Non-commutative Crepant Resolutions Research Articles

Mutations of noncommutative crepant resolutions in geometric invariant theory

Let X be a generic quasi-symmetric representation of a connected reductive group G. The GIT quotient stack X=[Xss(ℓ)/G]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {X}=[X^\ ext {ss}(\\ell )/G]$$\\end{document} with respect to a generic ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} is a (stacky) crepant resolution of the affine quotient X/G, and it is derived equivalent to a noncommutative crepant resolution (=NCCR) of X/G. Halpern-Leistner and Sam showed that the derived category Db(cohX)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extrm{D}}^{\ extrm{b}}}({\ ext {coh}}\\mathfrak {X})$$\\end{document} is equivalent to certain subcategories of Db(coh[X/G])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extrm{D}}^{\ extrm{b}}}({\ ext {coh}}[X/G])$$\\end{document}, which are called magic windows. This paper studies equivalences between magic windows that correspond to wall-crossings in a hyperplane arrangement in terms of NCCRs. We show that those equivalences coincide with derived equivalences between NCCRs induced by tilting modules, and that those tilting modules are obtained by certain operations of modules, which is called exchanges of modules. When G is a torus, it turns out that the exchanges are nothing but iterated Iyama–Wemyss mutations. Although we mainly discuss resolutions of affine varieties, our theorems also yield a result for projective Calabi-Yau varieties. Using techniques from the theory of noncommutative matrix factorizations, we show that Iyama–Wemyss mutations induce a group action of the fundamental group π1(P1\\{0,1,∞})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi _1(\\mathbb {P}^1\\,\\backslash \\{0,1,\\infty \\})$$\\end{document} on the derived category of a Calabi-Yau complete intersection in a weighted projective space.

Non-commutative resolutions of linearly reductive quotient singularities

ABSTRACT We prove the existence of non-commutative crepant resolutions (in the sense of Van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension 2, we relate these to resolutions of singularities provided by G-Hilbert schemes and F-blowups. As an application, we establish and recover results concerning resolutions for toric singularities, as well as canonical, log terminal and F-regular singularities in dimension 2.

A note on affine cones over Grassmannians and their stringy 𝐸-functions

We compute the stringy E E -function of the affine cone over a Grassmannian. If the Grassmannian is not a projective space then its cone does not admit a crepant resolution. Nonetheless the stringy E E -function is sometimes a polynomial and in those cases the cone admits a noncommutative crepant resolution. This raises the question as to whether the existence of a noncommutative crepant resolution implies that the stringy E E -function is a polynomial.

On the M2–Brane Index on Noncommutative Crepant Resolutions

On certain M-theory backgrounds which are a circle fibration over a smooth Calabi–Yau the quantum theory of M2 branes can be studied in terms of the K-theoretic Donaldson–Thomas theory on the threefold. We extend this relation to noncommutative crepant resolutions. In this case the threefold develops a singularity and classical smooth geometry is replaced by the algebra of paths of a certain quiver. K-theoretic quantities on the quiver representation moduli space can be computed via toric localization and result in certain rational functions of the toric parameters. We discuss in particular the case of the conifold and certain orbifold singularities.

Conic divisorial ideals and non-commutative crepant resolutions of edge rings of complete multipartite graphs

The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by k[Kr1,…,rn], where 1≤r1≤⋯≤rn. More concretely, we prove that the class group of k[Kr1,…,rn] is isomorphic to Zn if n=3 with r1≥2 or n≥4, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of k[Kr1,…,rn], called conic divisorial ideals. We describe conic divisorial ideals for certain Kr1,…,rn including all cases where k[Kr1,…,rn] is Gorenstein. Finally, we give a non-commutative crepant resolution (NCCR) of k[Kr1,…,rn] in the case where it is Gorenstein.

On the noncommutative Bondal–Orlov conjecture for some toric varieties

We show that all toric noncommutative crepant resolutions (NCCRs) of affine GIT quotients of “weakly symmetric” unimodular torus representations are derived equivalent. This yields evidence for a non-commutative extension of a well known conjecture by Bondal and Orlov stating that all crepant resolutions of a Gorenstein singularity are derived equivalent. We prove our result by showing that all toric NCCRs of the affine GIT quotient are derived equivalent to a fixed Deligne–Mumford GIT quotient stack associated to a generic character of the torus. This extends a result by Halpern–Leistner and Sam which showed that such GIT quotient stacks are a geometric incarnation of a family of specific toric NCCRs constructed earlier by the authors.

Dimer models and Hochschild cohomology

Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau–Ginzburg models dual to punctured Riemann surfaces. For a consistent dimer embedded in a torus, we explicitly compute the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. This includes a full characterization of the Batalin–Vilkovisky structure induced by the Calabi–Yau structure of the Jacobi algebra. We then compute the compactly supported Hochschild cohomology of the category of matrix factorizations for the Jacobi algebra with its canonical potential.

Noetherian criteria for dimer algebras

Let A be a nondegenerate dimer (or ghor) algebra on a torus, and let Z be its center. Using cyclic contractions, we show the following are equivalent: A is noetherian; Z is noetherian; A is a noncommutative crepant resolution; each arrow of A is contained in a perfect matching whose complement supports a simple module; and the vertex corner rings eiAei are pairwise isomorphic.

On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

The Abuaf-Ueda flop is a 7-dimensional flop related to G<sub>2</sub> homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.

