arxiv: 2603.25679 · v2 · submitted 2026-03-26 · 🧮 math.RA

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(n,d)-Coherent Rings

Rafael Parra

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Pith reviewed 2026-05-15 00:19 UTC · model grok-4.3

classification 🧮 math.RA

keywords coherent ringshomological algebraprojective dimensionfinitely presented modulesrelative homological theoryFP-injective modulesFP-flat modulesring characterizations

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The pith

Left (n,d)-coherent rings are characterized by closure and vanishing properties of the class of finitely n-presented modules with projective dimension at most d.

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

The paper develops relative homological algebra for the class FP_n^{≤d}(R) consisting of modules that are finitely n-presented and have projective dimension bounded by d. It proves that a ring is left (n,d)-coherent precisely when this class and its associated relative injective, projective, flat, and cotorsion modules satisfy a list of equivalent conditions involving Ext and Tor. A reader would care because these conditions give practical tests for generalized coherence without needing to check every finitely presented module directly. The results recover Costa's n-coherence when d is taken large enough and supply new tests for regularly coherent rings.

Core claim

The paper establishes several equivalent characterizations of left (n,d)-coherent rings in terms of the class FP_n^{≤d}(R) of finitely n-presented modules of projective dimension at most d together with the relative classes of FP_n^{≤d}-injective, FP_n^{≤d}-projective, FP_n^{≤d}-flat, and FP_n^{≤d}-cotorsion modules. These characterizations are obtained by developing the corresponding relative homological theory and showing that the ring satisfies the (n,d)-coherence condition if and only if the relative Ext and Tor functors behave in specific ways on this class.

What carries the argument

The class FP_n^{≤d}(R) of finitely n-presented modules of projective dimension at most d, which is used to define and test the relative homological properties that characterize the ring.

If this is right

When d is at least the global dimension of R or d equals infinity the characterizations reduce exactly to Costa's n-coherence.
New equivalent conditions are obtained for regularly coherent rings as a special case.
The relative FP_n^{≤d}-flat and FP_n^{≤d}-injective modules satisfy the usual closure properties under extensions, direct sums, and direct limits precisely when the ring is (n,d)-coherent.
Vanishing of relative Ext groups between modules in FP_n^{≤d}(R) and the relative injective class becomes a test for the ring property.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same pattern of relative classes could be used to define and study (n,d)-coherence over non-associative rings or rings without identity once the appropriate projective-dimension theory is fixed.
These characterizations may allow computational checks of coherence by restricting attention to a generating set of modules of bounded dimension rather than all finitely presented modules.
Connections to other finiteness conditions such as perfect rings or rings of finite global dimension become testable via the same relative module classes.

Load-bearing premise

The standard setup of homological algebra over associative rings with identity, including the existence of enough projective and injective modules, is assumed to hold.

What would settle it

A concrete ring R together with specific integers n and d for which the listed equivalent conditions on FP_n^{≤d}(R) and its relative module classes fail to hold while R is claimed to be (n,d)-coherent.

read the original abstract

We investigate finiteness conditions on modules of bounded projective dimension and their connection with generalized notions of coherence. For a ring $R$, we consider the class $\mathsf{FP}_n^{\le d}(R)$ of finitely $n$-presented modules of projective dimension at most $d$ and develop the corresponding relative homological theory. We establish several characterizations of left $(n,d)$-coherent rings in the sense of Mao and Ding [43], in terms of $\mathsf{FP}_n^{\le d}(R)$ and the associated classes of $\mathsf{FP}_n^{\le d}$-injective, $\mathsf{FP}_n^{\le d}$-projective, $\mathsf{FP}_n^{\le d}$-flat, and $\mathsf{FP}_n^{\le d}$-cotorsion modules. As a consequence, when $d \ge \gD(R)$ or $d=\infty$, we recover Costa's $n$-coherence [17] and obtain new characterizations of regularly coherent rings.

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.

