PropositionStatement: AI-adaptedProof: Literature-sourced

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Chambers label smooth minimal models and their NCCR categories

Statement

Use the complete-local isolated complex cDV smooth flopping setting and the finite arrangement $H_{Λ}$ of the cited local interfaces. Let $CM R$ consist of the zero module and the finite $R$ -modules of depth three, with $R$ -linear maps. Write $add M$ for summands of finite direct sums of $M$ . A generator has $R \in add M$ ; basic means a finite sum of pairwise nonisomorphic indecomposable modules. A cluster-tilting object $M \in CM R$ satisfies $add M = {L \in CM R : {Ext}_{R}^{1} (L, M) = 0} = {L \in CM R : {Ext}_{R}^{1} (M, L) = 0} .$ For a basic cluster-tilting generator $M = M_{i} \oplus \hat{M}$ with $M_{i}$ nonfree indecomposable, its mutation $ν_{i} M = M_{i}^{'} \oplus \hat{M}$ replaces $M_{i}$ by the unique other indecomposable complement, up to isomorphism, that leaves the module cluster-tilting.

Every chamber $C$ labels a minimal model $f_{C} : X_{C} \to Spec R$ and a basic cluster-tilting generator $M_{C}$ , with NCCR $Λ_{C} = {End}_{R} M_{C}$ . All these models are nonsingular projective crepant resolutions. Choose the compatible basic globally generated Van den Bergh vector bundles $V_{C}$ (finite locally free sheaves) in the convention of Hirano–Wemyss Setup 4.1, using the setting's fixed $V$ at the initial chamber, with $H^{0} (X_{C}, V_{C}) = M_{C}$ and ${End}_{X_{C}} V_{C} ≅ Λ_{C}$ . Their tilting equivalences are $E_{C} : = R {Hom}_{X_{C}} (V_{C}, -) : D^{b} (coh X_{C}) \tilde{\to} D^{b} (\mod Λ_{C}) .$ Use bounded derived categories, finite right modules and exact functors with the conventions of the cited definitions. Here $R Hom$ is total right-derived Hom; the existence of these bundles and equivalences is imported.

For an oriented adjacent pair $C \to D$ , a single exceptional curve flops and $M_{D} ≅ ν_{i} M_{C}$ at its corresponding nonfree summand. Fix this isomorphism and the induced endomorphism-ring identifications. Put $T_{C D} = {Hom}_{R} (M_{C}, M_{D})$ , a left $Λ_{D}$ , right $Λ_{C}$ bimodule via postcomposition and precomposition. The mutation equivalence is $t_{C D} : = R {Hom}_{Λ_{C}} (T_{C D}, -) : D^{b} (\mod Λ_{C}) \tilde{\to} D^{b} (\mod Λ_{D}) .$

Let $X_{C} \to Y_{i} \leftarrow X_{D}$ be the common contraction of that curve and its flop. For the ordinary fibre product $Z_{i} = X_{C} \times_{Y_{i}} X_{D}$ , write $p_{C} : Z_{i} \to X_{C}$ and $p_{D} : Z_{i} \to X_{D}$ for the projections. Choose the Bridgeland–Chen functor in the direction $B_{D C} : = R (p_{C})_{*} L p_{D}^{*} : D^{b} (coh X_{D}) \tilde{\to} D^{b} (coh X_{C}) .$ This is the Fourier–Mukai functor with structure-sheaf kernel $O_{Z_{i}}$ ; $L p_{D}^{*}$ and $R (p_{C})_{*}$ denote derived pullback and pushforward. With the compatible choices above, the precise comparison is the natural isomorphism $t_{C D} \circ E_{C} ≅ E_{D} \circ B_{D C}^{- 1} : D^{b} (coh X_{C}) ⟶ D^{b} (\mod Λ_{D}) .$ The inverse denotes a quasi-inverse equivalence. In particular, after transport, $E_{D}^{- 1} \circ t_{C D} \circ E_{C} ≅ B_{D C}^{- 1}$ .

