Separated monic correspondence of cotorsion pairs and semi-Gorenstein-projective modules

Abstract

Given a finite dimensional algebra $A$ over a field $k$, and a finite acyclic quiver $Q$, let $\Lambda =A{\otimes }_{k}kQ/I$, where $kQ$ is the path algebra of $Q$ over $k$ and $I$ is a monomial ideal. We show that $(\mathcal{X},\mathcal{Y})$ is a (complete) hereditary cotorsion pair in $A$-mod if and only if $(\mathrm{smon}(Q,I,\mathcal{X}),\mathrm{rep}(Q,I,\mathcal{Y}))$ is a (complete) hereditary cotorsion pair in $\Lambda$-mod. We also show that $A$ is left weakly Gorenstein if and only if so is $\Lambda$. Provided that $kQ/I$ is non-semisimple, the category ${}^{\perp }\Lambda$ of semi-Gorenstein-projective $\Lambda$-modules coincides with the category of separated monic representations $\mathrm{smon}(Q,I{,}^{\perp }A)$ if and only if $A$ is left weakly Gorenstein.

19 pages, we added a result of constructing left weakly Gorenstein triangular matrix rings (Proposition 5.4) and updated the proof of Theorem B in section 5