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-$\operatorname{Ext}^1$-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.

中文翻译：

---

覆盖和直接限制：基于反模块的方法

我们将逆模技术应用于 Enochs 关于覆盖和直接限制的猜想，无论是在分类倾斜上下文中还是在分类倾斜上下文中。在 $n$-tilting-cotilting 对应情况下，如果 $\mathsf A$ 是格罗腾迪克阿贝尔范畴并且相关的阿贝尔范畴 $\mathsf B$ 等价于拓扑环上的逆模范畴 $\mathfrak R$ 属于到某些四类拓扑环之一（例如，$\mathfrak R$ 是可交换的），则左倾类在 $\mathsf A$ 中覆盖当且仅当它在 $\mathsf A$ 中的直接限制下封闭，并且当且仅当拓扑环 $\mathfrak R$ 的所有离散商环都是完美的。更一般地，如果 $M$ 是满足某个望远镜 Hom 精确性条件的模块（例如，$M$ 是 $\Sigma$-pure-$\operatorname{Ext}^1$-self-orthogonal) 并且 $M$ 自同态的拓扑环 $\mathfrak R$ 属于某些七类拓扑环之一，则类 $\mathsf{Add}(M)$ 在直接极限下闭合当且仅当 $M$ 副本的每个可数直接极限都有一个 $\mathsf{Add}(M)$-cover，并且如果并且仅当 $M$ 具有完美分解时。总的来说，对于具有（共）核的加法类别 $\mathsf A$ 和在被加数下封闭的预覆盖类 $\mathsf L\subset\mathsf A$，对象 $N\in\mathsf A$ 具有 $\ mathsf L$-cover 当且仅当具有足够射影的阿贝尔范畴 $\mathsf B$ 中的某个对象 $\Psi(N)$ 具有射影覆盖。