Published paper

Computing upper cluster algebras

Jacob P. Matherne, Greg Muller

MSC 2020

13F6014Q99

Published version ↗Open DOI ↗arXiv record →

Publication metadata

Journal citation: International Mathematics Research Notices 2015(11), 3121–3149 (2015)
Publication date: 2014-03-09
DOI: 10.1093/imrn/rnu027 ↗
arXiv counterpart: arXiv:1307.0579v1
MSC 2020: 13F60, 14Q99
As specified in the publication (MSC 2010): 13F60, 14Q99 ↗Public author manuscript, first page (primary 13F60; secondary 14Q99)Both printed codes carry forward unchanged in the official conversion table.
Official MSC conversion: MSC 2010 → MSC 2020 ↗

Abstract

The paper develops constructive criteria and algorithms for producing presentations of upper cluster algebras, with guaranteed termination for totally coprime finitely generated upper cluster algebras and several explicit examples.

Role in dependence graphs

Proof-critical source

On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modules

This paper is included only for the following marked statement:

Lemmas 4.3.1 and 4.4.2 · IMRN pp. 3131–3133 / author manuscript pp. 8–10Algebraic Hartogs extension and its upper-cluster chart-union criterion.

AI-generated audit

Audit summary

Audited against an explicitly disclosed arXiv fallback

Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.

Exact reviewed source

arXiv:1307.0579v1 · explicit pre-revision fallback for the inaccessible IMRN version of record

Jacob P. Matherne and Greg Muller. Computing upper cluster algebras. International Mathematics Research Notices 2015 (2015), no. 11, 3121–3149. Reviewed in arXiv:1307.0579v1 because the IMRN version of record was not publicly downloadable.

The Oxford Academic version of record was subscription-gated during this audit. The only arXiv version, also mirrored as an author manuscript, predates the journal record's December 2013 revision; it is therefore an explicit pre-revision fallback, and this report does not attribute its exact wording or defects to the inaccessible publisher version.

Open audited source ↗

Generated August 27, 2026

01Statements4 reported findingsNo statement issues found

In the reviewed pre-revision fallback, the algebraic Hartogs criteria, their upper-cluster specialization, and the finite-termination result are mathematically correct after a local repair to one displayed construction. The explicit example presentations are plausible consequences of the method, but their decisive computer calculations are not accompanied by reproducible certificates. Exact final-IMRN wording remains unverified.

Lemma 4.3.1Correct after an apparently sound local repair

The algebraic Hartogs equivalences have the stated Noetherian and normal hypotheses

arXiv:1307.0579v1, PDF pages 8–9

The forward chain uses Serre's criterion, the $S_2$ depth condition, and the local-cohomology obstruction to extending functions. For a normal Noetherian domain, the codimension-one converse is also correct, but the displayed element contains the gcd/lcm and exponent errors recorded in the proof audit. The corrected least-common-multiple construction gives a rational function regular away from the height-one component and proves the required converse.

arXiv:1307.0579v1 ↗

Lemma 4.4.2Correct after ancestor repair

The Hartogs criterion specializes correctly to a totally coprime upper cluster algebra

arXiv:1307.0579v1, PDF pages 9–10

Total coprimality identifies the upper cluster algebra with the initial-plus-adjacent upper bound, and Proposition 3.3.1 identifies that upper bound with regular functions away from the lower deep ideal. Applying Lemma 4.3.1 gives the equivalences. In particular, equality of the two global-function rings forces the omitted chart complement to have codimension at least two, which is the exact implication used in MNTY Proposition 4.6.

arXiv:1307.0579v1 ↗

Proposition 5.2.1 and Corollary 5.2.2No issue found

The saturation iteration terminates when the upper algebra is finitely generated

arXiv:1307.0579v1, PDF page 11

At each stage, the saturation adds all elements of the upper algebra with one additional power of the fixed cluster product in the denominator. Induction captures the intersection with every bounded denominator, and a finite generating set has a common denominator bound. The conclusion is appropriately conditional on finite generation and total coprimality.

