Mittag-Leffler Conditions, Gorenstein Modules and Homological Invariants

Abstract

In this paper, we investigate certain properties on the Mittag-Leffler conditions via set-theoretic methods. We establish that a strongly ${\aleph }_1$-presented module $M$ satisfying Ext${}_R^{\ge 1}(M,D^{({\aleph }_1)})=0$ belongs to the left orthogonal class of $\overline{D}$, where $\overline{D}$ is the definable of $D$. This yields the consequence that every ${\aleph }_1$-generated strongly Gorenstein projective module is Gorenstein flat. Furthermore, we investigate when a flat Gorenstein projective module is projective. Finally, for any two-sided ${\aleph }_1$-coherent ring $R$, we prove the identity $\mathrm{silp}R+\mathrm{silp}{R}^{\mathrm{op}}=\mathrm{spli}R+\mathrm{spli}{R}^{\mathrm{op}}$.

We revised the paper to make it more readable