On the left perpendicular category of the modules of finite projective dimension

TL;DR

This paper explores properties of commutative noetherian local rings through the lens of the left perpendicular category of modules with finite projective dimension, linking regularity to the existence of specific test modules.

Contribution

It provides new characterizations of regular local rings and semidualizing modules using the structure of the left perpendicular category of finitely generated modules.

Findings

A local ring is regular iff a strong test module for projectivity with finite projective dimension exists.

Characterizations of properties of rings via the left perpendicular category.

Results connecting semidualizing modules to ring properties.

Abstract

In this paper, we characterize several properties of commutative notherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application we prove that a local ring is regular if (and only if) there exists a strong test module for projectivity having finite projective dimension. We also obtain corresponding results with respect to a semidualizing module.