Generating toric noncommutative crepant resolutions

Bocklandt, Rafael

doi:10.1016/j.jalgebra.2012.03.040

Cited by 31 publications

(34 citation statements)

References 30 publications

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“…Our work corrects a result of Yasuda [Yas10], and is closely related to recent work of Higashitani and Nakajima [HN17]. A different approach to the existence of non-commutative crepant resolutions for three dimensional Gorenstein toric rings in low dimension can be found in [Bro12]; see also [Boc12]. As mentioned earlier, our results overlap considerably with work of Van den Bergh andŠpenko, although their machinery is quite different and less constructive; see [ŠVdB17a], [ŠVdB17b].…”

Section: Introductionsupporting

confidence: 89%

Non-commutative resolutions of toric varieties

Faber

Muller

Smith

2019

Advances in Mathematics

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Let R be the coordinate ring of an affine toric variety. We prove, using direct elementary methods, that the endomorphism ring End R (A), where A is the (finite) direct sum of all (isomorphism classes of) conic R-modules, has finite global dimension equal to the dimension of R. This gives a precise version, and an elementary proof, of a theorem ofŠpenko and Van den Bergh implying that End R (A) has finite global dimension. Furthermore, we show that End R (A) is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field k of prime characteristic, we show that the ring of differential operators D k (R) has finite global dimension. mension. 1 This theorem is due to Roos in characteristic zero [Roo72, Cha74], and Paul Smith in characteristic p [Smi87].

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Section: Introductionsupporting

confidence: 89%

Non-commutative resolutions of toric varieties

Faber

Muller

Smith

2019

Advances in Mathematics

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“…In this section, we will apply the mutations of MM modules introduced in subsection 2.2 to modules giving NCCRs of three dimensional Gorenstein toric singularities. Although the basic idea is the same as [Boc2], we will discuss details for the sake of completeness, and for the convenience of the readers.…”

Section: Mutations Of Dimer Models and Splitting MM Generatorsmentioning

confidence: 99%

“…In order to show this theorem, we discuss the relationship between the mutation of QPs associated with dimer models and that of splitting MM generators arising from dimer models in Section 4 along the idea as in [Boc2]. After that we will consider the mutation of splitting MM generators for toric singularities associated with reflexive polygons.…”

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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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“…Theorem 3.9 (see e.g., [Bro,IU2,Boc2]). Suppose that (Q, W Q ) is the QP associated with a consistent dimer model Γ and P(Q, W Q ) is the complete Jacobian algebra.…”

Section: Figure 3 Extremal Perfect Matchingsmentioning

confidence: 99%

Semi-steady non-commutative crepant resolutions via regular dimer models

Nakajima¹

2019

Algebraic Combinatorics

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A consistent dimer model gives a non-commutative crepant resolution (= NCCR) of a 3dimensional Gorenstein toric singularity. In particular, it is known that a consistent dimer model gives a class of NCCRs called steady if and only if it is homotopy equivalent to a regular hexagonal dimer model. Inspired by this result, we detect another nice property on NCCRs that characterizes square dimer models. We call such NCCRs semi-steady NCCRs, and study their properties.

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Generating toric noncommutative crepant resolutions

Cited by 31 publications

References 30 publications

Non-commutative resolutions of toric varieties

Non-commutative resolutions of toric varieties

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

Semi-steady non-commutative crepant resolutions via regular dimer models

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