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Publication metadata

Journal citation
Advances in Mathematics 453, 109822 (2024)
Publication date
2024-09
arXiv counterpart
arXiv:2209.11622v2
MSC 2020
13F60, 53D17, 16G30, 17B37
As specified in the publication (MSC 2020)
13F60, 53D17, 16G30, 17B37 ↗Version-of-record first page (primary 13F60; secondary 53D17, 16G30, 17B37)

Abstract

The paper identifies an explicit Zariski-open torus orbit of symplectic leaves for every finitely generated upper cluster algebra with a GSV Poisson structure and relates it to fully Azumaya loci of root-of-unity upper quantum cluster algebras.

Role in dependence graphs

Proof-critical source

On Lagrangianity of pp-supports of holonomic DD-modules and qq-DD-modules

This paper is included only for the following marked statements:

  • Proposition 4.6 · Advances in Mathematics article 109822, pp. 24–25The singular locus and the complement of the initial-plus-adjacent cluster charts have codimension at least two.
  • Theorem 4.1 (Theorem A) · Advances in Mathematics article 109822, pp. 20–26The smooth locus away from non-inverted frozen divisors is one open torus orbit of symplectic leaves.

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Exact reviewed source

Version of record · Advances in Mathematics 453 (2024), Article 109822

Greg Muller, Bach Nguyen, Kurt Trampel, Milen Yakimov. Poisson geometry and Azumaya loci of cluster algebras. Advances in Mathematics 453 (2024), Article 109822.

Open audited source ↗
Generated August 27, 2026
01Statements4 reported findingsSuspected incorrect statements

The Poisson-side Theorem A and the construction of the root-of-unity Poisson order are correct. The upper inclusion in the quantum-side Theorem B/Theorem 6.1 is false: a noncentral non-inverted frozen generator need not lower the PI degree after it is set to zero. An explicit strict mixed-quantum-torus example gives fully Azumaya boundary points excluded by the theorem.

Theorem A / Theorem 4.1No issue found

The open torus orbit of symplectic leaves is correctly identified

Version-of-record pages 4 and 20–26 · Theorems 4.1, 4.5, and 4.7

Compatibility gives a log-canonical Poisson tensor of constant generic rank on every cluster torus, while the kernel torus supplies enough transverse vector fields. Their wedge with the top Poisson power is the canonical logarithmic anticanonical section. Total coprimality and normality put the complement of the initial and adjacent charts in codimension at least two, so that section cannot acquire a zero on the smooth all-frozen-inverted locus. Its nonvanishing makes one subtorus orbit of a maximal symplectic leaf open and equal to the whole smooth locus. The two printed symbol errors in this proof have the unique corrections recorded below.

Theorems 3.1 and 5.2No issue found

The central classical form and Poisson-order construction are supported

Version-of-record pages 15–18 and 27–30 · Theorem 3.1, Proposition 3.3, and Theorem 5.2

For odd root order coprime to the symmetrizer, taking central powers follows the classical exchange relations, and the strict integral compatible form propagates through mutation. Specialization of the generic quantum upper cluster algebra has the stated principal kernel when Aϵ=UϵA_\epsilon=U_\epsilon; the commutator divided by the parameter produces derivations whose restriction to the central classical form is exactly the GSV bracket. The seedwise formulas agree on the intersection.

Theorem B / Theorem 6.1, upper inclusionSuspected incorrect statement

A noncentral frozen boundary can still be fully Azumaya

Version-of-record pages 5 and 31–37 · Theorem 6.1 and Propositions 6.4–6.5

Let the root order be ℓ=3\ell=3, take no exchangeable variables, set inv={1,3}\mathrm{inv}=\{1,3\} and ninv={2}\mathrm{ninv}=\{2\}, and use the integral skew form Λ=(011−100−100).\Lambda=\begin{pmatrix}0&1&1\\-1&0&0\\-1&0&0\end{pmatrix}. The empty exchange matrix is compatible, the seed is strict, and Aϵ=UϵA_\epsilon=U_\epsilon is the finitely generated mixed quantum torus generated by X1±1,X2,X3±1X_1^{\pm1},X_2,X_3^{\pm1}. Its PI degree is 33 because Λ‾\overline\Lambda has rank 22. The vector e2−e3e_2-e_3 lies in ker⁡Λ‾\ker\overline\Lambda, but e2e_2 does not, so X2X_2 is noncentral and 2∈nc2\in\mathrm{nc}. Now fix a central fiber with X13=a≠0X_1^3=a\ne0, X23=0X_2^3=0, and X33=c≠0X_3^3=c\ne0. The normal nilpotent X2X_2 annihilates every simple module of the fiber. Modulo X2X_2, the fiber is the central fiber of the nondegenerate two-dimensional quantum torus on X1,X3X_1,X_3, hence is M3(C)M_3(\mathbb C). Every simple module therefore has dimension 33, equal to the PI degree, so this point is fully Azumaya although it lies in V(x2)V(x_2). This contradicts A⊆Y∖V(xnc)\mathcal A\subseteq Y\setminus V(x_{\mathrm{nc}}).

Verified weaker upper boundCorrect after substantive scope repair

The argument works after replacing noncentrality by actual PI-degree drop

Version-of-record pages 33–36 · Propositions 6.4–6.5

For Hj=⨁i≠jZeiH_j=\bigoplus_{i\ne j}\mathbb Ze_i, Proposition 6.4 correctly computes the boundary simple dimension as [Hj:ker⁡(Λ‾j)]\sqrt{[H_j:\ker(\overline\Lambda_j)]}. If one replaces nc\mathrm{nc} by the set of indices for which [Hj:ker⁡(Λ‾j)]<[ZN:ker⁡(Λ‾)],[H_j:\ker(\overline\Lambda_j)]<[\mathbb Z^N:\ker(\overline\Lambda)], the same irreducibility and closedness argument proves the corresponding upper bound on the fully Azumaya locus. The printed criterion ej∉ker⁡Λ‾e_j\notin\ker\overline\Lambda is weaker and does not imply this strict inequality.

