Certain Non-projective Modules in Tilting Theoretic Arguments

Abstract

Let $Lambda$ be an artin $R$-algebra for a commutative artinian ring $R$, and let $T$ be a tilting $Lambda$-module with $Gamma={ m End}_Lambda(T)$. It is well-known that one may attach to $T$ a couple of torsion pairs $(FT(T), FF(T))$ and $(FX(T), FY(T))$, respectively in the categories of finitely generated $Lambda$- and $Gamma$- modules. In usual tilting theoretic arguments, it sometime happens that one needs to know whether or not a prescribed $Gamma$-module is projective. In this article we will argue that if $T$ is a separating tilting module then for any $YinFT(T)$, $Hom_Lambda( au(T), Y)$ is either zero or certainly not a projective $Gamma$-module provided $Y$ has no indecomposable direct summands in ${ m add},T$; here $ au$ is reserved to denote the usual Auslander-Reiten translation of $Lambda$ and ${ m add},T$ denotes the additive closure of $T$.