Self-orthogonal tau-tilting modules and tilting modules

Abstract

Let $\Lambda$ be an artin algebra and $T$ a $\tau$-tilting $\Lambda$-module. We prove that $T$ is a tilting module if and only if ${\rm Ext}_Λ^{i}(T,\Fac T)=0$ for all $i\ge 1$, where $\Fac T$ is the full subcategory consisting of modules generated by $T$. Consequently, a $\tau$-tilting module $T$ of finite projective dimension is a tilting module if and only if ${\mathrm{Ext}}_{\Lambda }^i(T,T)=0$ for all $i\ge 1$. Moreover, we also give an example to show that a support $\tau$-tilting but not $\tau$-tilting module $M$ of finite projective dimension satisfying ${\mathrm{Ext}}_{\Lambda }^i(M,M)=0$ for all $i\ge 1$ need not be a partial tilting module.

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References in corpus (1)

• $\tau$-Tilting Finite Cluster-Tilted Algebras