Tilting modules and Gorenstein rings

Abstract

In his pioneering work [30], Miyashita extended classical tilting theory to the setting of finitely presented modules of finite projective dimension over an arbitrary ring. More recently, tilting theory has been generalized to arbitrary modules of finite projective dimension [2], [3], [6], et al. Encouraged by this developement, in the present paper, we apply infinite dimensional tilting theory to the setting of modules over Iwanaga-Gorenstein rings. We start with pointing out the central role played by resolving subcategories of modR for an arbitrary ring R. In Theorem 2.2 we prove that these subcategories correspond bijectively to tilting classes of finite type in ModR, as well as to cotilting classes of cofinite type in RMod. This generalizes a classical result of Auslander-Reiten [9] characterizing cotilting classes in modR over an artin algebra R. Moreover, it provides for a bijection between certain cotilting left modules and tilting right modules which extends the duality between finitely generated cotilting and tilting modules over artin algebras. We then consider Iwanaga-Gorenstein rings, that is (not necessarily commutative) two-sided noetherian rings of finite selfinjective dimension on both sides. We show that the first finitistic dimension conjecture holds true for any Iwanaga-Gorenstein ring (Theorem 3.2). We prove that Gorenstein injectivity, and ∗This work was started while the first author was a Ramon y Cajal Fellow at the Universitat Autonoma de Barcelona. The research of the second author was partially supported by the DGESIC (Spain) through the project PB98-0873, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. Third author supported by grant SAB2001-0092 of Secretaria de Estado de Educacion y Universidades MECD at CRM IEC Barcelona, and by MSM 113200007.