Semi-orthogonal decompositions of GIT quotient stacks

If G is a reductive group acting on a linearized smooth scheme X then we show that under suitable standard conditions the derived category \({{{\mathcal {D}}}}(X^{ss}{/}G)\) of the corresponding GIT quotient stack \(X^{ss}{/}G\) has a semi-orthogonal decomposition consisting of derived categories of coherent sheaves of rings on \(X^{ss}{/\!\!/}G\) which are locally of finite global dimension. One of the components of the decomposition is a certain non-commutative resolution of \(X^{ss}{/\!\!/}G\) constructed earlier by the authors. As a concrete example we obtain in the case of odd Pfaffians a semi-orthogonal decomposition of the corresponding quotient stack in which all the parts are certain specific non-commutative crepant resolutions of Pfaffians of lower or equal rank which had also been constructed earlier by the authors. In particular this semi-orthogonal decomposition cannot be refined further since its parts are Calabi–Yau. The results in this paper complement results by Halpern–Leistner, Ballard–Favero–Katzarkov and Donovan–Segal that assert the existence of a semi-orthogonal decomposition of \({{{\mathcal {D}}}}(X/G)\) in which one of the parts is \({{{\mathcal {D}}}}(X^{ss}/G)\).

The acyclic closure of an exact category and its triangulation

For any exact category A with splitting idempotents, a maximal exact category T(A) containing A as a biresolving subcategory, is constructed. Important types of exact categories, including n-tilting torsion classes, categories of Cohen-Macaulay modules over a Cohen-Macaulay order, or categories of Gorenstein projectives, are shown to be of the form T(A). The quotient category T(A)/A in the sense of Grothendieck always exists and carries a triangulated structure. More generally, it is proved that any biresolving subcategory A of an exact category M gives a triangulated localization M/A. For example, the unbounded derived category D(A) of an exact category A is obtained directly, with no passage through a homotopy category of complexes. As applications, some recent developments related to Gorenstein projectivity, non-commutative crepant resolutions, singularity categories, and Cohen-Macaulay representations are extended and improved in the new framework. For example, the concept of non-commutative resolution of a noetherian Frobenius category is extended to arbitrary exact categories, which leads to an overarching connection with representation dimension of exact categories and n-tilting.

Non-commutative crepant resolutions for some toric singularities. II

Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec $R$ admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec $R$ using standard toric methods. Our proof does not use dimer models.

Holography, matrix factorizations and K-stability

Placing D3-branes at conical Calabi-Yau threefold singularities produces many AdS5/CFT4 duals. Recent progress in differential geometry has produced a technique (called K-stability) to recognize which singularities admit conical Calabi-Yau metrics. On the other hand, the algebraic technique of non-commutative crepant resolutions, involving matrix factorizations, has been developed to associate a quiver to a singularity. In this paper, we put together these ideas to produce new AdS5/CFT4 duals, with special emphasis on non-toric singularities.

Non-Commutative Crepant Resolutions for Some Toric Singularities I

Abstract We give a criterion for the existence of noncommutative crepant resolutions (NCCRs) for certain toric singularities. In particular, we recover Broomhead’s result that a three-dimensional toric Gorenstein singularity has an NCCR. Our result also yields the existence of an NCCR for a four-dimensional toric Gorenstein singularity, which is known to have no toric NCCR.

Gorenstein modifications and \mathds{𝑄}-Gorenstein rings

Let R R be a Cohen–Macaulay normal domain with a canonical module ω R \omega _R . It is proved that if R R admits a noncommutative crepant resolution (NCCR), then necessarily it is Q \mathds {Q} -Gorenstein. Writing S S for a Zariski local canonical cover of R R , a tight relationship between the existence of noncommutative (crepant) resolutions on R R and S S is given. A weaker notion of Gorenstein modification is developed, and a similar tight relationship is given. There are three applications: non-Gorenstein quotient singularities by connected reductive groups cannot admit an NCCR, the centre of any NCCR is log-terminal, and the Auslander–Esnault classification of two-dimensional CM-finite algebras can be deduced from Buchweitz–Greuel–Schreyer.

Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions

In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.

Hori-mological projective duality

Kuznetsov has conjectured that Pfaffian varieties should admit noncommutative crepant resolutions which satisfy his Homological Projective Duality. We prove half the cases of this conjecture by interpreting and proving a duality of nonabelian gauged linear sigma models proposed by Hori.

Noncommutative quasi-resolutions

The notion of a noncommutative quasi-resolution is introduced for a noncommutative noetherian algebra with singularities, even for a non-Cohen-Macaulay algebra. If A is a commutative normal Gorenstein domain, then a noncommutative quasi-resolution of A naturally produces a noncommutative crepant resolution (NCCR) of A in the sense of Van den Bergh, and vice versa. Under some mild hypotheses, we prove that(i)in dimension two, all noncommutative quasi-resolutions of a given noncommutative algebra are Morita equivalent, and(ii)in dimension three, all noncommutative quasi-resolutions of a given noncommutative algebra are derived equivalent. These assertions generalize important results of Van den Bergh, Iyama-Reiten and Iyama-Wemyss in the commutative and central-finite cases.

Non-commutative resolutions of toric varieties

Let R be the coordinate ring of an affine toric variety. We prove, using direct elementary methods, that the endomorphism ring EndR(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 of Špenko and Van den Bergh implying that EndR(A) has finite global dimension. Furthermore, we show that EndR(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 Dk(R) has finite global dimension.

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

A consistent dimer model gives a non-commutative crepant resolution (= NCCR) of a 3-dimensional 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.