# Pan-American Advanced Studies Institute (PASI)

## Commutative Algebra and Its Interactions with Algebraic Geometry, Representation Theory, and Physics

Centro de Investigación en Matemáticas (CIMAT)
Guanajuato, Mexico
May 14–25, 2012

organized by
Ragnar-Olaf Buchweitz
José Antonio de la Peña
Srikanth B. Iyengar
Sarah Witherspoon

update 27 August 2012: Notes from the lecture series are becoming available.

• Avramov, Support varieties for algebras
• Iyama, Auslander-Reiten theory and cluster tilting for maximal Cohen-Macaulay modules: lecture notes
• Leuschke, Maximal Cohen-Macaulay modules and non-commutative desingularizations: lecture notes, problem session notes
• Lupercio, Mini Course on Topological Quantum Field Theories
• Murfet, Bicategories and matrix factorisations

Thanks to the notetakers!

update 24 March 2013: As a follow-up activity, there was a focus period on Matrix Factorizations at the Mathematical Sciences Research Institute, in 2013. The notes from most of the lectures there are posted here. Many thanks to Eleonore Faber for providing these notes.

This two-week program at CIMAT is targeted at advanced graduate students, postdocs, and early career faculty from North and South America working in the fields of commutative algebra, algebraic geometry, representation theory, and theoretical physics. The program will highlight emerging applications and interactions among these fields and will facilitate international collaboration. Experts will give lecture series at the advanced graduate level and supervise problem sessions. Complementing the lecture series will be research talks given by both senior and younger mathematicians, and a poster session.

Specific topics to be covered at the advanced studies institute include matrix factorizations and maximal Cohen-Macaulay modules, cluster algebras and cluster categories, cohomological theory of support varieties, and topological quantum field theories and Frobenius algebras. These areas are at the intersection of several important and rapidly developing fields of mathematics, each subject benefitting from advances in the others. At the heart of it all is commutative algebra that provides a guide book and some of the critical tools used in research on these seemingly diverse topics.

Full support of young mathematicians (recent PhDs and advanced graduate students) from the Americas is expected through funding by the National Science Foundation (USA) and the National Council for Science and Technology (Mexico).

Registration for this program is now closed. For accepted participants, practical information (concerning lodging and travel, for example) about the PASI can be found here. Contact pasi2012@math.unl.edu with inquiries.

May 14–18, 2012
The first week will consist of lecture series and problem sessions. Speakers and topics are:

• Luchezar L. Avramov (University of Nebraska-Lincoln, USA): Support varieties for algebras

In 1971 Quillen introduced a completely new point of view in the study of cohomology of groups, and used it to establish conjectures of Atiyah and others concerning resolutions of the trivial representation. His ideas were extended first to arbitrary finite dimensional representations of finite groups, then to modules over an increasing number of algebraic systems (restricted Lie algebras, algebraic groups, classes of commutative rings…). In each one of these situations a geometric object, such as an algebraic variety, was associated to a purely algebraic object. The resulting theories present three distinct aspects: Construction of geometric invariants, proof of their properties, and applications to problems in the area of origin.

In the lectures, constructions and applications will be illustrated through specific examples. The focus will be on the properties of cohomological support varieties. They are known to be remarkably similar in all contexts, but their proofs often heavily rely on techniques specific to the objects being studied. Recent work suggests that general algebraic structures may be at play, and proofs of some basic properties can be obtained through simple arguments that handle all cases simultaneously.

• Osamu Iyama (University of Nagoya, Japan): Auslander-Reiten theory and cluster tilting for maximal Cohen-Macaulay modules

Representation theory of (maximal) Cohen-Macaulay modules is initiated by Auslander for a certain class of non-commutative algebras called orders [A1,A2][I, section 3.1]. There are two important cases: One is commutative Cohen-Macaulay rings [Y], and another is finite dimensional algebras over fields. Recently a new connection between representation theories of these two types of rings is found via tilting and cluster tilting.

Tilting theory is one of the basic tools in representation theory, and enables us to control equivalences of derived categories. Recently many equivalences between stable categories of graded Cohen-Macaulay modules and derived categories of finite dimensional algebras were
found (e.g. [KST,IT]).

