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Homological rigidity of quiver representations over $\mathbb{F}_1$

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Pith's one-line read Quiver representations over the field with one element have global dimension at most 2, classified exactly by whether the quiver is a point, bipartite, or neither.

desk verdict Universal gldim ≤ 2 for F1-quiver reps, with a clean orientation classification; the combinatorial lift is the real contribution and it holds up. read the letter →

arxiv 2607.10253 v1 pith:2IM2B3D7 submitted 2026-07-11 math.RT math.RA

Changjian Fu , Liang Yang , Zhiyuan Zeng This is my paper · ORCID
classification math.RTmath.RA MSC 16G2018G1514A23
keywords F1-representationsquiverYonedaextensionsglobaldimensionproto-exactcategorybipartitenilpotentrepresentations
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

Over ordinary fields the global dimension of a quiver representation category can be large, but over the virtual field F1 the category is non-additive and classical homological tools fail. This paper proves that every higher Yoneda extension group vanishes after degree two, for every quiver (finite or infinite) and for both ordinary and nilpotent representations. The global dimension is therefore always 0, 1 or 2. The precise value is read off from the orientation alone: a single vertex gives dimension 0, a bipartite quiver gives dimension 1, and any non-bipartite quiver (one containing an oriented cycle or a linearly oriented A3) gives dimension 2. The result supplies a rigid, orientation-only dictionary that replaces the usual derived-category machinery and makes the Euler form of the associated Hall algebra well-defined.

What carries the argument

Lemma 3.1, a combinatorial lifting that replaces the octahedral axiom: given a sequence L o M↠N exact at M and N, one constructs an auxiliary representation fW on the pointed sets Li⊕Ni whose maps Wα are defined by a three-case rule using the duals of the original maps; the resulting short exact sequence L↣fW↠N forces every 3-extension class to vanish.

What would settle it

Exhibit a single quiver (finite or infinite) and a concrete triple of representations for which the case-by-case construction of Lemma 3.1 fails to yield an F1-linear map, or for which a non-split 3-extension class survives.

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Extended reading notes

Core claim

For an arbitrary quiver Q the global dimension of the category of F1-representations is at most 2, and likewise for the full subcategory of nilpotent representations. When Q is connected the dimension equals 0 if Q is a single vertex, equals 1 if Q is bipartite but not a single vertex, and equals 2 if and only if Q is non-bipartite.

Load-bearing premise

The three-case combinatorial rule that defines the arrow maps of the auxiliary representation must always produce genuine F1-linear maps, even for infinite quivers or multi-arrow configurations.

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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

0 major / 5 minor

Summary. The paper studies the non-additive category rep(Q,F1) of quiver representations over the virtual field F1 (and its nilpotent subcategory). Using Yoneda’s construction of Ext groups as pointed sets, it proves that Ext^n vanishes for all n≥3 for an arbitrary quiver Q (finite or infinite). Consequently gldim rep(Q,F1)≤2 and gldim rep(Q,F1)^nil≤2 (Theorem 1.2). For connected Q the global dimension is completely classified: it equals 0 precisely when Q is a single vertex, 1 when Q is bipartite but not a single vertex, and 2 when Q is non-bipartite (i.e., contains an oriented cycle or a linear A3 subquiver) (Theorem 1.3). The key technical step is a combinatorial lifting construction (Lemma 3.1) that replaces the octahedral axiom and produces a short exact sequence realizing any exact sequence of length three as equivalent to zero. Classification then follows from exact embedding/restriction functors for subquivers together with base cases for A2, A3 and oriented cycles.

Significance. The result is a genuine rigidity theorem in a setting where classical homological algebra is unavailable. The universal bound gldim≤2 for every quiver, including infinite ones, is unexpected and sharply contrasts with the classical global dimension of path algebras. The classification by orientation structure (bipartite versus non-bipartite) is clean and complete. The combinatorial lift of Lemma 3.1 is an original non-additive substitute for the octahedral axiom and is of independent interest for proto-exact categories. The proofs are fully written and self-contained for the vanishing statement; the classification rests on transparent functors plus previously published base cases. The work therefore settles a natural question raised by earlier computations and supplies a solid foundation for further study of Hall algebras and Euler forms over F1.

minor comments (5)
  1. Remark 3.2 asserts uniqueness of the lift fW up to isomorphism but only sketches the argument; a one-sentence expansion (that the three-case formula is forced by commutativity of diagram (3.1) once the underlying pointed sets are fixed as L_i⊕N_i) would make the claim fully rigorous.
  2. Section 4.1: the functors are called “embedding/retriction”; correct the typo “retriction” to “restriction” throughout.
  3. Lemma 4.1 and the subsequent maps ι_i, Res_i are stated only for i=1,2; a brief remark that the same argument works for all n would clarify that the functors remain exact in higher degrees (even though higher Ext already vanish).
  4. In the definition of W_α (Lemma 3.1) the dual maps g^t and f^t are used without an explicit reminder of their domain/codomain; a parenthetical reference back to §2.1 would help readers less familiar with F1-linear maps.
  5. Acknowledgements mention AI tools for conceptual inspiration of Lemma 3.1; while transparent, the journal may prefer a shorter, more conventional formulation.

