Schedule for: 19w5161 - Tilting Theory, Singularity Categories, & Noncommutative Resolutions

Following work of Buchweitz, one defines Tate-Hochschild cohomology of an algebra A to be the Yoneda algebra of the identity bimodule in the singularity category of bimodules. We show that Tate-Hochschild cohomology is canonically isomorphic to the ordinary Hochschild cohomology of the singularity category of A (with its canonical dg enrichment). In joint work with Zheng Hua, we apply this to prove a weakened version of a conjecture by Donovan-Wemyss which states that a complete isolated cDV singularity is determined by the derived equivalence class of the contraction algebra associated with a resolution.

Joint work with Ragnar-Olaf Buchweitz and Colin Ingalls. The classical McKay correspondence relates irreducible representations of a finite subgroup $G$ of $SL(2,\mathbb{C})$ to exceptional curves on the minimal resolution of the quotient singularity $\mathbb{C}^2/G$. Maurice Auslander observed an algebraic version of this correspondence: let $G$ be a finite subgroup of $SL(2,K)$ for a field $K$ whose characteristic does not divide the order of $G$. The group acts linearly on the polynomial ring $S=K[x,y]$ and then the so-called skew group algebra $A=G*S$ can be seen as an incarnation of the correspondence. In particular, $A$ is isomorphic to the endomorphism ring of $S$ over the corresponding Kleinian surface singularity. Auslander's isomorphism holds more generally for small finite subgroups $G$ of $GL(n,K)$, that is, $G$ does not contain any (pseudo-)reflections. The goal of this work is to establish a similar result when $G$ in $GL(n,K)$ is a finite group generated by reflections, assuming that the characteristic of $K$ does not divide the order of the group. Therefore we will consider a quotient of the skew group ring $A=S*G$, where $S$ is the polynomial ring in $n$ variables. We show that our construction yields a generalization of Auslander's result, and moreover, a noncommutative resolution of the discriminant of the reflection group $G$. In particular, we obtain a correspondence between the nontrivial irreducible representations of $G$ and certain maximal Cohen--Macaulay modules over the discriminant, and we can identify some of these modules, namely the so-called logarithmic (co-)residues of the discriminant.

Exceptional collections frequently arise in algebraic and symplectic geometry. Since the work of Beilinson on coherent sheaves on projective space, exceptional collections have been used to construct derived equivalences. In representation theory, exceptional collections appear for the class of quasi-hereditary algebras. In the first talk I will give an overview over the theory of exceptional collections and quasi-hereditary algebras. In a second talk, I will focus on an approach to quasi-hereditary algebras using exact Borel subalgebras, corings, and A-infinity Koszul duality.

The lectures will summarise the joint work with Michel Van den Bergh on the noncommutative resolutions of quotient singularities for reductive groups. The standard results about non-commutative resolutions of quotient singularities for finite groups can be naturally extended to arbitrary reductive groups. We will outline the "algorithm" and discuss several examples. Moreover, we will try to relate the noncommutative resolutions to their commutative counterparts.

Under a non-commutative noetherian scheme we understand a ringed space ${\mathbb X} =(X,{\mathcal A})$, where $X$ is a classical commutative noetherian scheme and ${\mathcal A}$ is a sheaf of ${\mathcal O}_X$-algebras, coherent as ${\mathcal O}_X$-module. I am going to state and then sketch the proof of the Morita theorem in this context, which tells when two categories of (quasi-) coherent sheaves on two non-commutative schemes ${\mathbb X}$ and ${\mathbb Y}$ are equivalent. The key ingredient of the proof uses properties of indecomposable injective quasi-coherent sheaves on ${\mathbb X}$. As an application, we obtain a new proof of a conjecture of Caldararu about Morita equivalences of Azumaya algebras on quasi-projective varieties. Another application provides a handleable criterion for two non-commutative projective curves to be Morita equivalent. This is a joint work with Yuriy Drozd.

Given an A-module M, and a dimension vector d, one can define a quiver Grassmannian, a projective algebraic variety parametrising the d-dimensional A-submodules of M. A famous result in geometric representation theory, obtained by several different authors, states that every projective variety X (over an algebraically closed field of characteristic zero) is isomorphic to such a quiver Grassmannian. In this talk I will explain how, at least in certain cases, one can use this algebraic description to construct a desingularisation of X. The construction is representation-theoretic, involving a tilt of an endomorphism algebra in mod(A), and the desingularising variety is again described in terms of quiver Grassmannians. This talk is based on joint work with Julia Sauter, in which we extend methodology of Crawley-Boevey and Sauter, and of Cerulli Irelli, Feigin and Reineke.

Joint work with Ragnar-Olaf Buchweitz and Colin Ingalls. The classical McKay correspondence relates irreducible representations of a finite subgroup $G$ of $SL(2,\mathbb{C})$ to exceptional curves on the minimal resolution of the quotient singularity $\mathbb{C}^2/G$. Maurice Auslander observed an algebraic version of this correspondence: let $G$ be a finite subgroup of $SL(2,K)$ for a field $K$ whose characteristic does not divide the order of $G$. The group acts linearly on the polynomial ring $S=K[x,y]$ and then the so-called skew group algebra $A=G*S$ can be seen as an incarnation of the correspondence. In particular, $A$ is isomorphic to the endomorphism ring of $S$ over the corresponding Kleinian surface singularity. Auslander's isomorphism holds more generally for small finite subgroups $G$ of $GL(n,K)$, that is, $G$ does not contain any (pseudo-)reflections. The goal of this work is to establish a similar result when $G$ in $GL(n,K)$ is a finite group generated by reflections, assuming that the characteristic of $K$ does not divide the order of the group. Therefore we will consider a quotient of the skew group ring $A=S*G$, where $S$ is the polynomial ring in $n$ variables. We show that our construction yields a generalization of Auslander's result, and moreover, a noncommutative resolution of the discriminant of the reflection group $G$. In particular, we obtain a correspondence between the nontrivial irreducible representations of $G$ and certain maximal Cohen--Macaulay modules over the discriminant, and we can identify some of these modules, namely the so-called logarithmic (co-)residues of the discriminant.

We report on a work jointly started with Ragnar Buchweitz about the pull back of a tilting bundle T to the total space of the canonical line bundle Y. Let X be an algebraic variety with a tilting bundle T, then we have a criterion when its pull back to Y is also a tilting bundle. This is closely related to my previous work on distinguished tilting sequences and generalizes the results therein. Morover, we can compute the endomorphism ring of the pull back tilting bundle as the higher preprojective algebra. This leads to a geometric construction of those algebras. The construction needs tilting bundles T with endomorphism algebra of global dimension dimX. In this talk we consider several examples for surfaces and compute the possible global dimensions of A.

Exceptional collections frequently arise in algebraic and symplectic geometry. Since the work of Beilinson on coherent sheaves on projective space, exceptional collections have been used to construct derived equivalences. In representation theory, exceptional collections appear for the class of quasi-hereditary algebras. In the first talk I will give an overview over the theory of exceptional collections and quasi-hereditary algebras. In a second talk, I will focus on an approach to quasi-hereditary algebras using exact Borel subalgebras, corings, and A-infinity Koszul duality.

Quasi-hereditary algebras were introduced by L. Scott in the context of the work of E. Cline, B. Parshall and L. Scott on highest weight categories arising in the representation theory of Lie algebras and algebraic groups. The notion of a quasi-hereditary algebra depends on a partial order given to the set of simple modules. In particular an algebra may be quasi-hereditary for one partial order but not for another one, even in the hereditary case. After recalling basic definitions, we will introduce an equivalence relation on the set of all partial orders giving a quasi-hereditary algebra, calling the equivalence classes quasi-hereditary structures. In the case of the path algebra of an equioriented quiver of type A, we will classify all its quasi-hereditary structures in terms of tilting modules, highlighting its nice combinatorial properties. Then we will generalise this classification to any orientation. As a complementary example we will discuss a class of quiver algebras with a unique quasi-hereditary structure. Time permitting, we will introduce a partial order on the set of all quasi-hereditary structures and give some examples. This is joint work in progress with Yuta Kimura and Baptiste Rognerud.

The lectures will summarise the joint work with Michel Van den Bergh on the noncommutative resolutions of quotient singularities for reductive groups. The standard results about non-commutative resolutions of quotient singularities for finite groups can be naturally extended to arbitrary reductive groups. We will outline the "algorithm" and discuss several examples. Moreover, we will try to relate the noncommutative resolutions to their commutative counterparts.

We show that the bounded derived category of many singular projectve varieties (with isolated singularities) does not admit tilting objects (and more generally no semiorthogonal decompositions of "Kawamata-type“). Using singularity categories, most of our obstructions come from cluster theory via NCCRs and also from negative K-theory. Our approach is inspired by recent work of Kawamata. This is joint work with N.Pavic and E.Shinder.

Fukaya categories of marked Riemann surfaces are now understood to be described combinatorially in terms of (graded) gentle algebras. In this talk I will explain how Iyama's higher Auslander algebras of type A relate to Fukaya categories of symmetric products of disks. I will also explain how to leverage this relationship to provide an alternative proof of a result of Beckert concerning certain derived equivalences between higher Auslander algebras of type A in different dimensions. This is part of joint work in progress with Tobias Dyckerhoff and Yanki Lekili.

Factorization homology provides a means of integrating algebraic structures over a space equipped with suitable geometric decorations thus producing interesting invariants. We will discuss how to use this framework to integrate higher Auslander algebras of Dynkin type A over a framed surface X and explain how the resulting invariants relate to Fukaya categories of symmetric products of X. This is based on joint work in progress with G. Jasso and Y. Lekili.

Consider a finitely generated normal commutative algebra R over a field K. A non-commutative resolution of singularities of Spec R is a (non-commutative) R-algebra A with finite global dimension of the form End(M) where M is some finitely generated reflexive R-module. The existence of a non-commutative resolution for a commutative ring R places strong conditions on R, such as rational singularities. In this talk, we discuss how in prime characteristic, the Frobenius can be used to construct non-commutative resolutions of nice enough rings. We conjecture that for a strongly F-regular ring R, End(F_*R) is a non-commutative resolution of R, where F_*R denotes R viewed as an R-module via restriction of scalars from Frobenius. We prove this conjecture when R is the coordinate ring of an affine toric variety. We also show that for toric rings, the ring of differential operators D(R) has finite global dimension (joint with Eleonore Faber and Greg Muller).

Tilting theory provides a fascinating link between the representation theory of finite dimensional algebras and algebraic geometry. Traditionally, it is approached from the algebraic geometry side by seeking tilting bundles on projective stacks. However, in studying representation theory, it is much more natural to start with a finite dimensional algebra and ask how one might attempt to construct a projective stack which is derived equivalent to it. In this talk, we look at two moduli stacks which address this question, the moduli of refined representations and tensor stable representations. The key is to incorporate data corresponding to the monoidal structure of the category of coherent sheaves on the derived equivalent stack. This is joint work with Tarig Abdelgadir and Boris Lerner.

The triangulated category of graded matrix factorizations for an exceptional unimodal singularity is known to have a tilting object by Kajiura-Saito-Takahashi and Lenzing-de la Pena. If we deform the singularity, then we lose the grading, which can be recovered by adding one more variable to the defining polynomial. The triangulated category of graded matrix factorizations of the resulting four-variable polynomial no longer has a tilting object, but has a classical generator, whose endomorphism algebra is the degree 2 trivial extension of the endomorphism algebra of the tilting object of the original category. In the talk, we will discuss the moduli space of A-infinity structures on this graded algebra, and its relation to 1. the positive part of the universal unfolding of the exceptional unimodal singularity, 2. the moduli space of K3 surfaces, and 3. homological mirror symmetry. If the time permits, we also discuss higher-dimensional generalizations and iterated singularity categories (i.e., singularity categories of singularity categories of ...) of non-isolated singularities. This is a joint work with Yanki Lekili.

We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, hereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated as an algebra. These results are deduced from a general sufficient condition for smoothness. This is joint work with Alexey Elagin and Valery Lunts.

Let Q be a finite quiver. A potential is called analytic if it is an infinite sum of cycles whose (complex) coefficients are bounded by a geometric series. Quiver with analytic potentials appear naturally in the deformation theory of sheaves on complex Calabi-Yau 3-folds. I will briefly survey the differential calculus of quivers with analytic potentials. It turns out that analyticity is preserved under mutations. We will construct a perverse sheaf of vanishing cycles on the moduli stack finite dimensional modules over Jacobi algebra for any iterated mutations, which can be used to define the refined Donaldson-Thomas invariants. This is a joint work with Bernhard Keller.

Let T be a triangulated category with coproducts and let X be a set of compact objects. Then X generates a certain t-structure, and in particular describes explicitly a left adjoint to the inclusion of the coaisle. Unfortunately, it does not make much sense to consider the naive dual of this setup; cocompact objects rarely appear in categories which occur naturally. Motivated by this, we introduce a weaker version of cocompactness called 0-cocompactness, and show that in a triangulated category with products these objects cogenerate a t-structure. As an application, we provide explicit right adjoints between certain homotopy categories (i.e. ``big'' singularity categories in the sense of Krause). Moreover, under the presence of a relative Serre functor we show how we can get 0-cocompact objects from compact ones. This is joint work with Steffen Oppermann and Torkil Stai.

We introduce the notion of a Nakayama functor relative to an adjunction, generalizing the classical Nakayama functor for a finite-dimensional algebra. We show that it can be characterized in terms of an ambidextrous adjunction of monads and comonads. We also study this concept from the viewpoint of Gorenstein homological algebra. In particular we obtain a generalization of the equality of the left and right injective dimension for a finite-dimensional Iwanaga-Gorenstein algebra, and for a module category we show that this property can also be characterized by the existence of a tilting module.

Our goal is to find conditions on a noetherian AS-regular algebra $A$ and an idempotent $e\in A$ for which the graded singularity category $\mathsf{Sing}^{\mathsf{gr}}(eAe)$ admits a tilting object. Of particular interest is the situation in which $A$ is a graded skew-group algebra $S\#G$, where $S$ is the polynomial ring in $n$ variables and $G < SL(n,k)$ is finite, and $e = \frac{1}{|G|}\sum_{g\in G} g$, so that $eAe\cong S^G$. A tilting object was found by Amiot, Iyama and Reiten in the case where $A$ has Gorenstein parameter $1$. Generalizing the work of Iyama and Takahashi, Mori and Ueyama obtained a tilting object in $\mathsf{Sing}^{\mathsf{gr}}(S^G)$, provided that $S$ is a noetherian AS-regular Koszul algebra generated in degree $1$ and $G$ has homological determinant $1$. In this talk, we will discuss certain silting objects and then specialise to the setting in which the Beilinson algebra is a levelled algebra, giving a generalisation of the result of Mori and Ueyama.

For a 3-fold flopping contraction from X to the spectrum Spec(R) of a complete local Gorenstein ring (R,m) with terminal singularity at m, we give a description of a distinguished connected component of the (normalized) space of Bridgeland stability conditions on certain triangulated categories associated to the flopping contraction. More precisely, we show that the connected component is a regular covering space of the complement of the complexification of a hyperplane arrangement associated to the 3-fold flop. We also determine the autoequivalence groups of the triangulated categories. As an application of these results, we determine the Stringy K ̈ahler Moduli Space (SKMS) for all smooth irreducible 3-fold flops. This is a joint work with Michael Wemyss.

In this abstract $K$ denotes a field of char $K = 0$ and $Q$ denotes a finite acyclic quiver. Recall that the preprojective algebra $\Pi(Q) = K\overline{Q}/(\rho )$ of a quiver $Q$ is the path algebra $K \overline{Q}$ of the double quiver $\overline{Q}$ of $Q$ with the mesh relation $\rho=\sum_{\alpha \in Q_{1}} \alpha \alpha^{*} - \alpha^{*} \alpha$. It is an important mathematical object having rich representation theory and plenty of applications. In this joint work with M. Herschend, we study a cubical generalization $\Lambda= \Lambda(Q) := K \overline{Q}/([a, \rho] \mid a \in \overline{Q}_{1})$ where $[-,+]$ is the commutator. We note that our algebra $\Lambda$ is a special case of algebras $\Lambda_{\lambda, \mu}$ introduced by Etingof-Rains, which is a special case of algebras $\Lambda_{P}$ introduced by Cachazo-Katz-Vafa. However, our algebra $\Lambda$ of very special case has intriguing properties, among other things it provides the universal Auslander-Reiten triangle for $K Q$. We may equip $\Lambda$ with a grading by setting $\deg \alpha = 0, \deg \alpha^{*} : =1$ for $\alpha \in Q_{1}$. We introduce an algebra to be $A = A(Q) := \begin{pmatrix} K Q & \Lambda_{1} \\ 0 & K Q \end{pmatrix}$ where $\Lambda_{1}$ is the degree $1$-part of $\Lambda$. Our results combining with other existing results show that the algebras $A(Q)$ and $\Lambda (Q)$ are one-dimensional higher versions of $K Q$ and $\Pi (Q)$.

We study the scaling dimension (or the similarity dimension) of the perfect derived category of a smooth compact dg algebra called the Serre dimension. It is expected that the infimum of the Ikeda-Qiu’s global dimsion function on the space of stability conditions also gives another “good” notion of dimension. One of our results is that its infimum is always greater than or equal to the Serre dimension. Motivated by the ADE classification of the 2-dimensional N=2 SCFT with \widehat{c}<1, we also give a characterization of the derived category of Dynkin quivers in terms of the Serre dimension and the global dimension function. This is a joint work in progress with Kohei Kikuta and Genki Ouchi.

A fuchsian singularity is classically attached to a finitely generated, cocompact subgroup $G$ of the automorphism group of the hyperbolic plane $\mathbb{H}$. It is the $\mathbb{Z}$-graded algebra of $G$-invariant differentials (automorphic forms) on $\mathbb{H}$. My talk has the following aims: (1) Extend the concept to algebraically closed fields of arbitrary characteristic. (2) Discuss their relationship to mathematical objets of a different nature. (3) Provide a ring-theoretic characterization of fuchsian singularities. (4) Explore their singularity categories.