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Global dimensions of local geodesic ghor algebras

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Pith's one-line read For a geodesic ghor algebra on a genus g surface, each cyclic localization has global dimension at most 2g+1, attaining equality exactly over the noetherian locus of the center.

desk verdict Plausible, genuinely new main theorem, but the proof as written does not get from projective dimensions of simples to global dimension, and the exactness of the constructed resolution is asserted rather than shown. read the letter →

arxiv 2412.04511 v1 pith:DOMH4SNU submitted 2024-12-04 math.RA

Karin Baur , Charlie Beil This is my paper · ORCID
classification math.RA MSC 16G2016S3816S50
keywords dimeralgebramodelJacobianquiverwithpotentialgaugetheorygeodesicghorcycliclocalizationglobaldimension
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

This paper proves a sharp homological bound for a class of noncommutative algebras built from dimer quivers on compact surfaces. For any geodesic ghor algebra on a smooth genus $g\ge 1$ surface, every cyclic localization at a maximal ideal of the cycle algebra has global dimension at most $2g+1$, which is also the Krull dimension of the center. The bound is an equality precisely when the localization point lies in the noetherian locus of the center; outside that locus the global dimension is strictly smaller. The result matters because it recovers, for higher-genus surfaces, the torus phenomenon in which the global dimension of a localization matches the Krull dimension of the center.

What carries the argument

The load-bearing object is a geodesic ghor algebra $A$ on a surface, along with its cycle algebra $S$, its center $R$, and the noetherian locus $U_{S/R}$ where $R$ and $S$ coincide locally. The argument is carried by the explicit projective complex (6.4), assembled from two pieces: a Koszul-like exterior part whose basis elements are linearly independent classes in $H_1(\Sigma)$ (with the cycle $\sigma$ adjoined when the simple module has full dimension vector) and whose differential uses differences $s_j - \tilde{s}_j e_i$ attached to geodesic cycles; and a generalized part modelled on the classical torus resolution that handles homotopic paths with zero scalars. The paper's Theorem 5.7 identifies the kernel of the augmentation map as being generated by exactly these elements, which makes (6.4) a projective resolution and yields the length bound.

What would settle it

Take a geodesic ghor algebra on a genus $2$ surface, fix a simple module with full dimension vector whose annihilator is a maximal ideal $\mathfrak{n}$ in $\operatorname{Max} S$, and compute the homology of the complex (6.4) at the first projected step: if the elements $s_j - \tilde{s}_j e_i$ fail to generate $\mathfrak{n}$, or if the complex has nonzero homology anywhere, the projective dimension is not $5$ and the theorem fails. A cheaper check is to find a single maximal ideal in the noetherian locus where a different choice of geodesic representatives changes the length of the minimal resolution.

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Extended reading notes

Core claim

The central claim is that the global dimension of a cyclic localization of a geodesic ghor algebra is controlled by the topology of the underlying surface. On a surface obtained by identifying opposite sides and vertices of a convex $2N$-gon, the paper proves that every cyclic localization at a maximal ideal of the cycle algebra has global dimension at most $N+1 = \dim R = \dim S$, with equality if and only if (for $N\ge 3$) the maximal ideal lies in the noetherian locus $U_{S/R}$. For a smooth genus $g$ surface this becomes global dimension at most $2g+1 = \operatorname{rank} H_1(\Sigma)+1$. The proof builds explicit projective resolutions of simple modules: a Koszul-like complex indexed by independent classes in $H_1(\Sigma)$, extended by a generalized version of the classical torus resolution for homotopic paths, with the first syzygy identified by the paper's Theorem 5.7. Full-dimension simple modules force the resolution to have length $N+1$ over the noetherian locus, while smaller dimension vectors shorten it to length at most $\max\{3,N\}$.

Load-bearing premise

The proof leans on the assertion, stated rather than demonstrated, that the geodesic cycles chosen to represent a homology basis generate the maximal ideal of the localization and form a regular sequence there, which is what makes the constructed complex a genuine projective resolution.

