标题： 平坦覆盖猜想对于独异点作用 显示英文标题

标题： The Flat Cover Conjecture for Monoid Acts

摘要： 我们证明，对于任何右可逆的幺半群 $S$，平坦覆盖猜想在(右)作用的范畴中成立，前提是平坦的 $S$-作用在稳定的Rees扩张下是封闭的。 该论证表明，在此类范畴中，类 $\mathcal{F}$-Mono (具有平坦Rees商的$S$-作用单射) 是由小生成的，回答了Bailey和Renshaw的问题。 但 $\mathcal{SF}$-Mono (具有 \emph{强烈地} 平坦Rees商的$S$-作用单射) 的由小生成性似乎要强得多，因为我们证明它意味着存在不可分解的强平坦作用的大小上限。 类似地，$\mathcal{U}_{\mathcal{F}}$的纤维生成（具有平坦补集的单位单态射）意味着不可分解平坦作用的大小有界。 关键工具是根据$\mathcal{M}$的“几乎处处”有效性对单态射类$\mathcal{M}$的纤维生成的新表征。 显示英文摘要

摘要： We prove that the Flat Cover Conjecture holds for the category of (right) acts over any right-reversible monoid $S$, provided that the flat $S$-acts are closed under stable Rees extensions. The argument shows that the class $\mathcal{F}$-Mono ($S$-act monomorphisms with flat Rees quotient) is cofibrantly generated in such categories, answering a question of Bailey and Renshaw. But cofibrant generation of $\mathcal{SF}$-Mono ($S$-act monomorphisms with \emph{strongly} flat Rees quotient) appears much stronger, since we show it implies that there is a bound on the size of the indecomposable strongly flat acts. Similarly, cofibrant generation of $\mathcal{U}_{\mathcal{F}}$ (unitary monomorphisms with flat complement) implies a bound on the size of indecomposable flat acts. The key tool is a new characterization of cofibrant generation of a class $\mathcal{M}$ of monomorphisms, in terms of ``almost everywhere" effectiveness of $\mathcal{M}$.