Persistence of homology over commutative noetherian rings

TL;DR

This paper introduces new classes of noetherian local rings where vanishing of certain homological functors at high degrees implies modules have finite projective or injective dimension, advancing understanding of homological properties over these rings.

Contribution

It identifies novel classes of noetherian local rings with specific homological vanishing properties that imply finiteness of projective or injective dimensions of modules.

Findings

Vanishing of Tor at high degrees implies finite projective dimension.

Vanishing of Ext at high degrees implies finite projective or injective dimension.

New classes of rings with these homological properties are described.

Abstract

We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $i\gg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $i\gg 0$ implies that $M$ has finite projective dimension or finite injective dimension.