A dimer ABC:

Bocklandt, Raf

doi:10.1112/blms/bdv101

Cited by 24 publications

(34 citation statements)

References 76 publications

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“…Especially, we will show that we can obtain NCCRs of a three dimensional Gorenstein toric singularity from dimer models that satisfy the consistency condition. For more results concerning dimer models, we refer to a survey article [Boc5] and references quoted in this section.…”

Section: Nccrs Arising From Consistent Dimer Modelsmentioning

confidence: 99%

Mutations of Splitting Maximal Modifying Modules: The Case of Reflexive Polygons

Nakajima

2017

International Mathematics Research Notices

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It is known that every three dimensional Gorenstein toric singularity has a crepant resolution. Although it is not unique, all crepant resolutions are connected by repeating the operation "flop". On the other hand, this singularity also has a non-commutative crepant resolution (= NCCR) which is constructed from a consistent dimer model. Such an NCCR is given as the endomorphism ring of a certain module which we call splitting maximal modifying module. In this paper, we show that all splitting maximal modifying modules are connected by repeating the operation "mutation" of splitting maximal modifying modules for the case of toric singularities associated with reflexive polygons.− → M J and we call this sequence the exchange sequence. We define the right mutation of M at J as µ + J (M ) := M J c ⊕ K, and define the left mutation of M at J as µ − J (M ) := (µ + J (M * )) * . Here, we collect some properties of the mutations of MM modules.Proposition 2.7 (see [IW2, Section 6]). Let R be a complete local normal d-sCY ring, M = ⊕ i∈I M i be a basic modifying R-module. Suppose that J is a subset of I. Then, we have the following.(1) µ + J and µ − J are mutually inverse operations. That is,M )) = M and µ + J (µ − J (M )) = M up to additive closure. (2) µ + J (M ) and µ − J (M ) are modifying R-modules. (3) If M gives an NCCR, then µ + J (M ) and µ − J (M ) also give NCCRs. (4) End R (M ), End R (µ + J (M )) and End R (µ − J (M )) are derived equivalent. (5) Assume that d = 3, if M is an MM module, then so are µ + J (M ) and µ − J (M ). (6) Assume that d = 3, if M is an MM module and J = {k}, then µ + J (M ) ∼ = µ − J (M ), hence we denote it by µ k (M ).

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Section: Nccrs Arising From Consistent Dimer Modelsmentioning

confidence: 99%

Mutations of Splitting Maximal Modifying Modules: The Case of Reflexive Polygons

Nakajima

2017

International Mathematics Research Notices

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“…(2.7) ?eq:graph in Zigzag paths play an important role in the literature on dimer models and are used to formulate consistency conditions [5,6,7,11,12]. Because not all dimer models which come from Zhegalkin zebra functions do satisfy these consistency conditions we will not say more about zigzag paths.…”

Section: The (Co)homology Of γmentioning

confidence: 99%

“…Theorem 3.7. The algebra homomorphisms Theorem 3.20 in [5] states that Jac([F] Λ ) is a non-commutative crepant resolution of the 3dimensional Gorenstein singularity Spec Z(Jac([F] Λ )) if the quiver Γ Λ with superpotential [F] Λ is cancellative.…”

Section: Weight Realizations and Jacobi Algebramentioning

confidence: 99%

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Zhegalkin Zebra Motives Digital Recordings of Mirror Symmetry

Stienstra¹

2018

SIGMA

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Zhegalkin zebra motives are tilings of the plane by black and white polygons representing certain F 2 -valued functions on R 2 . They exhibit a rich geometric structure and provide easy to draw insightful visualizations of many topics in the physics and mathematics literature. The present paper gives some pieces of a general theory and a few explicit examples. Many more examples will be shown in the forthcoming article "Zhegalkin zebra motives: algebra and geometry in black and white".

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“…see the suspended pinch point example considered in [7,Example B.8.7] where the choice of zero vertex as the vertex currently labelled 1 or 2 yields two crepant resolutions that are not isomorphic as varieties;…”

Section: Remark 53mentioning

confidence: 99%

Multigraded linear series and recollement

2017

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Given a scheme Y equipped with a collection of globally generated vector bundles E 1 , . . . , E n , we study the universal morphism from Y to a fine moduli space M(E) of cyclic modules over the endomorphism algebra of E :This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup G ⊂ GL(2, k), every sub-minimal partial resolution of A 2 k /G is isomorphic to a fine moduli space M(E C ) where E C is a summand of the bundle E defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid's recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution.B Alastair Craw a.craw@bath.ac.uk

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A dimer ABC:

Cited by 24 publications

References 76 publications

Mutations of Splitting Maximal Modifying Modules: The Case of Reflexive Polygons

Mutations of Splitting Maximal Modifying Modules: The Case of Reflexive Polygons

Zhegalkin Zebra Motives Digital Recordings of Mirror Symmetry

Multigraded linear series and recollement

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