On Gorenstein Projective, Injective and Flat Dimensions — a Functorial Description with Applications

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions rarely come with practical and robust criteria for finiteness, even over commutative noetherian local rings. Indeed, over such a ring (R,m, k), projectivity of a finitely generated module M is equivalent to ExtR(M, k) = 0, while verification of socalled Gorenstein projectivity a priori requires computation of infinitely many cohomology groups: ExtmR (M,R) and Ext m R (HomR(M,R), R). In this paper we vastly enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the noncommutative realm, k–algebras with a dualizing complex.