On Modules Detecting the Finiteness of Homological Dimensions

I will discuss a class of finitely generated modules over commutative Noetherian rings, known as test modules, which are defined via the vanishing of Ext and Tor functors. A module M is called a test module (for a homological dimension H-dim) if, for every finitely generated module N, the vanishing of all higher Tor(M, N) implies that H-dim(N) is finite. Test modules are abundant in the literature, and such modules have been studied since the 1970s. For example, a more recent work of Bhatt–Iyengar–Ma explores perfectoid algebras over local rings that act as test modules in a broader sense.

A classical result of Auslander–Buchsbaum–Serre shows that if H-dim denotes projective dimension, then a local ring R is regular provided it admits a test module M with H-dim(M) finite. In a similar vein, one result I will discuss is joint work with Sather-Wagstaff, which shows that if H-dim denotes Gorenstein dimension, then R is Gorenstein provided it admits a test module M with H-dim(M) finite.