Desk Editor's Note private letter to a colleague

The paper adds a projective dimension bound to n-coherence and gives relative characterizations, but the advance stays incremental within existing homological algebra.

read the letter

This paper defines FP_n^{≤d}(R) as the finitely n-presented modules with projective dimension at most d, then uses that class to characterize left (n,d)-coherent rings via the associated relative injective, projective, flat, and cotorsion modules. It recovers Mao-Ding (n,d)-coherence and Costa's n-coherence when d is large or infinite, and it supplies some fresh descriptions of regularly coherent rings along the way. The setup follows the usual pattern of relative homological algebra, relying on enough projectives and injectives plus standard closure properties, which keeps the claims consistent with prior work. The characterizations appear to be the main new output, stated directly in terms of the new class rather than retrofitted quantities. The approach is straightforward and avoids obvious circularity. The main limitation is scope: this is a technical refinement that organizes finiteness conditions with an extra parameter, but it does not introduce new methods or tackle wider questions in ring theory. The proofs likely use expected tools from the literature, so any gaps would probably be minor rather than load-bearing. Readers already working on coherence conditions or relative dimensions over rings will find the equivalences useful for reference. Someone outside that niche will not get much from it. The paper shows clear engagement with the existing results and deserves peer review as a competent, grounded extension even if the overall impact remains narrow.

Referee Report

0 major / 3 minor

Summary. The paper defines the class FP_n^{≤d}(R) consisting of finitely n-presented left R-modules of projective dimension at most d. It develops the associated relative homological algebra (FP_n^{≤d}-injective, FP_n^{≤d}-projective, FP_n^{≤d}-flat, and FP_n^{≤d}-cotorsion modules) and establishes several characterizations of left (n,d)-coherent rings in the sense of Mao-Ding. As a special case (d ≥ gl.dim(R) or d = ∞) the results recover Costa's n-coherence and yield new characterizations of regularly coherent rings.

Significance. If the stated equivalences hold, the work supplies a uniform framework that interpolates between classical coherence and n-coherence while incorporating a projective-dimension bound. The relative classes introduced are natural and the recovery of prior results is a clear strength; the characterizations may prove useful for studying rings of finite global dimension or regularly coherent rings.

minor comments (3)

[§2] §2, Definition 2.3: the precise meaning of 'finitely n-presented' when the projective dimension bound is imposed should be stated explicitly (e.g., whether the n-presentation is required to consist of modules already of pd ≤ d).
[Theorem 3.4] Theorem 3.4 (and analogous statements in §4): the proof that the relative Ext functor vanishes on FP_n^{≤d} modules would benefit from an explicit reference to the closure properties used, rather than citing only the general theory in [17] and [43].
[Introduction] Notation: the symbol gl.dim(R) is used without prior definition in the abstract and introduction; a brief reminder in §1 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. No specific major comments appear in the provided report, so there are no individual points to address point-by-point at this stage. We will incorporate any minor editorial or typographical changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces the class FP_n^{≤d}(R) via the standard definition of finitely n-presented modules of projective dimension ≤ d, then derives characterizations of left (n,d)-coherent rings (in the Mao-Ding sense) as equivalent conditions on the associated relative FP_n^{≤d}-injective, -projective, -flat, and -cotorsion classes. These are standard homological-algebra theorems resting on the existence of enough projectives/injectives and closure properties under the given finiteness conditions; no equation or claim reduces by construction to a fitted parameter, self-referential definition, or self-citation chain. External citations to Mao-Ding [43] and Costa [17] supply the target notion of coherence and the special case d=∞, but are not load-bearing for the new relative characterizations themselves. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates entirely within standard ring and module theory; no free parameters, no new postulated entities, and axioms are the usual ones of associative unital rings and abelian categories of modules.

axioms (1)

standard math R is an associative ring with identity and Mod(R) is the category of left R-modules.
Basic setup invoked throughout the abstract and all such papers.

pith-pipeline@v0.9.0 · 5462 in / 1233 out tokens · 59281 ms · 2026-05-15T00:19:06.098953+00:00 · methodology

discussion (0)

Reference graph

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