Statement references

Complete-local smooth threefold flops and their slice data

Generic slicing identifies flop stability walls with restricted ADE walls

Derived category of an abelian category

Exact functor between triangulated categories

Natural isomorphism

External input

Hirano–Wemyss §1.3 and §4.1 supply the module definitions, mutation and chamber dictionary. Wemyss survey §4.1 and Theorem 4.3 supply the minimal-model correspondence and the smooth/NCCR conclusions under the stated hypotheses. The initial smooth resolution supplies an NCCR; isolatedness makes the finite chamber graph exhaustive.

Hirano–Wemyss Setup 4.1 uses the globally generated bundle and its pushdown, matching the setting's $V$ and $M$ ; §1.3 uses right modules. With these geometric tilting identifications, Notation 4.3 defines exactly $T_{C D}$ and $t_{C D}$ above, and Remark 4.4 supplies their natural identification with the inverse single-curve flop. This published comparison in the globally generated convention is the input for the displayed square. Donovan–Wemyss §1.4 specifies its structure-sheaf kernel convention. These comparison and equivalence statements are imported, without reconstructing their proofs.

Wemyss (2.A), Notation 2.10 and Lemma 2.12 instead use the dual extension bundle: his $V_{X}$ is the setting's $T = V^{\lor}$ , rather than $V$ . Theorem 4.2 states the comparison in that convention. The comparison for the globally generated $V_{C}$ is supplied by Hirano–Wemyss Remark 4.4 in its Setup 4.1; no identification of $V_{C}$ with Wemyss's $V_{X}$ , or deduction of a derived-functor square from the slice supplier's stability-wall duality, is used.

Wemyss's §1.6 requires each complete-local base to be the completion of a finitely generated $C$ -algebra at a maximal ideal. The Statement of Generic slicing identifies flop stability walls with restricted ADE walls supplies exactly this conclusion for the common base: $R ≅ \hat{B_{n}}$ as complete-local $C$ -algebras, for a finitely generated $C$ -algebra $B$ and maximal ideal $n$ . Every model $X_{C}$ here is over this same $R$ , so this completion requirement in the Wemyss theorem cited by Hirano–Wemyss Remark 4.4 is satisfied. The globally generated-bundle comparison itself is imported from that remark in its stated convention.

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Hirano–Wemyss, Faithful actions from hyperplane arrangements · §1.3, printed p.3; Setup 4.1, printed pp.8–9; §4.1, Notation 4.3 and Remark 4.4, printed p.9

CM R consists of finite maximal Cohen–Macaulay modules, and add M consists of summands of finite sums of M. Cluster-tilting means both Ext-one orthogonals equal add M; mutation uniquely replaces one indecomposable summand by a different one while preserving cluster-tilting. Chambers index basic cluster-tilting modules and simple walls give mutation. Setup 4.1 uses the globally generated tilting bundle V and M=f_*V. Notation 4.3 defines T_CD=Hom_R(M_C,M_D) and t_CD=RHom_{Lambda_C}(T_CD,-) for right modules. Remark 4.4 identifies this simple-wall mutation functor with the inverse single-curve flop functor under the geometric tilting equivalences of Setup 4.1. This supplies the comparison in the globally generated convention used here.

Hypotheses: Complete-local threefold flopping contraction with nonsingular source over an algebraically closed characteristic-zero field, using the basic globally generated Van den Bergh bundle, its pushdown and right endomorphism-ring modules. Here R is isolated complex cDV, every corresponding minimal model is nonsingular, and only nonfree summands and their single-curve flops are used.

Reading limits: In this adjudication read §1.3, Setup 4.1 and §4.1 through Notation 4.3 and Remark 4.4 in the online PDF text. Remark 4.4 is used as the published comparison in its own globally generated convention, with the geometric equivalences in Setup 4.1; its reliance on Wemyss 4.2 is not reconstructed. No independent proof of the bundle, Auslander–McKay or comparison results, or PDF image inspection, is claimed.

