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 Λ=A⊗kkQ/I, where kQ is the path algebra of Q over k and I is a monomial ideal. This paper is devoted to studying the separated monic representations over the bounded quiver (Q,I) over a subcategory X of A-mod, denoted by smon(Q,I,X). We show that (X,Y) is a (complete) hereditary cotorsion pair in A-mod if and only if (smon(Q,I,X),rep(Q,I,Y)) is a (complete) hereditary cotorsion pair in Λ-mod. We also show that A is left weakly Gorenstein if and only if so is Λ. Provided that kQ/I is non-semisimple, the category ▪ of semi-Gorenstein-projective Λ-modules coincides with the category of separated monic representations ▪ if and only if A is left weakly Gorenstein.