Tilting theory and the finitistic dimension conjectures

Abstract

Let R R be a right noetherian ring and let P > ∞ \mathcal {P}^{>\infty } be the class of all finitely presented modules of finite projective dimension. We prove that findim R = n > ∞ R = n > \infty iff there is an (infinitely generated) tilting module T T such that pd T = n T = n and T ⊥ = ( P > ∞ ) ⊥ T ^\perp = (\mathcal P^{>\infty })^\perp . If R R is an artin algebra, then T T can be taken to be finitely generated iff P > ∞ \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.