Proof-critical dependence graph

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

A statement-restricted graph for the proved $D$-module, $q$-difference, and quantum-cluster assertions in arXiv:2504.19329v4. It follows every proof-critical non-book source recursively and stops at books or monographs. Historical antecedents that are reproved in the focal paper, definitional pointers, alternative citations, and the explicitly conjectural symplectic-singularity discussion are excluded.

Graph scope15 nodes16 proof-critical linksChecked August 27, 2026

Oriented proof graph

Dependence map

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Proof-critical links

Systèmes holonomes d’équations aux $q$ -différences→On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modulesHeadline lineageSource link partially checked

Sabbah supplies the integral symplectic coordinate automorphisms and the Bernstein inequality that make the induction in Theorem 3.3 work. The automorphisms compose only up to q-power rescalings, so the focal phrase that Sp(2r,Z) acts is valid only after absorbing these harmless rescalings when composing twists.

Citation location: Etingof arXiv v4, pp. 8–10, citations [S] to Subsection 1.3 and Corollary 2.1.2Verification note: The q-power composition correction is required. Sabbah Theorem 2.7.1(2)'s missing finite-lattice/same-localization qualification concerns background localization prose and is not an input to the proof of focal Theorem 3.3.

Theorem 3.3 $q$ -Bernstein transform making a holonomic $q$ - $D$ -module finite over the Laurent polynomial subalgebra.

Quantum cluster algebras→On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modulesHeadline lineageSource link partially checked

Quantum-seed mutation and the quantum Laurent phenomenon make the local quantum tori share one skew field and define the upper quantum cluster algebra used in the cluster application.

Citation location: Etingof arXiv v4, §3.8, p. 15, citation [BZ]Verification note: The generic-q BZ statements are correct after the recorded integral-center repair. Their specialization identifies the Frobenius centers across mutations only when gcd(N,d_k)=1 for every diagonal skew-symmetrizer entry d_k, equivalently when each q^{d_k} has full order N; the focal theorem omits that restriction. Odd N is only a sufficient convention-dependent hypothesis in MNTY, not an intrinsic requirement.

Theorem 3.14Lagrangianity of the glued $p$ -support for holonomic objects in the quantum upper-cluster Serre quotient.

Poisson geometry and Azumaya loci of cluster algebras→On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modulesHeadline lineageSource link checked

MNTY provide the codimension-two cluster-chart cover and identify the maximal-rank Poisson locus with the smooth open torus orbit of symplectic leaves used to interpret and glue the Lagrangian supports.

Citation location: Etingof arXiv v4, §3.8, p. 15, citation [MNTY] to Proposition 4.6 and Theorem A / Theorem 4.1

Theorem 3.14Lagrangianity of the glued $p$ -support for holonomic objects in the quantum upper-cluster Serre quotient.

Serre’s Problem on Projective Modules→On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modulesHeadline lineageSource-chain endpoint

Quillen–Suslin trivializes the polynomial vector bundle used in the differential connection reduction, while Swan’s Laurent-polynomial theorem turns local freeness into the freeness claimed in Proposition 3.2.

Citation location: Etingof arXiv v4, p. 2 footnote 2 and Proposition 3.2, p. 9, citation [La, Corollary 4.10]

Theorem 2.9Lagrangianity of the generic $p$ -support of a nonzero holonomic Weyl-algebra module.
Proposition 3.2Freeness of an $O$ -coherent holonomic $q$ - $D$ -module over a Laurent polynomial ring.

On the $p$ -supports of a holonomic $\mathcal{D}$ -module→On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modulesHeadline lineageSource link checked

Bitoun’s closed-immersion formula identifies the $p$ -support of the direct image with the cotangent pullback-and-embedding of the original $p$ -support; his cotangent Lemma A.0.6 then transfers symplectic-form vanishing and the dimension count.

Citation location: Etingof arXiv v4, pp. 5–6, citations [B], [vdB]; Bitoun Proposition 2.4.1, Inventiones Mathematicae pp. 787–788, and Lemma A.0.6, p. 817

Corollary 2.11Smooth-variety form of the Lagrangian $p$ -support theorem.

Cluster algebras III: Upper bounds and double Bruhat cells→Quantum cluster algebrasHeadline lineageSource link checked

The divisibility/intersection calculation in the quantum upper-bound proof explicitly follows the commutative case-by-case argument of BFZ Section 4; checking BFZ closes the compressed branch in BZ Proposition 5.7 and Lemma 5.8.

Citation location: Berenstein–Zelevinsky, Proposition 5.7 and Lemma 5.8, Advances in Mathematics pp. 423–426; BFZ Theorem 1.5 and Section 4

Theorem 5.1 and Corollary 5.2Quantum Laurent phenomenon and mutation invariance of the upper quantum cluster algebra.

Cluster algebras III: Upper bounds and double Bruhat cells→Poisson geometry and Azumaya loci of cluster algebrasHeadline lineageSource link checked

BFZ Corollary 1.7 identifies the upper cluster algebra with the finite initial-plus-adjacent upper bound under total coprimality, and Proposition 1.8 supplies total coprimality from full rank.

