arXiv:1501.01260v4
Bernstein inequality and holonomic modules
math.RT16W70
Abstract
In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that the generalized Bernstein inequality holds for simple modules and turns into equality for holonomic simples provided the algebraic fundamental groups of all leaves are finite. Under the same assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length or if the algebra in question is a quantum Hamiltonian reduction, then any holonomic module has finite length. This allows to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.
Dependence graphs
Proof lineage
Bernstein inequality and holonomic modules
Bounded statement-restricted first-order citation graph for arXiv:1501.01260v4. It has one focal node and fourteen selected direct ancestor nodes. It is not a full recursive closure: selected source proof citations are unexpanded/unverified, and superseded Gies is excluded because the focal manuscript re-proves its lemma. It does not repair T1.1.
Open dependence graph →AI-generated audit
Audit summary
Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.
Current report
Detailed mathematical audit
01Statements6 reported findingsSuspected unsupported statements
The current arXiv record says the proof of Theorem 1.1 has a crucial error. Theorem 1.1 and the printed Theorem 1.2 route are unsupported as presented. Theorem 1.3 has a separate §4 proof and is not downgraded merely through Theorem 1.1; only applications that additionally invoke Theorem 1.1 are blocked, and its selected source inputs are bounded below.
Theorem 1.1Suspected unsupported in current manuscript
Author-declared proof defect leaves the theorem unsupported
arXiv:1501.01260v4, statement PDF p. 2; proof PDF pp. 11–12; official metadata comment
The author reports a crucial error and v4 supplies no corrected proof or precise replacement. The audit records lack of establishment, not a counterexample or invented local flaw.
Official arXiv:1501.01260v4 ↗Theorem 1.2Suspected unsupported as presented
Printed proof relies directly on unrepaired T1.1
arXiv:1501.01260v4, statement PDF p. 3; proof PDF p. 12
The printed proof calls its first part a direct corollary of T1.1 and returns to that route for the other parts. The metadata expectation that it should hold is not a replacement proof.
Official arXiv v4 PDF ↗Theorem 1.3Independently proved, source-bounded
Section 4 independently proves the sufficient conditions
arXiv:1501.01260v4, statement PDF p. 3; proof PDF pp. 12–16
For (i) and (iii), §4 proves finite algebraic leaf fundamental groups and finite regular-bimodule length; for (ii), it proves finite regular-bimodule length. It invokes neither T1.1 nor T1.2 as a result. The lone textual T1.1 comparison in the RCA base case has a separate finite-codimension-ideal justification.
Official arXiv v4 PDF ↗Appendix Lemma 5.1(i)Suspected repairable printed defect
Domain and target notation require unique repair
arXiv:1501.01260v4, Appendix 1 PDF p. 17; TeX lines 1132–1145
The text says Z is contained in X but applies rho to Z, while the proof treats Z as inside a stratum in X-tilde. The coherent repair is Z contained in X-tilde. The phrase that rho(Z) is locally closed in undefined Y must be read with the relevant leaf target.
Official arXiv v4 PDF ↗Appendix Lemma 5.1(ii)Suspected repairable printed defect
Upstairs smooth-subvariety variable is shadowed
arXiv:1501.01260v4, Appendix 1 PDF p. 17; TeX lines 1147–1148
The proof reuses L for a smooth subvariety above a leaf and then invokes L's isotropicity ambiguously. It should quantify a new smooth W contained above L and compare tangent vectors with the downstairs isotropic L.
Official arXiv v4 PDF ↗Appendix Lemma 5.2Suspected repairable printed defects
Wrong cross-reference and module notation require repair
arXiv:1501.01260v4, Appendix 1 PDF p. 17; TeX lines 1161–1174
Part (i) needs Lemma 5.1(ii), not (i), for rho-inverse of the associated variety. Part (ii) must use calligraphic M in the derived global-sections expression. These are local repairs separate from the T1.1 notice.
Official arXiv v4 PDF ↗02Proofs4 reported findingsSuspected incomplete or incorrect proofs
The author-supplied defect notice controls T1.1. T1.2 inherits the printed break. Section 4 remains an independently audited, source-bounded route, while the Appendix 1 defects are disclosed rather than silently normalized.
T1.1 proofSuspected incorrect or incomplete as written
Exact defect was not specified
arXiv:1501.01260v4, PDF pp. 11–12; metadata comment
The proof closes through Lemma 3.7, Corollary 3.2, and Proposition 3.8, but the author does not locate the crucial error. The audit does not invent one.
Official arXiv:1501.01260v4 ↗T1.2 proofSuspected incomplete as written
Direct T1.1 route remains broken
arXiv:1501.01260v4, PDF p. 12
No alternative T1.2 proof is supplied in v4.
Official arXiv v4 PDF ↗T1.3 §4Further source verification needed
Independent conditional proof is retained
arXiv:1501.01260v4, §4 PDF pp. 12–16
§4 does not run through T1.1/T1.2. The Namikawa leafwise transfer and Crawley–Boevey generality are instead explicitly unresolved source applications.
Official arXiv v4 PDF ↗Appendix 1Suspected repairable but not silently corrected
Five localized repairable defects are disclosed
arXiv:1501.01260v4, Appendix 1 PDF p. 17
Lemma 5.1(i) has the Z-domain and undefined-target problems; 5.1(ii) shadows the upstairs variable; 5.2(i) cites the wrong part; and 5.2(ii) uses non-calligraphic M.
Official arXiv v4 PDF ↗03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.
04Sources3 reported findingsFurther source verification needed
The exact current arXiv v4 PDF/source archive is frozen. Twelve selected ordinary sources are frozen at their exact current versioned arXiv forms; Gabber's original remains unresolved. The graph is a bounded first-order shell, not a full recursive proof closure.
Review formExact arXiv fallback frozen
ArXiv v4 is the exact frozen review source
arXiv:1501.01260v4, official 18-page PDF and source archive
The Advances article is identifiable but direct VOR full-text retrieval for this audit was blocked. This report does not call arXiv v4 a VOR.
Official arXiv:1501.01260v4 ↗VOR wording differenceVersion difference disclosed
VOR and v4 T1.3(ii) wording differs
arXiv:1501.01260v4, T1.3 PDF p. 3 and §4.2 PDF pp. 14–15; Advances in Mathematics 308 (2017)
The VOR includes the stronger holonomic finite-length consequence for Hamiltonian reduction, while v4's theorem statement lists regular-bimodule finite length there. V4 §4.2 directly proves the stronger conclusion; the two forms are not conflated.
Publisher article record ↗Ordinary source shellBounded source closure
Every ordinary report now has its own source pin
arXiv:1501.01260v4, §§2–5 and Appendix 1; frozen source-pin table
No ordinary report points at the focal PDF. Gies is historical because the focal text re-proves its lemma; unaccessed Gabber is neither called terminal nor verified.
Official arXiv:1501.01260v4 ↗