness condition (e. g., M is Σ-pure-Ext -self-orthogonal) and the topological ring R of endomorphisms of M belongs to one of certain seven classes of topological rings, then the class Add(M) is closed under direct limits if and only if every countable direct limit of copies of M has an Add(M)-cover, and if and only if M has perfect decomposition. In full generality, for an additive category A with (co)kernels and a precovering class L ⊂ A closed under summands, an object N ∈ A has an L-cover if and only if a certain object Ψ(N) in an abelian category 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.