Citation location: MNTY Proposition 4.6, Advances in Mathematics article 109822, pp. 24–25, citation [1]

Proposition 4.6The singular locus and the complement of the initial-plus-adjacent cluster charts have codimension at least two.

Cluster algebras and Poisson geometry→Poisson geometry and Azumaya loci of cluster algebrasHeadline lineageSelf-contained endpoint

The proof of GSV Lemma 2.4 supplies transitivity of the compatible torus action on symplectic leaves inside every cluster torus, the local orbit input in MNTY Theorems 4.5 and 4.7.

Citation location: MNTY pp. 24 and 26, citations [19, Lemma 2.4]

Theorem 4.1 (Theorem A)The smooth locus away from non-inverted frozen divisors is one open torus orbit of symplectic leaves.

Algebraic geometry of Poisson brackets→Poisson geometry and Azumaya loci of cluster algebrasHeadline lineageSelf-contained endpoint

Polishchuk’s Corollary 2.4 makes the singular locus a union of symplectic leaves, which is required for the one-sided containment in MNTY Proposition 4.4.

Citation location: MNTY Proposition 4.4, p. 22, citation [34, Corollary 2.4]

Theorem 4.1 (Theorem A)The smooth locus away from non-inverted frozen divisors is one open torus orbit of symplectic leaves.

Cluster Algebra Structures on Poisson Nilpotent Algebras→Poisson geometry and Azumaya loci of cluster algebrasHeadline lineageSource-chain endpoint

Goodearl–Yakimov Proposition 2.3 turns each radical Poisson frozen-variable ideal into a union of symplectic leaves, so its complement is stable under the leaf stratification.

Citation location: MNTY Proposition 4.4, p. 22, citation [26, Proposition 2.3]

Theorem 4.1 (Theorem A)The smooth locus away from non-inverted frozen divisors is one open torus orbit of symplectic leaves.

Computing upper cluster algebras→Poisson geometry and Azumaya loci of cluster algebrasHeadline lineageSource link partially checked

The algebraic Hartogs lemma and its upper-cluster specialization show that the complement of the initial and adjacent toric charts has codimension at least two.

Citation location: MNTY Proposition 4.6, p. 25, citation [29, Lemmas 4.3.1 and 4.4.2]Verification note: The public pre-revision manuscript has a repairable gcd/lcm and exponent error in the displayed codimension-one construction; the corrected construction proves the implication MNTY uses. Exact IMRN wording remains unavailable.

Proposition 4.6The singular locus and the complement of the initial-plus-adjacent cluster charts have codimension at least two.

Commutative Algebra with a View Toward Algebraic Geometry→Poisson geometry and Azumaya loci of cluster algebrasHeadline lineageSource-chain endpoint

Serre’s criterion gives codimension at least two for each irreducible component of the singular locus of the normal upper-cluster variety.

Citation location: MNTY Proposition 4.6, p. 25, citation [11, Theorem 11.5(i)]

Proposition 4.6The singular locus and the complement of the initial-plus-adjacent cluster charts have codimension at least two.

Commutative Algebra with a View Toward Algebraic Geometry→Computing upper cluster algebrasHeadline lineageSource-chain endpoint

The commutative-algebra extension criterion across codimension-two subsets and Serre conditions are the textbook inputs behind the algebraic Hartogs formulation.

Citation location: Matherne–Muller §4.3, especially Lemma 4.3.1 and its citations to Eisenbud Theorems 11.5 and 18.15

Lemmas 4.3.1 and 4.4.2Algebraic Hartogs extension and its upper-cluster chart-union criterion.

The Weyl Algebra→Systèmes holonomes d’équations aux $q$ -différencesHeadline lineageSource-chain endpoint

Sabbah adapts Ehlers’ Bernstein-filtration injection and spectral-sequence method to the $q$ -commutative operator algebra before deriving Corollary 2.1.2.

Citation location: Sabbah Sections 1.5 and 2.1, citations [7], especially Theorem 2.1.1

Corollary 2.1.2Bernstein inequality for nonzero finitely generated modules over the $q$ -difference operator algebra.

Rings of Differential Operators→Systèmes holonomes d’équations aux $q$ -différencesHeadline lineageSource-chain endpoint

Björk’s homological-dimension framework is the cited model for Sabbah’s Ext-dimension bounds; together with localization it yields the $q$ -Bernstein inequality.

Citation location: Sabbah Section 1.5, citation [5, Chapter 2 §7], and Corollary 2.1.2

Corollary 2.1.2Bernstein inequality for nonzero finitely generated modules over the $q$ -difference operator algebra.

Algebraic $D$ -Modules→On the $p$ -supports of a holonomic $\mathcal{D}$ -moduleHeadline lineageSource-chain endpoint

The classical transfer-bimodule description for a closed immersion gives the local presentation that commutes with base change and lets Bitoun read off the induced map on the Frobenius-center support.

Citation location: Bitoun Proposition 2.4.1, Inventiones Mathematicae p. 788, citation [9, equation (4), p. 259]

Proposition 2.4.1Closed-immersion direct image transforms $p$ -support by the cotangent correspondence, and Lagrangianity transfers back to the original smooth variety.