Editorial extensions

If this is right

  • Every cyclic localization of a geodesic ghor algebra on a smooth genus $g$ surface is homologically finite, with global dimension at most $2g+1$.
  • A localization has maximal possible global dimension $2g+1$ exactly when its point lies in the noetherian locus, so the global dimension functions as a detector of that locus.
  • The $g=1$ (torus) case reproduces the known formula $\operatorname{gldim} A_m = 3 = \dim R$ for noetherian dimer algebras on a torus.
  • The explicit resolution gives a practical way to compute projective dimensions of simple modules: length $N+1$ for full-support simples and length at most $\max\{3,N\}$ otherwise.
  • The equality $\operatorname{gldim} A_n = \dim R = \dim S$ ties homological dimension to the rank of the first homology group plus one, making the bound topological rather than accidental.

Reading between the lines

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

  • If the equality statement is right, global dimension could serve as a homological definition of the noetherian locus for other nonnoetherian matrix rings that admit a depiction, not just ghor algebras.
  • The paper's conjecture that the strict inequality over the nonnoetherian locus is governed by a geometric height would turn the bound into a precise formula; the pinched-torus examples in Section 3 are natural test cases to compute explicitly.
  • The homology-indexed exterior construction suggests that algebras whose relations are governed by geodesic cycles on any surface should admit analogous resolutions, with length one plus the rank of the relevant homology group.
  • One could try to extend the result to surfaces with punctures or boundary, where the first homology is no longer free abelian of rank $2g$; the expected bound would involve the first Betti number and boundary data.
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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

3 major / 4 minor

Summary. The paper studies geodesic ghor algebras A on compact surfaces obtained by identifying opposite sides and vertices of a convex 2N-gon, with center R and cycle algebra S. Its main result, Theorem 6.3, asserts that for every maximal ideal n in Max S, the cyclic localization A_n has global dimension at most dim R = dim S = N+1, with equality precisely when n lies in the noetherian locus U_{S/R} (for N >= 3). The proof strategy is to construct, in Section 6, a finite projective resolution for each simple A_n-module, of length N+1 for modules of maximal dimension vector and of length at most max{3,N} otherwise, and to combine this with Proposition 3.2, which bounds projective dimensions of finite-dimensional indecomposable modules in terms of simple modules.

Significance. If correct, the theorem establishes a striking and genuinely new relation between homological dimension of noncommutative localizations and the topology of the underlying surface: gldim A_n <= rank H_1(Sigma)+1, with equality exactly over the noetherian locus. The paper also offers a concrete conjectural refinement via geometric height, and the adaptation of the Berenstein-Douglas complex to higher genus surfaces is a valuable idea. The authors are careful to situate the result within their earlier work on geodesic structure and depictions of the center. However, the proof as written contains several load-bearing gaps, so the main theorem is not yet established to the standard required for publication.

