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Stabilization of the Spread-Global Dimension

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Pith's one-line read The spread-global dimension of T x Q stabilizes once T is large enough, with an explicit linear bound for grids.

desk verdict Settles the spread-global dimension stabilization conjecture with a clean, coherent proof; the main caveat is an imported monotonicity theorem that should be stated explicitly. read the letter →

arxiv 2506.01828 v2 pith:PLE7CJR2 submitted 2025-06-02 math.RT math.ATmath.CO

Benjamin Blanchette , Justin Desrochers , Eric J. Hanson , Luis Scoccola This is my paper
classification math.RTmath.ATmath.CO MSC 16G1018G2055N31
keywords spreadexactstructureposetrepresentationsspread-globaldimensiongridposetsresolutionspersistencetheoryrelativehomologicalalgebraleftKanextension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the spread exact structure on representations of posets, where the projective objects are the indicator representations of convex connected subsets, called spreads. It proves that for any finite poset $Q$, the spread-global dimension of $T \times Q$ is bounded independently of $T$, settling a conjecture for total orders. The key quantitative statement says that for a grid poset $G$, the supremum over all finite total orders $T$ is already attained at $T = [1 + 4|G|]$. The paper also shows that over any grid poset, spread-projective representations are exactly spread-decomposable ones and every finitely presented representation admits a finite spread-resolution. This matters because it makes spread-Betti-type invariants well-defined for finitely presented modules over grids, a setting used in persistence theory.

What carries the argument

The load-bearing tool is the left Kan extension $\mathrm{Lan}_{\iota}$ along an aligned grid inclusion $\iota: Q \to P$, that is, a product of inclusions of totally ordered sets. Proposition B proves $\mathrm{Lan}_{\iota}$ is exact and fully faithful, maps spread-decomposable representations to spread-decomposable representations, and maps spread-approximations to spread-approximations. A second pillar is Proposition 4.6, a general bound on relative global dimensions when a finite family of exact fully faithful functors satisfies four closure conditions. In the proof of Theorem C, the functors are the left Kan extensions $\mathrm{Lan}_{\iota}$ for origin aligned grid inclusions $[\ell] \times G \to [k] \times G$, and radical approximations are controlled explicitly: Proposition 4.11 lists the spreads that can occur in a minimal spread-radical approximation, and Proposition 5.6 builds a subgrid of size $1 + 4|G|$ whose image contains any given spread together with its radical-approximation domain.

What would settle it

Take a finite grid $G$, set $k = 1 + 4|G|$, and compute $\mathrm{gl.dim}_{\mathrm{sprd}}([\ell] \times G)$ for some $\ell > k$; if it exceeds $\mathrm{gl.dim}_{\mathrm{sprd}}([k] \times G)$, Theorem C is false. For $G = [2]$ the decisive comparison is between $[10] \times [2]$ and $[9] \times [2]$, a finite computation over incidence algebras using minimal spread-radical approximations.

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Extended reading notes

Core claim

The paper's central claims are Theorem A and Theorem C. Theorem A says that over a grid poset, the projectives for the spread exact structure are precisely the spread-decomposable representations, and every finitely presented representation has a finite spread-resolution. Theorem C says that for every finite poset $Q$, $n_Q = \sup_T \mathrm{gl.dim}_{\mathrm{sprd}}(T \times Q)$ is finite, and if $G$ is a finite grid then $n_G = \mathrm{gl.dim}_{\mathrm{sprd}}([1 + 4|G|] \times G)$. The corollaries include a positive answer to Conjecture 1, namely $\mathrm{gl.dim}_{\mathrm{sprd}}([k] \times [2]) = 2$ for $k \geq 4$, and a weakening of Conjecture 2 with a linear stabilization constant $1 + 4m$. The paper also notes that Theorem A cannot extend from grids to all upper semilattices.

