Homological dimensions of rigid modules

Source abstract

We obtain various characterizations of commutative Noetherian local rings (R,m) in terms of homological dimensions of certain finitely generated modules. Our argument has a series of consequences in different directions. For example, we establish that R is Gorenstein if the Gorenstein injective dimension of the maximal ideal m of R is finite. Moreover, we prove that R must be regular if a single ExtRn(I,J) vanishes for some integrally closed m-primary ideals I and J of R and for some positive integer n.