On the other hand, the notion of cluster tilting modules was introduced in higher dimensional Auslander-Reiten theory [I,IY]. It played an important role in cluster theory which gives a connection between the structure of certain triangulated categories called cluster categories and combinatorics of Fomin-Zelevinsky cluster algebras (e.g. [R,K]). We give equivalences between stable categories of (ungraded) Cohen-Macaulay modules and cluster categories of finite dimensional algebras [AIR]. As a special case, we recover Auslander’s algebraic McKay correspondence between Kleinian singularities and path algebras of Dynkin quivers [A3].

We also discuss another important aspect of cluster tilting modules: We show that the endomorphism algebras of cluster tilting modules are non-singular orders in the sense of Auslander [A2]. In particular they give non-commutative crepant resolutions in the sense of Van den Bergh [V]. Applying tilting theory, we give derived equivalences between non-commutative crepant resolutions of certain fixed rings including rings in dimension three [IW1]. If time allows, we also discuss generalizations of these results [IW2]. [References]

• Graham Leuschke (Syracuse University, USA): Maximal Cohen-Macaulay modules and non-commutative desingularizations

This series of lectures will describe some recent work on non-commutative analogs of resolutions of singularities, with a focus on the contribution made by maximal Cohen-Macaulay modules. The first lecture will discuss the McKay Correspondence, a key motivating example, while in the second lecture I will explain the broad outlines of non-commutative desingularizations. In the third and fourth lectures, I will give Van den Bergh’s definition of a non-commutative crepant resolution and describe recent joint work with Buchweitz and Van den Bergh on constructing non-commutative crepant resolutions for varieties defined by minors of generic matrices. [References]

• Ernesto Lupercio (CINVESTAV, Mexico): Mini Course on Topological Quantum Field Theories

The purpose of this course is to introduce algebraists to the mathematics of quantum field theory in a self-contained manner.

Lecture 1: From Physics to Algebra.

Lecture 2: Categories and Functors.

Lecture 3: Frobenius Algebras.

Lecture 4: Constructing theories from geometry and topology.

Lecture 5: Recent developments. Coda.

• Daniel Murfet (University of California, Los Angeles, USA): Bicategories and matrix factorisations

We will present the basic theory of matrix factorisations in the setting of a bicategory $\mathcal{LG}$ whose objects are isolated hypersurface singularities and whose $1$-morphisms are matrix factorisations \cite{McNameethesis,cr0909.4381,cr1006.5609}. Using the language of bicategories and string diagrams we will formulate and prove rigorous statements about matrix factorisations following the physical intuition about two-dimensional Landau-Ginzburg models on worldsheets with defect lines \cite{ffrs0607247}.

The main aim will be to understand duality in this bicategory, that is, adjointness of $1$-morphisms and the explicit units and counits of these adjunctions, in terms of noncommutative differential forms and Atiyah classes \cite{Loday,dm1102.2957}. As a special case this will include a discussion of the Kapustin-Li formula and Serre duality in the homotopy category of matrix factorisations \cite{Kapustin03,m0912.1629,dyckmurf}. We will also address Dyckerhoff’s results on Hochschild (co)homology \cite{d0904.4713} the work of Polishchuk and Vaintrob on the Chern character and the Hirzebruch-Riemann-Roch theorem \cite{pv1002.2116} and shadows and traces \cite{ps0910.1306}. If time permits, we will sketch the connection of $\mathcal{LG}$ to the Khovanov-Rozansky homology of knots and $2$-representation theory \cite{kr0401268}, and recent results on matrix factorisation kernel algebras. [References]

There will also be two special lectures:

• Special Lecture I: CIMAT Colloquium, Wednesday, 16th May, 4 PM.
Speaker: Ragnar-Olaf Buchweitz
Title: Maximal Cohen-Macaulay modules over complete intersection rings
• Special Lecture II: Seminar on Differential Geometry, Thursday, 17th May, 5 PM.
Speaker: Ernesto Lupercio
Title: The moduli space of (non-commutative) toric varieties

May 21–25, 2012
The second week will include survey lecture series, research talks, and a poster session. A partial list of speakers is:

• Nicolás Andruskiewitsch (Academia Nacional de Ciencias, Córdoba, Argentina): From Hopf algebras to tensor categories

In the first part of the talk I will introduce Hopf algebras and discuss the classification scheme of finite-dimensional complex ones. An overview of the actual state of the art will be given. In the last part I will discuss how to obtain tensor categories from spherical Hopf algebras and how we hope to discover new examples in this way.