Load-bearing premise

The proof depends on two imported facts: monotonicity of spread-global dimension under full embeddings, and the theorem that every finite poset has finite spread-resolutions; if either fails, the reduction from arbitrary posets to grids breaks.

Editorial extensions

If this is right

  • Over any grid poset, the spread exact structure has spread-decomposable projectives and finite spread-resolutions for every finitely presented representation, so relative Betti numbers and Koszul coresolutions are defined for these modules.
  • For every $k \geq 4$, the spread-global dimension of $[k] \times [2]$ is $2$, resolving the ladder-morphism case that motivated the stabilization conjecture.
  • For every $m$, the spread-global dimension of $[k] \times [m]$ is constant once $k \geq 1 + 4m$.
  • For a finite grid $G$ and any total order $T$, including infinite ones, the spread-global dimension of $T \times G$ is finite and bounded by $n_G$.
  • For every finite poset $Q$, the quantity $n_Q$ is finite, so arbitrarily wide products of $Q$ with total orders cannot push the spread-global dimension to infinity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The constant $1 + 4|G|$ is an existence bound, and the paper's own data suggest it is not sharp: for $[k] \times [2]$ stabilization holds already at $k = 4$, while the general theorem only ensures it at $k = 9$. Testing intermediate grids could reveal a tighter general rule.
  • Lemma 3.13 and Proposition B make it possible in principle to compute spread resolutions over infinite grids by pulling the presentation back to a finite aligned subgrid; the paper does not present this as an algorithm.
  • The failure for upper semilattices recorded in Remark 1.3 suggests that positive answers for m-tame representations, if they exist, will require different tools; the paper leaves this as an open question.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the spread exact structure on categories of finitely presented poset representations. Its main results are Theorem A, stating that over any grid poset the spread-projective representations are exactly the spread-decomposable ones and every finitely presented representation has a finite spread-resolution; Proposition B, showing that left Kan extensions along aligned grid inclusions preserve spread-approximations; and Theorem C, proving that for every finite poset Q the quantity n_Q = sup_T gl.dim_sprd(T×Q) is finite, with the sharp explicit bound n_G = gl.dim_sprd([1+4|G|]×G) for finite grids G. The paper also derives Corollaries D and E, giving a positive answer to Conjectures 1 and 2 of [AENY23] with a linear stabilization constant, and Corollary F for infinite total orders. The proof strategy is modular: a general transfer theorem (Proposition 4.6) bounds relative global dimensions using radical approximations, and the explicit description of spread-radical approximations (Proposition 4.11) is combined with combinatorial lemmas about spreads in products T×G to construct the needed grid inclusions. The manuscript is carefully written, contains a transparent Remark 1.3 documenting a limitation of an earlier draft, and gives a concrete counterexample showing that Theorem A does not extend to arbitrary upper semilattices.

Significance. If the results are correct, they resolve the stabilization conjecture of [AENY23] in a strong form and give the first explicit linear bound on spread-global dimension for products of a fixed finite poset with a varying total order. Theorem A justifies the use of spread-Betti-type invariants for finitely presented representations of grid posets, which is directly relevant to multiparameter persistence. The paper is notable for its explicit constants, its clean reduction of the infinite-grid case to finite grids via Lemma 3.13 and Proposition B, and its honest documentation of the failure of the main theorem for upper semilattices. The internal proof chain is detailed and appears coherent; the main caveat is that several load-bearing inputs are imported from other papers, especially [AET25, Thm. 1.2] and [AENY23, Prop. 4.5], and the manuscript does not state their exact hypotheses.