Wemyss, A lockdown survey on cDV singularities · §1.3, printed p.3; §4.1, printed p.16; Theorem 4.3 and following paragraph, printed p.17

Minimal models correspond to maximal modifying generators; nonfree summands correspond to exceptional curves and mutation to flops. For isolated cDV R this graph covers every finite-arrangement chamber. If one minimal model is nonsingular then all are; if an NCCR exists then all associated maximal modification algebras are NCCRs.

Hypotheses: Complete-local complex cDV singularity; the whole-arrangement conclusion uses isolatedness and the smooth/NCCR conclusions use existence of a smooth crepant resolution/NCCR.

Reading limits: Read §1.3, §4.1 and Theorem 4.3 with its following paragraph in the online PDF text for this proposal. Import the stated correspondence and smooth/NCCR consequences, without reconstructing their proofs.

Wemyss, Flops and clusters in the homological minimal model programme · §1.6, printed p.9; Setup 2.9, equations (2.A)–(2.B), and Notation 2.10, printed p.12; Lemma 2.12, printed p.13; Theorem 4.2(1)–(2), printed p.21

Notation 2.10 defines V_X using the duals of the extension bundles in (2.B), with N=H^0(V_X); (2.A) and Lemma 2.12 give its tilting equivalence and endomorphism identification. In this dual-bundle convention, Theorem 4.2 identifies nu_I N with H^0(V_X+) and gives the naturally commutative mutation/inverse-Bridgeland–Chen-flop square. The globally generated-bundle comparison used in this item is imported separately from Hirano–Wemyss Remark 4.4, not obtained by identifying these two bundles.

Hypotheses: Over C, with complete-local bases that are completions of finitely generated C-algebras at maximal ideals (§1.6); projective crepant birational morphism to a normal Gorenstein base of dimension three, fibres of dimension at most one, terminal Gorenstein source, and a floppable subset I (equivalently dim_C Lambda_I finite). Theorem 4.2 uses V_X of Notation 2.10, the dual-bundle choice. The slice supplier Statement supplies the completion description of the common R. No direct application of this theorem with the globally generated V_C substituted for V_X is made.

Reading limits: In this adjudication read §1.6, Setup 2.9, (2.A), (2.B), Notation 2.10, Lemma 2.12 and the statement of Theorem 4.2 in the online PDF text. Checked the setting definition for its explicit V versus T=V-dual convention and read the slice supplier Statement and proof, including the existing completion argument in step 1.2. No reconstruction of the geometric comparison, bundle proofs or finite-determinacy theorem, and no image inspection, is claimed.

Donovan–Wemyss, Noncommutative deformations and flops · §1.4, printed p.5, first paragraph and displayed flop-equivalence pair

For a flop pair X and X-prime over X_con, both Bridgeland–Chen directions use the structure sheaf of X times_{X_con} X-prime as Fourier–Mukai kernel. This specifies the geometric flop-functor convention used here.

Hypotheses: The cited passage states the kernel convention for a flopping contraction of projective terminal Gorenstein threefolds. Only that convention is imported here; the complete-local comparison is supplied by Hirano–Wemyss Setup 4.1 and Remark 4.4, rather than inferred from global projectivity.

Reading limits: In this adjudication read the first paragraph and displayed equivalence pair of §1.4 in the online PDF text. Use only the structure-sheaf kernel convention, without importing the twist or flop–flop results or reconstructing a complete-local equivalence proof.

Sources

Hirano–Wemyss, Faithful actions from hyperplane arrangements · §1.3, printed p.3; Setup 4.1, printed pp.8–9; §4.1, Notation 4.3 and Remark 4.4, printed p.9
Wemyss, A lockdown survey on cDV singularities · §1.3, printed p.3; §4.1, printed p.16; Theorem 4.3 and following paragraph, printed p.17
Wemyss, Flops and clusters in the homological minimal model programme · §1.6, printed p.9; Setup 2.9, equations (2.A)–(2.B), and Notation 2.10, printed p.12; Lemma 2.12, printed p.13; Theorem 4.2(1)–(2), printed p.21
Donovan–Wemyss, Noncommutative deformations and flops · §1.4, printed p.5, first paragraph and displayed flop-equivalence pair