major comments (3)
  1. [Section 6, Theorem 6.3 (proof)] The proof says 'Follows from Proposition 3.2 and Theorem 6.2', but Proposition 3.2 bounds projective dimension only for finite-dimensional indecomposable A-modules. Global dimension is the supremum of projective dimensions over all left modules, and A_n is not shown to be left artinian (indeed it is nonnoetherian outside U_{S/R}). No argument is given that sup_M pd_{A_n}(M) equals the supremum over finite-dimensional simple modules; direct limits do not preserve projective dimension, so the desired inequality gldim A_n <= N+1 does not follow from the stated results. This is a load-bearing gap that requires an additional reduction or a different argument.
  2. [Section 6, Theorem 6.2 and complex (6.4)] Theorem 6.2 asserts that (6.4) is a projective resolution for every simple A_n-module, but exactness is not established. The maps (6.2) presuppose that for every alpha in T (or T^sigma) there is a geodesic cycle s_j in e_i A_n e_i with [s_j] = alpha, and for every (alpha_1, alpha_2) a cycle s homotopic to t_1 t_2; the paper does not prove that such representatives exist compatibly for the stated homology classes. Moreover, exactness of the Koszul-like part requires that the elements s_j - ~s_j e_i generate the maximal ideal n and form a regular sequence on S_n (or A_n); no regularity argument is supplied. The sentence 'Therefore ... (6.4) is a projective resolution by Theorem 5.7' invokes only the first-syzygy description and does not justify exactness at the higher syzygy modules.
  3. [Section 6, Theorem 6.3 equality statement] The 'if and only if' for equality is not supported by the cited results. Theorem 6.2 gives projective dimensions for simple modules in terms of their dimension vector, and Proposition 5.2 shows that a simple module with dimension vector 1_Q0 has annihilator in the noetherian locus. But the equality statement additionally requires that every n in U_{S/R} admits a simple A_n-module of dimension vector 1_Q0 with pd = N+1, and that n outside U_{S/R} forces pd < N+1 for all simple modules. Neither direction is proved or cited, so the equality claim in Theorem 6.3 is currently unsupported.
minor comments (4)
  1. [Section 3, first paragraph] 'filteration' should be 'filtration'.
  2. [Section 6, proof of Theorem 6.2] 'minimimal' should be 'minimal'.
  3. [Definition 2.3] The phrase 'a representative of p + ker eta' is ambiguous; a representative should be a path in Q, not an element of the quotient algebra.
  4. [Section 6, equations (6.3)] The paths u_j, v_j, r_j in (6.3) are not defined with enough precision; the reference to 'Figure 5.ii' should be made consistent with the figure labels, and the minimality conditions on these paths should be stated explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction of the main theorem to its inputs; the proof relies heavily on the authors' prior structural results, but none assumes the target result.

full rationale

I found no step in which a stated conclusion is equivalent to an input by construction, or in which a fitted parameter is renamed as a prediction. The central new objects—cyclic localizations A_n, the noetherian locus U_{S/R}, and the claimed bound gldim A_n ≤ dim R = dim S = N+1—are defined independently of the theorem. The heavy use of the same authors' earlier work ([2], [4], [5], [7], [8]) supplies Definition 2.3, the homology criterion (2.2), the identification of R with the center, and the description of the noetherian locus; these are parameter-free structural results that do not assume Theorem 6.3, so they are self-citations but not load-bearing circularity. The proof of Theorem 6.3 is terse ('Follows from Proposition 3.2 and Theorem 6.2'), and Theorem 6.2 asserts that (6.4) is a projective resolution without proving exactness or regularity; moreover Proposition 3.2 only bounds projective dimensions of finite-dimensional modules, so the passage to the global dimension of all A_n-modules requires an additional argument. These are substantive correctness gaps, not circular reductions: the resolution length N+1 and the equality with dim R are not wired into the definitions, and the theorem would succeed or fail independently of any fitted input. Accordingly the circularity score is 2, reflecting minor self-citation reliance rather than a derivation that reduces to its own assumptions.

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

No constants are fitted to data. The central claim rests on the geodesic hypothesis from [2] and on two unproved regularity/existence assertions in the construction of the resolution; the main theorem also inherits the uncountable-field hypothesis from [2, Theorem 4.7] without stating it.

