Tilting theory and the finitistic dimension conjectures

Angeleri-Hügel, Lidia; Trlifaj, Jan

doi:10.1090/S0002-9947-02-03066-0

Tilting theory and the finitistic dimension conjectures
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Trans. Amer. Math. Soc. 354 (2002), 4345-4358 Request permission

Abstract:

Let $R$ be a right noetherian ring and let $\mathcal {P}^{<\infty }$ be the class of all finitely presented modules of finite projective dimension. We prove that findim$R = n < \infty$ iff there is an (infinitely generated) tilting module $T$ such that pd$T = n$ and $T ^\perp = (\mathcal P^{<\infty })^\perp$. If $R$ is an artin algebra, then $T$ can be taken to be finitely generated iff $\mathcal P^{<\infty }$ is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalø.

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