Gorenstein test modules

Abstract

Let (R,ℳ) be a commutative Noetherian local ring. We exhibit certain modules T over R which test G-dimension of a finitely generated R-module M with finite G-dimension in the following sense: if Extjg(M,T) = 0 for all j ≥ i, where i is a positive integer, then G-dimR M < i. Modules with the property like T will be called Gorenstein test modules (G-test modules for short). It is known that R itself is a G-test module. We show that k, the residue field of R, also tests G-dimension. Some more examples of G-test modules are introduced.