The Flat Cover Conjecture for Monoid Acts

Abstract

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 of monomorphisms in terms of ``almost everywhere" effectiveness of the class.