Quasi-projective dimensions of complexes over rings

TL;DR

This paper extends the concept of quasi-projective dimension from modules to complexes over rings, establishing foundational properties and exploring implications for ring theory.

Contribution

It introduces a generalized quasi-projective dimension for complexes, compares it with projective dimension, and addresses open questions in ring theory.

Findings

Derived Auslander-Buchsbaum formula for complexes of finite quasi-projective dimension

Conditions under which a ring is a complete intersection based on quasi-projective dimension

Behavior of quasi-projective dimension under quotient by regular sequences

Abstract

Quasi-projective dimension of modules over associative rings is generalized in this paper to the one of complexes of modules. Basic properties of this dimension are established, including a comparison result with projective dimension and a derived Auslander-Buchsbaum formula for complexes of finite quasi-projective dimension. Several sufficient conditions are provided for a commutative noetherian local ring to be a complete intersection under the assumption that each finitely generated module has finite quasi-projective dimension. This provides some positive answers to an open question on quasi-projective dimension proposed by Gheibi-Jorgensen-Takahashi. Moreover, the behavior of quasi-projective dimension under taking the quotient of a commutative ring modulo a regular sequence is investigated, and some partial results toward the change-of-rings question on quasi-projective dimension are given.