Quasi-projective dimension

TL;DR

This paper introduces the quasi-projective dimension, a new homological invariant that generalizes projective dimension, and explores its properties and implications in module theory over certain rings.

Contribution

It defines the quasi-projective dimension and proves key properties, including its finiteness over regular local rings and the validity of classical formulas for modules with finite quasi-projective dimension.

Findings

Modules over quotient of regular local rings have finite quasi-projective dimension.

Auslander–Buchsbaum and depth formulas extend to modules with finite quasi-projective dimension.

Vanishing results for Tor and Ext hold for modules with finite quasi-projective dimension.

Abstract

In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the following. (1) Over a quotient of a regular local ring by a regular sequence, every finitely generated module has finite quasi-projective dimension. (2) The Auslander--Buchsbaum formula and the depth formula for modules of finite projective dimension remain valid for modules of finite quasi-projective dimension. (3) Several results on vanishing of Tor and Ext hold for modules of finite quasi-projective dimension.