• Matthew Ballard (University of Wisconsin, USA): Variation of GIT quotients and matrix factorizations

Let $G$ be a reductive algebraic group acting on affine space, $X=\mathbb{A}^n$, and assume we have a $G$-invariant polynomial, $w$. Mumford’s Geometric Invariant Theory gives a prescription for forming “good” quotients of $X$ by $G$. One chooses a linearization of the action, in this case a character, $\chi$, of $G$, produces the semi-stable locus, $X^{\mathrm{ss}}(\chi)$, and takes its quotient, $X //_{\chi} G$. The polynomial, $w$, descends to regular function on $X //_{\chi} G$ denoted by $w //_{\chi} G$, and one can study the categories of matrix factorizations, $\mathrm{mf}(X //_{\chi} G, w //_{\chi} G)$. We will show how, under some mild restrictions, different choices of $\chi$ lead to categories of matrix factorizations related by semi-orthogonal decompositions. This approach builds on ideas of Herbst, Hori, Page, Walcher, Segal, and Shipman and, after a slight level-up, provides a robust generalization of Orlov’s theorem relating the derived category of a complete intersection in projective space to the graded singularity category of its affine cone. All results in the talk come from joint work, arXiv:1203.6643, with David Favero (U. Wien) and Ludmil Katzarkov (U. Miami, U. Wien).

• Lars Winther Christensen (Texas Tech University, USA): Brauer-Thrall for totally reflexive modules

Let $R$ be a commutative noetherian local ring. If the category $\mathcal{G}(R)$ of totally reflexive modules over $R$ is representation finite, then it is either trivial—$R$ is the unique indecomposable module in the category—or $R$ is Gorenstein. This was proved in joint work with Piepmeyer, Striuli and Takahashi from 2007. Thus, if $R$ is not Gorenstein and allows a non-free totally reflexive module, then the category $\mathcal{G}(R)$ is representation infinite. I will discuss recent progress in our understanding of how complex the category is under these circumstances. This part of the talk is based on joint work with Jorgensen, Rahmati, Striuli, and Wiegand.

• Hailong Dao (University of Kansas, USA): Classifying resolving subcategories of mod R

Let $R$ be a commutative noetherian ring and $\mathrm{mod} R$ be the category of finitely generated $R$-modules. A subcategory of $\mathrm{mod} R$ is called resolving if it contains $R$ and is closed under taking extensions, syzygies and direct summands. In this talk I will describe a joint project with Ryo Takahashi that allows us to classify all resolving subcategories of $\mathrm{mod} R$ when $R$ is a locally complete intersection ring. The classification involves functions from $\mathrm{Spec} R$ to the non-negative integers.

• Toby Dyckerhoff (Yale University, USA): Higher Segal spaces

In this talk, based on joint work in progress with M. Kapranov, we will introduce the theory of 2-Segal spaces. The motivating example of a 2-Segal space is the Waldhausen S-construction of an exact (higher) category. In particular, this example explains why Hall algebras of abelian, exact, and triangulated categories are associative, but beyond that, the notion of a 2-Segal space captures higher coherences which have not been previously studied. As I will explain in the talk, these higher coherences are concretely useful in various ways: For example, (1) to any unital 2-Segal space, we can associate a higher monoidal Hall category, categorifying the classical Hall algebra construction, (2) to any cyclic 2-Segal space we can associate space-valued representations of the mapping class groups of marked Riemann surfaces.

• David Eisenbud (University of California-Berkeley, USA): The Shapes of Free Complexes (two lectures)

The Hilbert Polynomial is a fundamental invariant of a graded module over a polynomial ring. It is refined by the Betti table of the module, and, in a different way, by the cohomology table of a coherent sheaf on a projective space. It is natural to consider these same invariants for complexes.

In the last five years our knowledge of the possible values of these invariants has increased dramatically. This development was sparked by a group of conjectures made by the Swedish mathematicians Mats Boij and Jonas Soederberg, and the area goes by the name of Boij-Soederberg Theory. I will survey these developments in the first lecture. In the second lecture I will focus on the most recent developments, which extend the theory from the case of modules to complexes and from polynomial rings to more general rings.

• Dmitry Kerner (University of Toronto, Canada): On local and global determinantal representations.

Let A be a square matrix whose entries are linear forms (global case) or power series (local case). Then A is a determinantal representation of the corresponding hypersurface {$\det A = 0$}.

In the global case we characterize such determinantal representations in terms of particular sheaves on the hypersurface. In both cases we address the following decomposability question. Suppose $\det(A)$ is (locally/globally) reducible. Which conditions ensure that A is equivalent to a block-diagonal or upper-block-triangular matrix? (i.e. the corresponding sheaf or module is decomposable or an extension.)