Circularity Check

2 steps flagged · score 1.0 of 10

No significant circularity: universal vanishing is proved by an independent combinatorial construction; self-citations supply only base cases for the classification.

  1. self citation load bearing [Proposition 4.3 / proof of Theorem 1.3(3)]
    "By [FRY24, Theorem 3.10], Ext^{2}_{Q'}(-,-) eq0. It follows that Ext^{2}_Q(-,-) eq0 by Lemma 4.1. According to Theorem 1.2, gldim rep(Q,F_{1})=2."

    The non-vanishing of Ext^{2} for any quiver containing a linear A3 subquiver is imported from the authors' earlier paper FRY24 rather than re-proved. The citation is used only as a base case; the universal upper bound ≤2 is independent, so the circularity is minor.

  2. self citation load bearing [Lemma 4.4 / Proposition 4.6]
    "The following has been established in [FYZ26]. Lemma 4.4. For any i,j∈Q'_{0}, we have Ext^{2}_{Q'}(S_i,S_j)^{nil} ≅ N … By Lemma 4.5, Lemma 4.4 and Lemma 4.1, we conclude that Ext^{2}_Q(-,-) eq0."

    Non-vanishing of Ext^{2} for oriented cycles is taken from the authors' concurrent paper FYZ26. Again the citation supplies only a base case for the classification; the vanishing proof of Section 3 does not rely on it.

full rationale

The central claim (Theorem 1.2) that Ext^n vanishes for all n≥3 is established in Section 3 by an explicit non-additive lifting construction (Lemma 3.1) that replaces the octahedral axiom. The construction defines W_i = L_i ⊕ N_i and a three-case formula for each W_α using dual maps, then verifies F1-linearity by exhaustive case analysis on preimages; the argument never invokes prior Ext computations or fitted parameters and applies verbatim to infinite quivers. Classification (Theorem 1.3) uses exact embedding/restriction functors (Lemma 4.1) together with three base cases: Ext^2 eq 0 for the linear A3 (cited from FRY24), Ext^2 eq 0 for oriented cycles (cited from FYZ26), and Ext^1 eq 0 for A2 / Ext^1 = 0 for a single vertex (again FRY24). Those citations are load-bearing only for the exact values 0/1/2, not for the universal upper bound of 2; the logical dependence is therefore ordinary and non-circular. No self-definitional identities, fitted-input-as-prediction steps, or uniqueness theorems imported from the authors appear.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The paper works entirely inside the already-established proto-exact category of F1-representations. No free parameters are fitted. The only non-standard ingredients are the combinatorial lifting of Lemma 3.1 (an ad-hoc construction that replaces the octahedral axiom) and the definition of global dimension via Yoneda Ext as a pointed set. Background results on coefficient quivers, Krull–Schmidt and earlier Ext computations for A_n and cycles are taken from the literature.

assumptions (3)
  • domain assumption rep(Q,F1) is a proto-exact (equivalently proto-abelian) category in which monomorphisms and epimorphisms are admissible, so Yoneda’s construction of Ext^n as equivalence classes of exact sequences of length n+2 is well-defined as a pointed set.
    Invoked throughout Sections 2–3; taken from Dyckerhoff–Kapranov and earlier F1-representation papers.
  • domain assumption Krull–Schmidt and Jordan–Hölder hold in rep(Q,F1); coefficient quivers detect indecomposability and nilpotence.
    Used in Lemmas 2.1–2.2 and Corollary 2.3; cited from Szczesny and Jun–Sistko.
  • ad hoc to paper The three-case combinatorial formula for W_α on L_i ⊕ N_i (using dual maps g^t and f^t) yields an F1-linear map for every arrow, including when Q is infinite.
    Core of Lemma 3.1; verified by a three-case injectivity argument that is the load-bearing new construction.
invented entities (1)
  • combinatorial lift fW (the representation L_i ⊕ N_i with the three-case arrow maps of Lemma 3.1)
    purpose: Provides a short exact sequence that factors any partial exact sequence, replacing the octahedral axiom and forcing Ext^3 = 0.
    Explicitly constructed in the proof of Lemma 3.1; uniqueness up to isomorphism is claimed in Remark 3.2 but not given an independent external characterisation.

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Pith. "Pith review of Homological rigidity of quiver representations over $\mathbb{F}_1$." pith.science (2026). https://pith.science/paper/2IM2B3D7

@misc{pith2026260710253,
  author       = {Pith},
  title        = {Pith review of: Homological rigidity of quiver representations over $\mathbbF_1$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2IM2B3D7}},
  note         = {Machine review of arXiv:2607.10253}
}
abstract

We establish a homological rigidity phenomenon for the category of representations of quivers over the virtual field $\mathbb{F}_1$, which is inherently non-additive and does not admit classical homological algebra tools. We prove that all higher Yoneda extension groups vanish beyond degree two for arbitrary quivers, including infinite ones. Consequently, the global dimension of the category is universally bounded by 2. Moreover, we obtain a complete classification of quivers according to their homological dimension, which is determined solely by the underlying orientation structure.

Discussion (0). Continue with ORCID to comment.

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