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I have already knowed that selfinjective algebras have only trivial tilting modules,but besides this,is there any more?

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    $\begingroup$ Do you allow $\rm{pd}(T)<\infty$ or only $\rm{pd}(T) \leq 1$? $\endgroup$
    – Dag Oskar Madsen
    Commented Oct 26, 2012 at 6:37

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Local finite dimensional algebras have only trivial tilting modules since in that case all finitely generated modules have either zero or infinite projective dimension.

In particular it follows that commutative finite dimensional algebras have only trivial tilting modules.

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More general it is true for algebras having finitistic dimension eqaul to zero, which includes local and selfinjective algebras.

A positive answer to Finitistic dimension via tilting modules would give a classification of such algebras. Namely: An algebra has only trivial tilting modules iff it has finitistic dimension equal to zero.

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