assumptions (6)
  • domain assumption The ghor algebra A is geodesic in the sense of Definition 2.3: for every reduced cycle p, there is a set C[p] of parallel geodesic cycles covering Q0 and containing a cycle homologous to p.
    Imported from [2, Definition 2.6]; it implies ker η = ker τ, the homology/σ-criterion (2.2), and the syzygy generators in Section 5.
  • domain assumption Σ is obtained from a convex 2N-gon by identifying opposite sides and all vertices, so rank H1(Σ) = N and dim S = dim R = N+1.
    Sets the topological parameter N used throughout Section 6; the equality of Krull dimensions is cited from [2].
  • domain assumption The base field k is algebraically closed; the noetherian-locus depiction from [2, Theorem 4.7] used to identify Max R with Max S requires k uncountable, a hypothesis not restated in Theorem 6.3.
    Proposition 5.2 and the equality claim depend on the geometry of the noetherian locus; Lemma 5.1 uses Hilbert's Nullstellensatz.
  • ad hoc to paper For each homology class α in T (or T^σ) there exists a geodesic cycle s_j in e_i A_n e_i with [s_j] = α, and for every pair (α1, α2) a geodesic cycle homotopic to t1 t2; these representatives can be chosen compatibly over the polygon.
    This existence is needed to define δ_im and δ'_i2 in (6.2); it is asserted without proof or reference in Section 6.
  • ad hoc to paper The elements s - ~s e_i (for all cycles without cyclic subpaths) generate the maximal ideal n and form a regular sequence on S_n (or A_n), so the Koszul-like complex (6.4) is exact.
    This is the key step in Theorem 6.2's proof, stated in one sentence; no regularity argument is supplied.
  • ad hoc to paper Projective dimension of any module over A_n is controlled by finite-dimensional simple modules.
    Proposition 3.2 proves only the finite-dimensional indecomposable case; the paper uses it to conclude gldim for all modules without proof.

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Pith. "Pith review of Global dimensions of local geodesic ghor algebras." pith.science (2026). https://pith.science/paper/DOMH4SNU

@misc{pith2026241204511,
  author       = {Pith},
  title        = {Pith review of: Global dimensions of local geodesic ghor algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DOMH4SNU}},
  note         = {Machine review of arXiv:2412.04511}
}
abstract

A ghor algebra is a path algebra with relations of a dimer quiver in a compact surface. We show that the global dimension of any cyclic localization of a geodesic ghor algebra on a genus $g \geq 1$ surface is bounded above by $2g+1$.This number coincides with the Krull dimension of the center of the ghor algebra. We further show that the bound is an equality if and only if the point of localization sits over the noetherian locus of the center.

Figures

Figures reproduced from arXiv: 2412.04511 by the authors.

Figure 1
Figure 1. The dimer quiver Q and its perfect matchings for Example 2.2. Example 2.2. Consider the dimer quiver Q on a torus in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. A geodesic ghor algebra on a genus 2-surface Σ, obtained by identifying the opposite sides, and the corner vertices, of the oc￾tagon. (i) The two cycles p, q are geodesic, and homologous but not homotopic. (ii) A set of geodesic cycles r1, . . . , r8 that spans H1(Σ) over N. (iii) A set of parallel geodesic cycles C[p] = {p, q, r1, s1, s2} that covers Q0 and contains p. Let Q be a dimer quiver on Σ, and let A = kQ/ … view at source ↗
Figure 3
Figure 3. The support of some simple modules, drawn in blue, over a ghor algebra on a pinched torus. 4. Cyclic localizations Let A be a k-algebra with prime center R. The central localization of A at a prime ideal p ∈ Spec R is the tensor product (4.1) Ap := A ⊗R Rp. The algebras we consider here are (isomorphic to) matrix rings A = [Aij ]i,j ⊆ Mn(B), where B is an integral domain k-algebra that contains R. 8 Note that each A… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The setup for the Berenstein-Douglas complex and its gen￾eralization in (6.3). Here, aj , bj ∈ Q1 are arrows; uj , vj ∈ Q≥0 are paths; and bjujaj , bj−1vjaj are unit cycles. complex 0 → Aei ·[ b1 ··· bn ] −→ M bk∈eiQ1 Aet(bk) ·      u1 −v2 0 ··· 0 0 u2 −v3 0 0 0 u…
Figure 5
Figure 5. Figure 5: On a torus, commutation relations of cycles are homotopy relations. The cycles p, q ∈ eiAei are drawn on the torus cover R 2 , with each square a fundamental domain. V . In particular, annS V = n. Recall the set of paths Pi from Proposition 5.6. Set Q ′ 0 := {i ∈ Q0 | …

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