LAUSR

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.