The talk is based on arXiv:1009.2517 and arXiv:0906.3012.

• Henning Krause (Bielefeld University, Germany): Koszul, Ringel, and Serre duality for strict polynomial functors

Strict polynomial functors were introduced by Friedlander and Suslin in their work on the cohomology of finite group schemes. More recently, a Koszul duality for strict polynomial functors has been established by Chalupnik and Touze. In my talk I will give a gentle introduction to strict polynomial functors (via representations of divided powers) and will explain the Koszul duality, making explicit the underlying monoidal structure which seems to be of independent interest. Then I connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors.

• Helmut Lenzing (University of Paderborn, Germany): Kleinian and Fuchsian versus triangle singularities (two lectures)

In my first talk I will deal with the algebraic analysis of the singularity

$f=x_1^a + x_2^b +x_3^c,$
or equivalently $S=k[x_1,x_2,x_3]/(f)$, where $a,\ b$ and $c$ are integers $\geq2$. Here, $k$ denotes an algebraically closed field. By means of the rank one abelian group $\mathbb{L}$ on generators $\vec{x}_1$, $\vec{x}_2$ and $\vec{x}_3$ subject to relations $a\vec{x}_1=b\vec{x}_2=c\vec{x}_3=:\vec{c}$, the $k$-algebra $S$ becomes $\mathbb{L}$-graded. The element $\vec{c}$ will be called the canonical element of $\mathbb{L}$. By Serre construction, that is, forming the quotient category of all finitely generated graded modules by the finite length modules, we obtain the category $\mathrm{coh}\,\mathbb{X}$ of coherent sheaves on a weighted projective line $\mathbb{X}$ with three marked points of order $a,\ b$ and $c$, respectively. On the other hand, we can pass to the singularity category $\mathrm{Sing}^\mathbb{L}(S)$ in the sense of Buchweitz (1986) and Orlov (2005). This category is triangulated and comes in many different incarnations. For instance, $\mathrm{Sing}^\mathbb{L}(S)$ is equivalent to the quotient of the bounded derived category of all finitely generated graded $S$-modules by the subcategory of perfect complexes.

Further $\mathrm{Sing}^\mathbb{L}(S)$ is equivalent to the stable category $\underline{\mathrm{CM}}^\mathbb{L}(S)$ of all maximal graded Cohen-Macaulay modules. Since we are dealing with the hypersurface case, it is also equivalent to the (stable) category of matrix factorizations of $f$. For our purposes the most accessible form of $\mathrm{Sing}^\mathbb{L}(S)$ is due to the fact that $S$ is two-dimensional. Because of this the singularity category can be viewed as the stable category $\underline{\mathrm{vect}}\mathbb{X}$ of vector bundles on the weighted projective line $\mathbb{X}$, where stable refers to the formation of the factor category of the category of vector bundles modulo the two-sided ideal of all morphisms factoring through a direct sum of line bundles. By a variant of Orlov’s theorem (2005,2009) the derived category $D^b(\mathrm{coh}\,\mathbb{X})$ compares to the singularity category $\mathrm{Sing}^\mathbb{L}(S)$ in a way that depends on the Gorenstein parameter $a$ of $f$. For instance, for $a=0$ the two triangulated categories are equivalent; otherwise, depending on the sign of $a$, one category sits in the other as the perpendicular category with respect to an exceptional sequence. Further $a=1-2\chi_{\mathbb{X}}$, where $\chi_{\mathbb{X}}$ denotes the (orbifold) Euler characteristic of $\mathbb{X}$.

In my second talk, I will deal with Kleinian and Fuchsian singularities.
Let $\vec{\omega}$ denote the dualizing element of $\mathbb{L}$, defined as

