Commutative algebra · local Gorenstein rings · torsion-free modules · tensor products and rigidity

Huneke–Wiegand Conjecture

Collaboration beta

Must a finitely generated torsion-free module over a one-dimensional Gorenstein local domain be free whenever its tensor product with its dual has no torsion?

M \otimes_{R} M^{*} torsion-free \Rightarrow ? M free

Known results and sources

For a finitely generated torsion-free module over a one-dimensional Gorenstein local domain, the conjecture asks whether a torsion-free tensor product with its dual forces the module to be free.

Research problem

Exact mathematical statement

Let R be a one-dimensional Gorenstein local domain and let M be a finitely generated torsion-free R-module. Write M* = Hom_R(M,R). The Huneke–Wiegand conjecture asks whether

M \otimes_R M^* \text{ torsion-free} \quad\Longrightarrow\quad M \text{ free}.

The source also records the tensor condition as equivalent to self-extension rigidity in this torsion-free setting. The full conjecture remains open. A conditional maximal-conductor, rigid, two-generated route reaches exact package counts and finite linear-algebra checks, but those source-reported computations were not independently reproduced and do not establish the missing algebraic interfaces.

Problem infographic

Problem at a glance

Wide mathematical explainer showing a one-dimensional Gorenstein local domain, a finitely generated torsion-free module M and its dual entering a tensor product, and the tensor-torsion-free condition leading to an open freeness question; below, a separate conditional route starts from the two-generated rank-one fractional ideal I = R + Rx, records 13 and 21 capacity-surviving packages at multiplicities 27 and 28, and compares the shifted-product lengths ℓ(VW_z) = 12 and ℓ(VW_z) ≤ 11 at an unresolved gate. — For a finitely generated torsion-free module M, the conjecture asks whether torsion-freeness of M tensor M* forces M to be free. A separate source-reported route for a two-generated rank-one fractional ideal reaches capacity-surviving package counts and an unresolved shifted-coordinate-product comparison ℓ(VW_z) = 12 versus ℓ(VW_z) ≤ 11; it does not classify modules or solve the unrestricted problem.

Current mathematical picture

Where work on Huneke–Wiegand Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureAutomatic transfer of multiplicity-27 eliminations to multiplicity 28

The source computes length(J/J²) = 13 - Y at e = 27 but 14 - Y at e = 28, so linked Hilbert functions, tangent quotients, stretchedness, and equal-square conclusions require a fresh hypothesis audit. Exact package arithmetic and the length-12 contradiction budget remain useful once each algebraic interface is proved with the correct Delta-sensitive hypotheses.

Route status · Narrowed route

Main reductionCurrent reduction

Inside the inherited maximal-conductor, rigid, two-generated framework, the source records the conditional consequence b = 5 implies e(R) at least 28; it expressly denies that this is unconditional or solves Huneke–Wiegand.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeSupply or independently validate the G5 quotient-tail, descended-coordinate, joint-generation, and determinant hypotheses before promoting the candidate e = 27 closure.Task status · Ready to work on

Work mapped so far

Huneke–Wiegand Conjecture in numbers

8.8kretained lines of mathematical investigation8,816 in the current working snapshot

Argument development: 7,467 · 85%
Explored or eliminated routes: 416 · 5%
Computational analysis: 217 · 2%
Open obligations: 337 · 4%
Definitions and setup: 379 · 4%

8selected mapped statements2routes investigated4open questions4contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byCraig Huneke and Roger WiegandProposed1994Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Supply or independently validate the G5 quotient-tail, descended-coordinate, joint-generation, and determinant hypotheses before promoting the candidate e = 27 closure.

Suggested move: Audit Gate G5 from its five explicit non-elementary hypotheses through the two-stage symmetric-square defect argument, without treating the numerical census as proof of the graph interfaces.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeAutomatic transfer of multiplicity-27 eliminations to multiplicity 28

Route status · Narrowed route

Narrowed routeTreating the saturated Y = 8 package as the whole multiplicity-28 frontier

The controlling status lists 21 capacity-surviving e = 28 packages and says the full shell is not reduced to the saturated package. A complete transfer or direct-elimination audit across all 21 packages could close the conditional e = 28 shell while leaving the global rank-one and arbitrary-rank barriers explicit.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Supply or independently validate the G5 quotient-tail, descended-coordinate, joint-generation, and determinant hypotheses before promoting the candidate e = 27 closure.Suggested move: Audit Gate G5 from its five explicit non-elementary hypotheses through the two-stage symmetric-square defect argument, without treating the numerical census as proof of the graph interfaces.

Ready to work on

For a targeted saturated-package flag, prove a coordinate-compression estimate length(W_zG) at most 7 so the exact inequality 4 + 7 = 11 < 12 closes that branch.Suggested move: Run a Delta-sensitive transfer checklist on the Y = 4, 5, 6, and 7 and positive-lambda packages before reusing any e = 27 linked-geometry conclusion.

Ready to work on

All-package multiplicity-28 closure

A valid conditional e = 28 closure must eliminate or explicitly transfer every one of the 21 surviving packages, not just the six flags in the saturated package.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Do not promote any conditional two-generated closure to the unrestricted Huneke–Wiegand conjecture without resolving the rank-one and arbitrary-rank barriers.Suggested move: After the e = 28 shell is audited, keep the rank-one (3,3) branch and arbitrary-rank determinant/overring transfer as independent global research programs.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

The general conjecture remains open. Current primary literature continues to call it an open question while proving affirmative special cases for large classes of ideals, certain hypersurface-style settings, modules governed by almost-split sequences, and numerical semigroup rings generated by generalized arithmetic sequences. None of those results resolves every finitely generated torsion-free module over every one-dimensional Gorenstein local domain.

[6][3][5]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2024
Peer reviewedLanderos, O'Neill, Pelayo, Peña, Ren, and Wissman proved the relevant numerical-semigroup special case for generalized arithmetic sequence generators and explicitly retained the general conjecture as open.[6]
2018–2019
Peer reviewedSeveral works established affirmative cases for ideals, modules controlled by theta invariants, and modules whose almost-split sequences have sufficiently rich middle terms, without resolving the general…[3][4]
1994–1997
Historical sourceHuneke and Wiegand developed rigidity and torsion results for tensor products over hypersurfaces and local rings, giving the primary setting from which the named conjecture emerged.[1][2]

Mathematical neighborhood

Related results and reusable starting points

Current focusHuneke–Wiegand conjecture

Solved special caseIdeal case

A large class of ideals over arbitrary one-dimensional local domains satisfies the predicted tensor-dual torsion conclusion; this is narrower than arbitrary modules over arbitrary one-dimensional Gorenstein local domains.

[3]

Solved special caseGeneralized-arithmetic numerical semigroup families

The conjectural conclusion holds for the targeted monomial-ideal setting over numerical semigroup rings whose semigroups are generated by generalized arithmetic sequences.

[6]

Dependency or reductionTheta-invariant vanishing

Hochster's theta invariant supplies torsion conclusions under finite-projective-dimension hypotheses over a hypersurface presentation, linking the conjecture to intersection-style homological invariants.

[4]

Related problemSelf-extension rigidity and almost-split sequences

Over a one-dimensional Gorenstein local domain, nonvanishing self-extension results from almost-split sequences provide an Auslander–Reiten-theoretic affirmative case.

[5]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA proof-assistant development of one-dimensional Noetherian local domains, Gorenstein rings, finitely generated torsion-free and maximal Cohen–Macaulay modules, and algebraic duals.
Formalization targetFormal tensor products, torsion submodules, depth, rank, reflexivity, Ext and Tor, and the precise equivalence between tensor-dual torsion and self-extension rigidity under stated hypotheses.
Formalization targetFormal hypersurface and complete-intersection homological algebra, matrix factorizations, theta invariants, and almost-split sequences for the neighboring special cases.
Formalization targetA formal statement that separates the universal conjecture from ideal, numerical-semigroup, hypersurface, and representation-theoretic special cases.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

4 standing statements4 proposed statements4 open questions2 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction2 of 82
lemma2 of 82
equivalence1 of 81
computational claim1 of 81
negative result1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included

retained route statementDoes a torsion-free tensor with the dual force freeness?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementTensor-dual freeness questionintermediate
retained route statementExact package censusintermediate
retained route statementDelta-sensitive transfer barrierintermediate
retained route statementConditional two-generated lower boundintermediate
retained route statementLength-12 contradiction budgetintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureAutomatic transfer of multiplicity-27 eliminations to multiplicity 28reported failure
Useful failureTreating the saturated Y = 8 package as the whole multiplicity-28 frontierreported failure
Research targetSupply or independently validate the G5 quotient-tail, descended-coordinate, joint-generation, and determinant hypotheses before promoting the candidate e = 27 closure.open
Research targetFor a targeted saturated-package flag, prove a coordinate-compression estimate length(W_zG) at most 7 so the exact inequality 4 + 7 = 11 < 12 closes that branch.open
Research targetDo not promote any conditional two-generated closure to the unrestricted Huneke–Wiegand conjecture without resolving the rank-one and arbitrary-rank barriers.open
Research targetAll-package multiplicity-28 closureopen
ComputationThe source reports 53 raw and 13 capacity-surviving packages at multiplicity 27, and 71 raw and 21 capacity-surviving packages at multiplicity 28, plus exact tables and a bounded finite linear-algebra regression check.These source-reported computations were not independently reproduced and do not establish graph, determinant, genericity, compression, or shifted-coordinate identities. · reported unreproduced
Narrowed routeAutomatic transfer of multiplicity-27 eliminations to multiplicity 28The source computes length(J/J²) = 13 - Y at e = 27 but 14 - Y at e = 28, so linked Hilbert functions, tangent quotients, stretchedness, and equal-square conclusions require a fresh hypothesis audit. Exact package arithmetic and the length-12 contradiction budget remain useful once each algebraic interface is proved with the correct Delta-sensitive hypotheses.
Narrowed routeTreating the saturated Y = 8 package as the whole multiplicity-28 frontierThe controlling status lists 21 capacity-surviving e = 28 packages and says the full shell is not reduced to the saturated package. A complete transfer or direct-elimination audit across all 21 packages could close the conditional e = 28 shell while leaving the global rank-one and arbitrary-rank barriers explicit.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeSupply or independently validate the G5 quotient-tail, descended-coordinate, joint-generation, and determinant hypotheses before promoting the candidate e = 27 closure.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointSupply or independently validate the G5 quotient-tail, descended-coordinate, joint-generation, and determinant hypotheses before promoting the candidate e = 27 closure.

Huneke–Wiegand Conjecture · ready to start

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Research contextPrepared context for any AI agent

Must a finitely generated torsion-free module over a one-dimensional Gorenstein local domain be free whenever its tensor product with its dual has no torsion?

Exact question and boundaries
Current routes and known obstacles
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Sources and references6 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
Tensor products of modules and the rigidity of Tororiginal source · Craig Huneke, Roger Wiegand · Mathematische Annalen / Springer · 1994 · DOI 10.1007/BF01459794 · MR MR1282227 · accessed Aug 14, 2026
2
Tensor products of modules, rigidity and local cohomologypeer reviewed result · Craig Huneke, Roger Wiegand · Mathematica Scandinavica · 1997 · DOI 10.7146/math.scand.a-12871 · MR MR1612887 · accessed Aug 14, 2026
3
On the ideal case of a conjecture of Huneke and Wiegandpeer reviewed result · Olgur Celikbas, Shiro Goto, Ryo Takahashi, Naoki Taniguchi · Proceedings of the Edinburgh Mathematical Society / Cambridge University Press · 2019 · ARXIV 1710.07398 · DOI 10.1017/S0013091518000731 · accessed Aug 14, 2026
4
On the vanishing of theta invariant and a conjecture of Huneke and Wiegandpreprint · Olgur Celikbas · arXiv · 2018 · ARXIV 1805.04568 · accessed Aug 14, 2026
5
The Huneke-Wiegand conjecture and middle terms of almost split sequencespreprint · Toshinori Kobayashi · arXiv · 2019 · ARXIV 1907.02348 · accessed Aug 14, 2026
6
Families of numerical semigroups and a special case of the Huneke-Wiegand conjecturepeer reviewed result · Miguel Landeros, Christopher O'Neill, Roberto Pelayo, Karina Peña, James Ren, Brian Wissman · Communications in Algebra / Taylor & Francis · 2024 · ARXIV 2404.12519 · DOI 10.1080/00927872.2024.2362934 · accessed Aug 14, 2026

Important qualifications

The review covers the general tensor-dual torsion question and representative special cases; it is not an exhaustive bibliography of rigidity, trace ideals, numerical semigroups, or Auslander–Reiten theory.
The exact relation between the 1994 rigidity paper, the 1997 local-cohomology paper, and later standard formulations is summarized conservatively; no stronger historical priority claim is made.
No intake-packet theorem, packet status label, source-reported computation, attachment, embedded bibliography, or submitted URL was used as external-status authority.
No exact problem-level formalization was identified in the scoped review, but this does not establish nonexistence in every proof assistant or private project.
This metadata has no proof, review, acceptance, credit, visibility, publication, or deployment authority.

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