02Proofs5 reported findingsSuspected incomplete or incorrect proofs

The proof of Proposition 6.5 makes a false equality-case inference, invalidating the upper inclusion in Theorem 6.1. The Poisson-side proof of Theorem A is complete after two uniquely determined symbol corrections and a verified repair in the public Matherne-Muller fallback used by Proposition 4.6. The remaining root-of-unity specialization and lower-Azumaya arguments are internally consistent.

Proposition 6.5Suspected incorrect as written · counterexample propagates to Theorem 6.1

Equality of lattice indices does not force eje_j into the radical

Version-of-record pages 35–36 · proof of Proposition 6.5

Write K=ker⁡Λ‾K=\ker\overline\Lambda and Hj=⨁i≠jZeiH_j=\bigoplus_{i\ne j}\mathbb Ze_i. The proof uses [ZN:K]≥[Hj:K∩Hj]≥[Hj:ker⁡(Λ‾j)][\mathbb Z^N:K]\geq[H_j:K\cap H_j]\geq[H_j:\ker(\overline\Lambda_j)] and says equality in the first inequality implies ej∈Ke_j\in K. Equality actually says only ZN=Hj+K\mathbb Z^N=H_j+K. For the displayed ℓ=3\ell=3 counterexample, KK contains e2−e3e_2-e_3, so Z3=H2+K\mathbb Z^3=H_2+K, while e2∉Ke_2\notin K. Moreover, Λ2\Lambda_2 on H2=⟨e1,e3⟩H_2=\langle e_1,e_3\rangle is nondegenerate, and both the full and restricted indices equal 99. Thus the claimed strict PI-degree drop fails, exactly as the fully Azumaya boundary fiber demonstrates. No local textual repair preserves the stated set nc\mathrm{nc}.

Proof of Theorem ACorrect and complete after apparently sound source and typo repairs

The anticanonical and torus-orbit proof closes

Version-of-record pages 20–26 · Propositions 4.3–4.6 and Theorems 4.5–4.7

GSV Lemma 2.4 supplies the torus transversality on a cluster torus, Polishchuk Corollary 2.4 makes the singular locus Poisson, and BFZ supplies equality of the totally coprime upper bound with the upper cluster algebra. The public Matherne-Muller fallback used for the Hartogs converse prints gcd⁡\gcd where lcm⁡\operatorname{lcm} is required and an incorrect power of aka_k; choosing the least common multiple and rewriting the valuation witness with (fk/ak)d/νk(f_k/a_k)^{d/\nu_k} gives the needed negative-valuation element on every localization. Hence Proposition 4.6's codimension-two statement survives, and the nonvanishing anticanonical-section argument establishes Theorem A.

Theorems 3.1, 5.2, and the lower inclusion in Theorem 6.1No issue found in the reviewed proof

The root-of-unity specialization and lower-bound chain is complete

Version-of-record pages 15–18 and 27–33 · Theorems 3.1 and 5.2 and Proposition 6.3

The oddness and coprimality hypotheses are retained at every mutation, the strict lift fixes the compatible integral matrices, and the specialization kernel is principal on the upper algebra under Aϵ=UϵA_\epsilon=U_\epsilon. Brown-Gordon constancy along symplectic leaves, the torus action, and localization to a root-of-unity quantum torus identify every smooth all-frozen-inverted fiber with an Azumaya quantum-torus fiber of the full PI degree. This proves the lower inclusion independently of the defective upper-bound argument.

Equations (4.5) and Theorem 4.7(i)Suspected typo

Two symbols in the Poisson-side proof have unique corrections

Version-of-record pages 24–25 · equation (4.5) and proof of Theorem 4.7(i)

For the arbitrary fixed basis ν1,…,νn(B~)\nu_1,\ldots,\nu_{n(\widetilde B)} of ker⁡(B~⊤)\ker(\widetilde B^\top), equation (4.5) must span the orbit tangent space with all n(B~)n(\widetilde B) vector fields, not only the first N−2rN-2r; one then selects an independent N−2rN-2r subset θ0\theta_0, exactly as the next sentence requires. In Theorem 4.7(i), Theorem 4.5 shows that χθ\chi_\theta is nonzero on the intersection of the smooth locus with the union of the cluster charts, not on its set-theoretic complement; the printed ∖\setminus must be ∩\cap. The following codimension-two argument then proves nonvanishing on the complement. Both repairs are mechanically forced and leave Theorem A unchanged.

Propositions 6.5 and 6.6 notationSuspected typo

The two final dimension displays use the wrong ambient group or relation sign

Version-of-record pages 36–37 · final displays in Propositions 6.5 and 6.6

At the end of Proposition 6.5, the restricted index must be [Hj:ker⁡(Λ‾j)][H_j:\ker(\overline\Lambda_j)], not [ZN:ker⁡(Λ‾j)][\mathbb Z^N:\ker(\overline\Lambda_j)]; this notation repair does not fix the substantive strictness failure above. In Proposition 6.6, the preceding line proves equality of the full and restricted indices when ej∈ker⁡Λ‾e_j\in\ker\overline\Lambda, so the final printed << must be ==. With that correction, Proposition 6.6's claim that such central boundary points are fully Azumaya follows.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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