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 $\mathsf{A}$ 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 $\mathrm{\Sigma }$-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 $\mathsf{A}\mathsf{d}\mathsf{d}(M)$ is closed under direct limits if and only if every countable direct limit of copies of $M$ has an $\mathsf{A}\mathsf{d}\mathsf{d}(M)$-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 $\mathrm{\Psi }(N)$ 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.