$\vec{\omega}=\vec{c}- \sum_{i=1}^3\vec{x}_i.$
Assuming $\chi_{\mathbb{X}}\neq0$, we define a $\mathbb{Z}$-graded algebra $R$ by putting $R_n=R_{-n\vec{\omega}}$ (resp.\ $R_n=R_{n\vec{\omega}}$) if $\chi_{\mathbb{X}}>0$, respectively $latex \chi_{\mathbb{X}}<0$. In the first case, we obtain the Kleinian, in the second case we obtain the Fuchsian singularities. (In the latter case one also has to allow more than three weights.) In both cases, the Serre construction yields the same weighted projective line given by the weight triple $latex (a,b,c)$. The Orlov context can be applied as before, with the only change to replace the system of all line bundles (=triangle case) by the system of all twisted line bundles $latex \mathcal{O}(n\vec{\omega})$, $latex n\in\mathbb{Z}$, from a single Auslander-Reiten orbit of line bundles (=Kleinian, resp.\ Fuchsian case). Though, technically, this looks to be a small change, the properties of the attached stable categories change significantly. While for both cases the stable categories in question have tilting objects, their endomorphism ring have quite different properties. Most importantly, in the triangle case all stable categories are fractionally Calabi-Yau and, in particular, have a periodic Coxeter transformation. By contrast, in the Fuchsian case this happens only for a finite set of weight triples, in particular, for the members of Arnold's strange duality list.

This is joint work with J.A. de la Pe\~na (2007), resp. with D. Kussin and H. Meltzer (2012). Related investigations in the Kleinian and Fuchsian case are due to H. Kajiura, K. Saito and A. Takahashi (2007).

• Claudia Miller (Syracuse University, USA): Restrictions on modules of finite projective dimension

Modules of finite projective dimension play a large role in commutative algebra, most notably in the homological conjectures, and yet interesting examples, such as the famous finite length counterexample given by Dutta, Hochster and MacLaughlin, can be difficult to find. To motivate part of why this is so, I will describe restrictions that the ring structure can impose on modules of finite length and finite projective dimension. In joint work with Avramov, Buchweitz, and Iyengar, the main conclusion is that such modules cannot be too small, where size here is measured by the Loewy length. In this talk, I will focus on the portion of the paper dealing with modules rather than complexes, where, under certain conditions on the ring, a simpler and larger lower bound can be given in terms of the Castelnuovo-Mumford regularity of the ring. Over a graded complete intersection ring this translates to a simple expression in terms of the degrees of the defining equations and extends a result of Ding for hypersurfaces.

• Julia Pevtsova (University of Washington, USA): Modules of constant Jordan type and their generalizations

Let $G$ be a finite group (scheme) with the group algebra $kG$ for a field $k$ of positive characteristic p. A module of constant Jordan type over $G$ is a module that exhibits the same behavior when restricted to various subalgebras of $kG$ isomorphic to $k[t]/t^p$. I’ll discuss properties of these modules and their generalizations, some open problems and recent progress towards them, and the connections between modules of constant Jordan type and the geometry of the projective variety $\mathrm{Proj} H^*(G,k)$. For many of my examples, I’ll concentrate on the case of an elementary abelian p-group, for which the geometric connections lead to a correspondence between modules of constant Jordan type and vector bundles on projective spaces (and between generalized modules of constant Jordan type and bundles on Grassmannians). This is joint work with J. Carlson and E. Friedlander.

• Alexander Polishchuk (University of Oregon, USA): Matrix factorizations and cohomological field theories, parts I and II (two lectures)

I will describe our joint work with Arkady Vaintrob on an algebraic version of Fan-Jarvis-Ruan theory. This theory is an analog of Gromov-Witten theory in which a target space is replaced by a quasihomogeneous isolated hypersurface singularity. In the case of simple singularities of type A it corresponds to the intersection theory on the moduli space of higher spin curves, and constitutes a framework for the famous Witten conjectures, proved by Faber-Shadrin-Zvonkine. In my talks I will explain the algebro-geometric construction of the relevant cohomological field theory based on the theory of matrix factorizations. The crucial construction of an analog of the virtual fundamental class involves derived categories of matrix factorizations in a global setting.

• Liana Şega (University of Missouri, USA): On the linearity defect of the residue field

Given a local commutative ring, the linearity defect of the residue field is an invariant that measures how far the residue field and its syzygies are from having a linear resolution. Motivated by a positive known answer in the graded case, we study the question of whether the linearity defect of the residue field, if finite, needs to be zero. We give answers in special cases, and we discuss several interpretations and refinements of the question.

• Ryo Takahashi (Shinsu University, Japan): Dimensions of derived categories and analogues for module categories

In this talk, we will study the dimensions (in the sense of Rouquier) of derived categories and sigularity categories of commutative noetherian rings, and also consider some analogues for module categories. This talk is based on joint work with Hailong Dao.

• Mark Walker (University of Nebraska-Lincoln, USA): Matrix factorizations in higher codimension

There is a well-known isomorphism joining the stable category (also known as the singularity category) of modules over a hypersurface and the homotopy category of matrix factorizations. For a complete intersection of codimension $c > 1$, a theorem of Orlov allows one to relate the stable category to the homotopy category of a generalization of the notion of matrix factorizations to “twisted” matrix factorizations over schemes. I will talk about joint work with Jesse Burke in which we apply this theory to the study of modules over complete intersections and give applications to the study of stable support sets. I will also speculate on future applications involving the Hochschild homology, Chern characters, and “Riemann-Roch like” formulas in this context.

Short talks

• Abhishek Banerjee (Ohio State University, U.S.A.): Noetherian schemes over symmetric monoidal categories

Let $A$ be a commutative monoid object in a symmetric monoidal category $(\mathbf C,\otimes,1)$ satisfying certain conditions and let $\mathcal E(A)=Hom_{A-Mod}(A,A)$. If the subobjects of $A$ satisfy a certain compactness property, we say that $A$ is Noetherian. We study the localisation of $A$ with respect to any $s\in \mathcal E(A)$ and define the quotient $A/\mathscr I$ of $A$ with respect to any ideal $\mathscr I\subseteq \mathcal E(A)$. We use this to develop appropriate analogues of the basic notions from usual algebraic geometry (such as Noetherian schemes, irreducible, integral and reduced schemes, function field, the local ring at the generic point of a closed subscheme, etc) for schemes over $(\mathbf C,\otimes,1)$ . Our notion of a scheme over a symmetric monoidal category $(\mathbf C,\otimes,1)$ is that of Toën and Vaquié.

• Hanno Becker (University of Bonn, Germany): Models for singularity categories

In this talk I will outline the construction of model category enhancements for singularity categories of dg-rings. I will start by recalling the connection between abelian model structures and cotorsion pairs and then show how the desired models can be constructed. Time permitting I will show how Krause’s recollement for the stable derived category can be constructed from the model categorical perspective and/or describe a concrete Quillen equivalence between matrix factorizations and a certain singular model structure on the category “curved” mixed complexes.

• Jesse Burke (University of Bielefeld, Germany): Constructing free resolutions from non-affine matrix factorizations

Using the description of the singularity category of a complete intersection as a homotopy category of non-affine matrix factorizations, we show how every non-affine matrix factorization gives rise to a (possibly non-minimal) free resolution of a syzygy of the corresponding module. In particular this construction recovers the standard resolutions introduced by Shamash and Eisenbud. This is joint work with Mark Walker.

• Ana Ros Camacho (Hamburg): A physicists’ view of Landau-Ginzburg models
• Lucas David-Roesler (University of Connecticut, U.S.A.): Constructing algebras from partially triangulated surfaces

I will introduce a method for constructing algebras from certain partially triangulated Riemann surfaces. When the surface is the disc with marked points on the boundary this will allow me to construct a model for the module category of iterated tilted algebras of Dynkin type A. For a general surface with boundary these algebras arise as the endomorphism algebra of a partial cluster-tilting object.

• Kosmas Diveris (Syracuse University, U.S.A): The Generalized Auslander-Reiten Condition for the Bounded Derived Category

We consider the bounded derived category of a Noetherian ring. We give a version of the Generalized Auslander-Reiten Condition for the derived category that is equivalent to the classical statement for the module category and is preserved under derived equivalences. This extends a recent result of J. Wei concerning the stability of the Generalized Auslander-Reiten Condition under tilting equivalences between Artin algebras. This is joint work with M. Purin.

• Cesar Galindo Martinez (Bogota, Colombia): Noncommutative Galois extensions and representations of finite groups

This talk is about the relation between noncommutative Galois extensions of field by a finite group G and the representation theory of G over k. I will present the clasification of Galois objects for a finite group, their forms and some aplications to isocategorical groups and real forms of semisimple Hopf algebra.

• Brian Pike (University of Toronto, Canada): The two meanings of matrix factorization

At this conference, the term ‘matrix factorization’ refers to a property of certain maximal Cohen-Macaulay modules. However, various ‘matrix factorizations’ are widely used in numerical linear algebra. I will discuss a connection between the two areas which occurs in the study of linear free divisors.

• Carlos Segovia Gonzalez (Bogota, Colombia): An approximation to the number of subgroups of a finite group

An unsolved problem in geometric group theory is to give an explicit formula for the number of subgroups of a (finite) non-abelian group. Even for abelian groups, this is a complicated task. There are families of groups which admit a nice description of this number. For example the number of divisors of the integer $n$, denoted by $\tau(n)$, is the number of subgroups of the cyclic group $\mathbb{Z}_n$. For the Dihedral group $D_{2n}$, Stephan A. Cavior proved that the number of subgroups is given by $\tau(n)+\sigma(n)$, where $\sigma(n)$ is the sum of the divisors. Similarly, for the dicyclic groups $Dic_n$, the number of subgroups of $Dic_n$ coincides with $\tau(2n)+\sigma(n)$.

In this talk we will see an approach to this problem with the study of the monoid of all possible deformations of the trivial principal $G$-bundle, over the circle, inside a discretization of the space of all principal $G$-bundles over surfaces. This monoid is isomorphic to a semi-direct product $M_G\rtimes G$ where the monoid $M_G$ is defined as a quotient monoid with a equivalence relation generated by some constraints given by the definition of a $G$-Frobenius algebra. The monoid $M_G$ is a finitely generated abelian monoid without torsion and consequently $M_G$ is the direct sum of a finite number of copies of the natural numbers $\mathbb{N}$. Let $r^G$ be the number of generators of $M_G$. We will see that this number is an approximation to the number of subgroups of the group $G$.

• Chelsea Walton (University of Washington, U.S.A.): Quantum binary polyhedral groups and their actions on quantum planes

We discuss a classification of quantum analogues of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$, and study their actions on quantum polynomial rings (i.e. Artin-Schelter regular algebras) of global dimension two. This is a preliminary report; joint work with Kenneth Chan, Ellen Kirkman, and James Zhang.

Notes from the focus period on Matrix Factorizations, held at the Mathematical Sciences Research Institute, in the Spring semester of 2013.

1. Ragnar-Olaf Buchweitz, Homological algebra on a complete intersection, with an application to group representations: Notes 1 and Notes 2
1. Juergen Herzog, Linear MCM modules over strict complete intersections
2. Bernd Ulrich, Criteria for Gorensteiness
2. Daniel Murfet, Matrix factorizations and TFTs
3. Graham Leuschke, Knörrer periodicity and finite MF representation type
4. David Eisenbud, Matrix factorizations of higher codimension
5. Matrix Factorization day lectures:
1. Matthew Ballard, Orlov spectra: bounds and gaps
2. Nils Carqueville, Orbifold completion of defect bicategories
3. Hanno Becker, Matrix Factorizations in Knot Theory
4. Ana Ros Camacho, Landau-Ginzburg/CFT correspondence via Temperley-Lieb categories
6. Luchezar Avramov, Constructing modules with prescribed cohomology

References for Iyama’s talks (back):

• [AIR] C. Amiot, O. Iyama, I. Reiten, Stable categories of Cohen-Macaulay modules and cluster categories, arXiv:1104.3658.
• [A1] M. Auslander, Functors and morphisms determined by objects. Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), pp. 1–244. Lecture Notes in Pure
Appl. Math., Vol. 37, Dekker, New York, 1978.
• [A2] M. Auslander, Isolated singularities and existence of almost split sequences. Representation theory, II (Ottawa, Ont., 1984), 194–242, Lecture Notes in Math., 1178, Springer, Berlin, 1986.
• [A3] M. Auslander, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293 (2) (1986) 511–531.
• [KST] H. Kajiura, K. Saito, A. Takahashi, Matrix factorization and representations of quivers. II. Type ADE case, Adv. Math. 211 (2007), no. 1, 327–362.
• [K] B. Keller, Cluster algebras, quiver representations and triangulated categories. Triangulated categories, 76–160, London Math. Soc. Lecture Note Ser., 375, Cambridge Univ. Press, Cambridge, 2010.
• [I] O. Iyama, Auslander-Reiten theory revisited. Trends in representation theory of algebras and related topics, 349–397, EMS Ser. Congr. Rep., Eur. Math. Soc., Zurich, 2008.
• [IT] O. Iyama, R. Takahashi, Tilting and cluster tilting for quotient singularities, arXiv:1012.5954.
• [IW1] O. Iyama, M. Wemyss, On the Noncommutative Bondal-Orlov Conjecture, arXiv:1101.3642, to appear in J. Reine Angew. Math.
• [IW2] O. Iyama, M. Wemyss, Auslander-Reiten Duality and Maximal Modifications for Non-isolated Singularities, arXiv:1007.1296.
• [IY] O. Iyama, Y. Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), no. 1, 117–168.
• [R] I. Reiten, Cluster categories. Proceedings of the International Congress of Mathematicians. Volume I, 558–594, Hindustan Book Agency, New Delhi, 2010.
• [V] M. Van den Bergh, Non-commutative crepant resolutions, in: The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749–770.
• [Y] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Math. Soc. Lecture Note Ser., vol. 146, Cambridge Univ. Press, Cambridge, 1990.