major comments (2)
  1. [Section 5.2, proof of Theorem C] The first reduction from an arbitrary finite poset Q to a containing grid G, and the case ℓ≤k in the second part of Theorem C, both invoke [AET25, Thm. 1.2] to conclude inequalities such as gl.dim_sprd(T×Q) ≤ gl.dim_sprd(T×G). This theorem is not stated in the paper, and its precise hypotheses are not given. Since the finiteness of n_Q for arbitrary finite Q, the monotonicity in ℓ, and Corollaries D–F all rest on this result, the authors should state the theorem verbatim and explicitly verify that its hypotheses hold for full embeddings of product posets under the spread exact structure. In particular, if the theorem requires an additional condition such as summand-injectivity of interval covers, that verification is essential; without it, the grid case is proved but the uniform statement for arbitrary finite posets is not.
  2. [Section 3, proofs of Proposition B and Theorem A] The proof of Proposition B invokes [AET25, Thm. 3.14] to conclude that padding along a convex full inclusion preserves spread-approximations, and the proof of Theorem A invokes [AENY23, Prop. 4.5] for the existence of finite spread-resolutions over finite posets. Both are external results that are load-bearing for the existence part of Theorem A and for the grid case of Theorem C. The paper should either state these results in the background section or give a precise statement with all hypotheses, and should confirm that [AENY23, Prop. 4.5] applies verbatim to the spread exact structure as defined here, since notation and conventions for interval representations vary across the literature.
minor comments (4)
  1. [Section 5.2, first paragraph of the proof of Theorem C] In the chain of inequalities, the expression 'gl.dimsprd([k]×Q)' should read 'gl.dimsprd([k]×G)', since the equality in the previous line is n_G = gl.dimsprd([k]×G).
  2. [Proposition 4.6] The statement of Proposition 4.6 uses the notation inj(Γ) before Γ is defined; Γ = End_Λ(⊕_{X∈X} X) is only introduced in the proof. The definition should be moved into the statement or given immediately before it.
  3. [Lemma 3.8, proof] The proof says 'For the second statement' twice; the second occurrence should be 'For the first statement', since it begins the argument for upset representations.
  4. [Corollary D, proof] The proof relies on [AENY23, Ex. 4.10] for the equality g_2(4)=...=g_2(9)=2; adding one sentence explaining how that example computes these values would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all load-bearing inputs are external theorems or independently verified structural results, and no claimed prediction reduces to a fit or definition.

full rationale

Walked the claimed derivation chain. Theorem C's reduction of an arbitrary finite poset Q to a grid uses the monotonicity inequality gl.dimsprd(T×Q) ≤ gl.dimsprd(T×G), imported from [AET25, Thm. 1.2], and Theorem A's base case uses the finite-poset finite-resolution result [AENY23, Prop. 4.5]. These are external, parameter-free statements whose assumptions do not include the stabilization conjecture, so importing them is evidence, not circularity. The only same-authors citation in the main proof is [BBH25, Prop. 6.8] (via Proposition 4.11) describing spread-irreducible morphisms; it is a structural input about morphisms, not a disguised version of the theorem being proved, and it does not determine the linear constant 1+4|G|. Proposition 4.6 is a general transfer bound whose hypotheses are verified internally by Proposition B, Lemma 3.3, Propositions 5.2 and 5.6, and Lemma 5.7; no fitted parameter is later relabelled as a prediction, and no equation reduces to its own input. Any concern about the unproved monotonicity theorem is a correctness risk about an external hypothesis, not a circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central theorems rest on several imported results: finite-poset finite spread-resolutions, monotonicity, the spread-irreducible classification, and a small computation for [k]×[2]. These are stated as prior literature rather than re-proved. No free parameters or invented entities appear. The bound's constant k=1+4|G| is explicit and proof-derived, not fitted.

