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Generating toric noncommutative crepant resolutions. (English) Zbl 1263.14006
The concept of a noncommutative crepant resolution (NCCR) of a singularity was introduced by Van den Bergh. An algebra acts as a substitute of an ordinary crepant resolution. This generalization stems from the fact that, for 3-dimensional terminal Gorenstein singularities, the derived category of representations of the substitute algebra is equivalent to the derived category of coherent sheaves of a commutative crepant resolution. The substitute algebra has the form $$A=\text{End}_R(T)$$ where $$T$$ is a reflexive $$R$$-module, and can be seen as a path algebra of a quiver $$Q$$ with relations. The vertices corresponds to the direct summands of $$T$$, and the set of arrows is a basic set of maps between them. Defining the dimension vector $$\alpha$$ which assigns to a vertex the rank of the corresponding summand of $$T$$, the singularity $$\text{Spec}(R)$$ can be recovered as the moduli space parameterizing $$\alpha$$-dimensional semisimple representations, and often a commutative resolution can be constructed by constructing the moduli space of stable $$\alpha$$-dimensional representations, for some stability condition.
The author goes through the basic theory of crepant resolutions in a very nice way, so that the generalization to the noncommutative situation follows easily: A not necessarily commutative algebra $$A$$ is a noncommutative crepant resolution of the (commutative) singularity $$R$$ if
(1) $$A\cong\text{End}(T)$$, where $$T$$ is a finitely generated reflexive $$R$$-module
(2) $$A$$ is homologically homogeneous, i.e., alle simple $$A$$-modules have the same projective dimension.
Van den Berg proved that, with respect to derived categories, in dimension 3, these algebras behave like crepant resolutions.
Now, in the 3-dimensional Gorenstein case, the NCCR’s can be constructed using the concept of a maximal modification algebra (MMA): An algebra $$A$$ is called a modification algebra if it is of the form $$\text{End}(T)$$ with $$T$$ a reflexive module, and $$A$$ is Cohen-Macaulay as an $$R$$-module. An NCCR is always an MMA, and in dimension 3, when $$R$$ is a Gorenstein singularity, all NCCR’s are MMA’s.
To introduce the toric aspect, the author gives a very neat introduction to toric algebraic geometry. Then the noncommutative crepant resolutions can be obliged to carry an additional toric structure: All summands of $$T$$ is supposed to have rank $$1$$ and are graded (by the toric variety $$M$$ in question). Then $$\text{End}(T)$$ is called a toric noncommutative crepant resolution.
The interesting problem is to classify all possible toric noncommutative crepant resolutions of a given singularity. The author gives an algorithm to construct all such for any toric 3-dimensional Gorenstein singularity. This is (basically) done by using the algorithm to construct quivers $$Q$$ that are also dimers, and then letting the algebra be the corresponding Jacobi algebra, which is the quiver algebra of $$Q$$ with some particular “dimer relations”. The toric conditions are strongly related to the representations of the quiver.
A method given by Craw and Quintero-Velez is used to embed the quivers of the toric NCCR algebras inside a real $$3$$-torus, such that the relations of the quiver algebra are precisely the homotopy relations. When the singularity is Gorenstein, the quiver can be projected to a 2-torus to obtain a dimer model.
The algorithm has to do with embedding the quivers into a torus, working on the varieties, and taking the corresponding quivers back to algebras (NCCR).
The algorithm is proven to be effective, in particular on the example of singularities from reflexive polygons and abelian quotients of the conifold. Then all dimer models corresponding to such a singularity are connected by mutations. Of course, generalizations of the algorithm to non toric, not Gorenstein, or higher dimensions are considered.
The article is very pedagogical, stringent, and it is a really good introduction to NCCR.

##### MSC:
 14A22 Noncommutative algebraic geometry 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 18E30 Derived categories, triangulated categories (MSC2010) 14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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