Self-orthogonal τ-tilting modules and tilting modules

Abstract

Let Λ be an Artin algebra and T a τ -tilting Λ-module. We prove that T is a tilting module if and only if Ext Λ i ( T , Fac T ) = 0 for all i ≥ 1 , where Fac T is the full subcategory consisting of modules generated by T . Consequently, a τ -tilting module T of finite projective dimension is a tilting module if and only if Ext Λ i ( T , T ) = 0 for all i ≥ 1 . Moreover, we also give an example to show that a support τ -tilting but not τ -tilting module M of finite projective dimension satisfying Ext Λ i ( M , M ) = 0 for all i ≥ 1 need not be a partial tilting module.