arXiv:1307.0579v1 ↗

Propositions 6.3.1, 7.1.1, and 7.2.1Not independently reproduced

The example presentations rely on unarchived computer calculations

arXiv:1307.0579v1, PDF pages 14–18

For the larger examples, the manuscript lists generators, relations, and Laurent identities but says only that a computer verifies the relation ideal and the decisive saturation equality. No software, input, output, Gröbner basis, or checkable certificate accompanies the source. This does not affect the general Hartogs lemmas used by the dependency graph, but it limits independent verification of those three explicit presentations from the paper alone.

arXiv:1307.0579v1 ↗

02Proofs3 reported findingsSuspected incomplete or incorrect proofs

The proof of Lemma 4.3.1 contains a malformed codimension-one construction: a greatest common divisor is used where a least common multiple is required, and the displayed denominator has the wrong exponent. Both errors have a direct repair, so the lemma and the graph-critical Lemma 4.4.2 remain valid. Several example proofs also omit their computer certificates.

Lemma 4.3.1Suspected incorrect as written · an apparently sound repair

The codimension-one display uses nonintegral exponents

arXiv:1307.0579v1, PDF page 9 · final paragraph of the proof

Write $\nu_i=\nu(a_i)>0$ for generators of the height-one prime and choose $a_1$ with $\nu_1=1$ . The manuscript sets $d=\gcd(\nu_i)$ , so $d=1$ , but then uses exponents $d/\nu_i$ , which need not be integers; it also prints a factor with $a_k^d$ in the wrong place. Set instead $d=\operatorname{lcm}(\nu_i)$ and define $x=a_1^{-d}\prod_{i>1}g_i^{d/\nu_i}$ . Since $a_k=(f_k/g_k)a_1^{\nu_k}$ , for every $k>1$ this rewrites as $x=(f_k/a_k)^{d/\nu_k}\prod_{i>1,\,i\ne k}g_i^{d/\nu_i}\in S[a_k^{-1}]$ . It is also in $S[a_1^{-1}]$ , while $\nu(x)=-d<0$ , so $x\notin S$ . This proves the intended converse and preserves Lemma 4.4.2.

arXiv:1307.0579v1 ↗

Large example computationsFurther verification needed: audit trail incomplete

Critical ideal and saturation checks are asserted without certificates

arXiv:1307.0579v1, proofs of Propositions 6.3.1, 7.1.1, and 7.2.1

Each proof reduces correctness to a relation-ideal calculation and a saturation equality and then records only that a computer verifies them. Those are finite algebraic checks, but the source gives no reproducible computation. The hand-written Laurent identities establish the required inclusions; the missing certificates concern the claimed presentations and final equality for the examples, not the general method or the lemmas imported by MNTY.

arXiv:1307.0579v1 ↗

Exact-form coverageExplicit pre-revision fallback

The reviewed source predates a recorded journal revision

arXiv and Oxford publication metadata

The only arXiv version is dated July 2013, while the journal metadata records a revision on December 27, 2013 before acceptance. The exact IMRN file was not freely retrievable. Therefore the display error and missing computational artifacts are findings about the reviewed fallback; this audit does not claim that the inaccessible version of record retains the same wording.

Computing upper cluster algebras

Publication metadata

Abstract

Role in dependence graphs

On Lagrangianity of ppp-supports of holonomic DDD-modules and qqq-DDD-modules

Audit summary

arXiv:1307.0579v1 · explicit pre-revision fallback for the inaccessible IMRN version of record

The algebraic Hartogs equivalences have the stated Noetherian and normal hypotheses

The Hartogs criterion specializes correctly to a totally coprime upper cluster algebra

The saturation iteration terminates when the upper algebra is finitely generated

The example presentations rely on unarchived computer calculations

The codimension-one display uses nonintegral exponents

Critical ideal and saturation checks are asserted without certificates

The reviewed source predates a recorded journal revision

On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modules