On the reducing projective dimension of the residue field

TL;DR

This paper investigates the reducing projective dimension of the residue field in local rings, establishing finiteness in many cases and exploring properties and examples related to this invariant.

Contribution

It answers affirmatively that the residue field has finite reducing projective dimension for a broad class of local rings and introduces new modules with finite reducing dimension despite infinite projective dimension.

Findings

Residue field has finite reducing projective dimension in many local rings.

Constructed examples of modules with infinite projective but finite reducing dimension.

Analyzed properties of reducing dimensions under local ring homomorphisms.

Abstract

In this paper we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya-Celikbas and Araya-Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension, and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings.