Almost free modules, perfect decomposition and Enochs's conjecture

Abstract

Given a module $X$ and a regular cardinal $\hat{I}º$ we study various notions of $(\hat{I}º,\mathrm{Add}(X))$-freeness and $(\hat{I}º,\mathrm{Add}(X))$-separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial $\hat{I}{º}^{+}$-generated, $(\hat{I}{º}^{+},\mathrm{Add}(X))$-free and $(\hat{I}{º}^{+},\mathrm{Add}(X))$-separable module. Our construction allows $\hat{I}º$ to be singular thus extending \cite[Theorem~4.7]{CortesGuilTorrecillas}. Bearing on similar set-theoretic assumptions, we characterize when every module $X$ has a perfect decomposition. As a subproduct we show that Enoch's conjecture for classes $\mathrm{Add}(X)$ is consistent with ZFC -- a fact first proved by Å aroch \cite{Saroch}.