assumptions (6)
  • domain assumption Every representation of a finite poset admits a finite spread-resolution (AENY23, Prop. 4.5).
    Used as the finite base case in Theorem A and as the source of finiteness for the spread-global dimension over [k]×Q in Theorem C.
  • domain assumption Spread-global dimension is monotone under full poset inclusions (AET25, Thm. 1.2).
    Load-bearing for the embedding argument in the first paragraph of Theorem C and for the inequalities in Corollaries D and E.
  • domain assumption Left Kan extension along an inclusion of an upset (padding functor) preserves spread-approximations (AET25, Thm. 3.14).
    Needed to finish Proposition B when the image of an aligned grid inclusion is a proper upset of the target grid.
  • domain assumption The morphisms between spread representations that are spread-irreducible are classified in BBH25, Prop. 6.8.
    This classification is used in Proposition 4.11 to describe the domain of a minimal spread-radical approximation, which is then bounded in Proposition 5.6.
  • domain assumption The values gl.dim_sprd([k]×[2])=2 hold for k=4,...,9 (AENY23, Ex. 4.10).
    Provides the finite check from which Corollary D extracts the exact stabilization value 2.
  • standard math Projectivization reduces add(X)-resolutions to projective resolutions over End(⊕X) (Proposition 2.18, from ARS95).
    Foundation for the relative global dimension bounds in Section 4.

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Pith. "Pith review of Stabilization of the Spread-Global Dimension." pith.science (2026). https://pith.science/paper/PLE7CJR2

@misc{pith2026250601828,
  author       = {Pith},
  title        = {Pith review of: Stabilization of the Spread-Global Dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PLE7CJR2}},
  note         = {Machine review of arXiv:2506.01828}
}
read the original abstract

Motivated by constructions from applied topology, there has been recent interest in the homological algebra of linear representations of posets, particularly in the context of homological algebra relative to non-standard exact structures. A prominent example is the spread exact structure on the category of representations of a fixed poset, in which the indecomposable projectives are the spread representations (that is, the indicator representations of convex and connected subsets). The spread-global dimension is known to be finite for finite posets and not uniformly bounded on the collection of all Cartesian products between two arbitrary finite total orders. It was conjectured in [AENY23] that the spread-global dimension is uniformly bounded on the collection of all Cartesian products between a fixed finite total order and an arbitrary finite total order. We provide a positive answer to this conjecture and, more generally, prove that the spread-global dimension is uniformly bounded on the collection of all Cartesian products between a fixed finite poset and an arbitrary finite total order. In doing so, we also establish the existence of finite spread-resolutions for finitely presented representations of arbitrary grid posets.

Figures

Figures reproduced from arXiv: 2506.01828 by the authors.

Figure 1
Figure 1. Dots represent the finite grid poset P = [31] × [19]. Note that in all figures in this paper, the poset order of a two-dimensional grid poset increases from left to right, and from bottom to top. The subsets S1, S2 ⊆ P are spreads, and the union S = S1 ∪ S2 is convex but not connected. Illustrated are also the minimal elements, maximal elements, covers, and cocovers of S, as well as ↑S \S and ↓S \S. Lemma 2.6 descri… view at source ↗
Figure 2
Figure 2. Left. A finite grid poset Q = [7]×[3] and a spread ↑A \ ↑B ⊆ Q. Right. A finite grid poset P = [9] × [3]. The crosses denote the image of an origin aligned grid inclusion ι : Q → P (Section 2.8), which is in particular an upper semilattice morphism (Lemma 2.7). Note that P = ↑ιQ, and that ⌊−⌋ι is a retraction for ι, as in Lemma 3.1. 3. Left Kan extension of spread-approximations In this section, we prove Proposition… view at source ↗
Figure 3
Figure 3. A spread-radical approximation over a grid poset [6] × [4]. The five spreads on the left constitute the summands in the domain of the radical approximation. The summands in the top row correspond to surjective spread-irreducibe morphisms, and each one corresponds to a cover of the spread. The summands in the bottom row correspond to injective spread￾irreducible morphisms, and each corresponds to a minimum of the spr… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The spread S in P = [9]×[7] serves as an example throughout Sections A and 5. We also consider the spreads V := S ⊔ (↓x \ ↓S) and W := S \ ⟨a, s⟩, corresponding to cases (1) and (2) of Proposition 4.11, respectively. Proposition 5.6. Let T be a finite total order, let …

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    1976

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