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arXiv · 2407.20363

Almost free modules, perfect decomposition and Enochs's conjecture

Manuel Cortés-Izurdiaga·Alejandro Poveda
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Abstract

Given a module $X$ and a regular cardinal $\kappa$ we study various notions of $(\kappa,\mathrm{Add}(X))$-freeness and $(\kappa,\mathrm{Add}(X))$-separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial $\kappa^+$-generated, $(\kappa^+,\mathrm{Add}(X))$-free and $(\kappa^+,\mathrm{Add}(X))$-separable module. Our construction allows $\kappa$ 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 \v{S}aroch \cite{Saroch}.

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Manuel Cortés-Izurdiaga, Alejandro Poveda. 2024-07-29. Almost free modules, perfect decomposition and Enochs's conjecture. https://arxiv.org/abs/2407.20363

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