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arxiv: 2604.09279 · v1 · submitted 2026-04-10 · 🧮 math.RA · math.AC

Quasi-projective dimensions of complexes over rings

Hongxing Chen , Jiangsheng Hu , Xiaoyan Yang This is my paper

Pith reviewed 2026-05-10 16:49 UTC · model grok-4.3

classification 🧮 math.RA math.AC
keywords dimensionquasi-projectivecomplexescommutativefinitemodulesquestionring
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The pith

Quasi-projective dimension is extended to complexes with a derived Auslander-Buchsbaum formula and conditions characterizing complete intersection rings via finite dimensions on all finitely generated modules.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Quasi-projective dimension measures how far a module is from being quasi-projective, a property related to direct summands of free modules. The authors lift this notion to chain complexes of modules, which are sequences of modules connected by maps. They establish that this new dimension behaves similarly to the classical projective dimension in many ways and derive a version of the Auslander-Buchsbaum formula that relates the dimension to depth and other invariants for complexes of finite quasi-projective dimension. For commutative noetherian local rings, they prove that if every finitely generated module has finite quasi-projective dimension, then under additional hypotheses the ring must be a complete intersection. This gives partial answers to a question posed by earlier researchers. They also examine how the dimension changes when the ring is quotiented by a regular sequence.

Core claim

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.

Load-bearing premise

The generalization of quasi-projective dimension to complexes preserves enough of the module-theoretic properties to allow the comparison result, the derived Auslander-Buchsbaum formula, and the sufficient conditions for complete intersections to hold over commutative noetherian local rings.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript generalizes the quasi-projective dimension of modules over associative rings to complexes of modules. It establishes basic properties including a comparison result with projective dimension and a derived Auslander-Buchsbaum formula for complexes of finite quasi-projective dimension. For commutative noetherian local rings, it provides several sufficient conditions under which the ring is a complete intersection whenever every finitely generated module has finite quasi-projective dimension, yielding positive answers to an open question of Gheibi-Jorgensen-Takahashi. It further investigates the behavior of the dimension under quotients by regular sequences and supplies partial results toward the change-of-rings question.

Significance. If the central results hold, the work extends a homological invariant to the derived setting in a manner that preserves key comparison and formula properties, enabling applications to complexes in derived categories. The sufficient conditions for complete intersections directly address an open question and supply concrete criteria linking finite quasi-projective dimension of all finitely generated modules to ring structure; this is a substantive contribution to the homological characterization of commutative rings. The treatment of regular sequences and partial change-of-rings results adds incremental but useful information on functoriality. The manuscript supplies explicit definitions, basic properties, and the required comparison and formula results before specializing to the commutative local case.

minor comments (2)
  1. The statement of the open question from Gheibi-Jorgensen-Takahashi in the introduction would benefit from a direct quotation or precise formulation so that the positive answers are immediately visible to readers.
  2. In the section treating quotients by regular sequences, the partial results on change-of-rings could be accompanied by a brief remark on why a full change-of-rings theorem fails, even if only by reference to a known counterexample in the module case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive recommendation of minor revision. The provided summary accurately reflects the scope and results of the paper.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines quasi-projective dimension for complexes, proves comparison with projective dimension and a derived Auslander-Buchsbaum formula using standard homological algebra techniques, then derives sufficient conditions for complete intersection rings from the finite quasi-projective dimension assumption on modules. These steps rely on explicit definitions and direct proofs rather than self-referential loops, fitted parameters renamed as predictions, or load-bearing self-citations. The open question addressed is from independent authors (Gheibi-Jorgensen-Takahashi). No step reduces by construction to its inputs; the central claims have independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard background from homological algebra over associative rings and commutative noetherian local rings. No new free parameters, invented entities, or ad-hoc axioms are introduced in the summary.

axioms (2)
  • standard math Quasi-projective modules and their dimensions satisfy the usual properties over associative rings that allow generalization to complexes.
    The entire generalization rests on this background from module theory.
  • domain assumption Commutative noetherian local rings admit well-defined notions of complete intersection and regular sequences compatible with homological dimensions.
    Invoked for the characterization results and change-of-rings investigation.

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Reference graph

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