References for Leuschke’s talks (back):

• Maurice Auslander, “On the purity of the branch locus”. Amer J Math. 1962; 84: 116–125.
• Maurice Auslander, “Rational singularities andalmost split sequences”. Trans Amer Math Soc. 1986; 293(2): 511–531.
• Alexei Bondal and Dmitry Orlov, Derived categories of coherent sheaves. Proceedings of the International Congress of Mathematicians, Vol. II.; 2002, Beijing; Higher Ed. Press; 2002. p. 47–56.
• Ragnar-Olaf Buchweitz, GJL, and Michel Van den Bergh, “Non-commutative desingularization of determinantal varieties, I: Maximal minors”, Invent. Math. 182 (2010), no. 1, 47–115, DOI:10.1007/s00222-010-0258-7 http://arxiv.org/abs/0911.2659
• Ragnar-Olaf Buchweitz, GJL, and Michel Van den Bergh, “Non-commutative desingularization of determinantal varieties,II: Arbitrary minors” http://arxiv.org/abs/1106.1833
• Jürgen Herzog, “Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen–Macaulay-Moduln”. Math Ann. 1978; 233(1): 21–34.
• Mikhail Kapranov, “On the derived categories of coherent sheaves on some homogeneous spaces”. Invent Math. 1988; 92(3): 479–508.
• GJL, “Non-commutative crepant resolutions: scenes from categorical geometry”, to appear in Progress in Commutative Algebra 1: Combinatorics and Homology, pp 293–361. http://arxiv.org/abs/1103.5380
• GJL and Roger Wiegand, “Cohen-Macaulay Representations,” to appear from Amer. Math. Soc. Draft link.
• John McKay, “Graphs, singularities, and finite groups”. Proceedings of The Santa Cruz Conference on Finite Groups; 1979; University of California, Santa Cruz, CA. vol. 37 of Proceedings of symposia in pure mathematics. Providence, RI: American Mathematical Society; 1980. p. 183–186.
• Michel Van den Bergh, “Non-commutative crepant resolutions (with some corrections)”. In: The legacy of Niels Henrik Abel. Berlin: Springer; 2004. p. 749–770. [the updated (2009) version on arXiv has some minor corrections over the published version].
• Michel Van den Bergh, “Three-dimensional flops and noncommutative rings”. Duke Math J. 2004; 122(3): 423–455.
• Jerzy Weyman and Gufang Zhao, “Noncommutative desingularization of orbit closures for some representations of $GL_n$” http://arxiv.org/abs/1204.0488v1

References for Murfet’s talks (back):

• {cr0909.4381} N. Carqueville and I. Runkel, On the monoidal structure of matrix bi-factorisations, J. Phys. A: Math. Theor. \textbf{43} (2010), 275401, arXiv:0909.4381.
• {cr1006.5609} N. Carqueville and I. Runkel, Rigidity and defect actions in Landau-Ginzburg models, Comm. Math. Phys. \textbf{310} (2012) 135–179, arXiv:1006.5609.
• {d0904.4713} T. Dyckerhoff, Compact generators in categories of matrix factorizations, Duke Math. J. \textbf{159} (2011), 223–274, arXiv:0904.4713.
• {dyckmurf} T. Dyckerhoff and D. Murfet, The Kapustin-Li formula revisited, arXiv:1004.0687.
• {dm1102.2957} T. Dyckerhoff and D. Murfet, Pushing forward matrix factorisations, arXiv:1102.2957.
• {ffrs0607247} J. Fr\”ohlich, J. Fuchs, I. Runkel, and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B \textbf{763} (2007), 354–430, hep-th/0607247.
• {k1004.2307} A. Kapustin, Topological {F}ield {T}heory, {H}igher {C}ategories, and {T}heir {A}pplications, arXiv:1004.2307.
• {Kapustin03} A. Kapustin and Y. Li, D-branes in {L}andau-{G}inzburg models and algebraic geometry, J. High Energy Phys. (2003), no. 12, 005, 44 pp.
• {kr0401268} M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. \textbf{199} (2008), 1–91, math/0401268.
• {Loday} J.-L. Loday, Cyclic homology, Springer, 1997.
• {McNameethesis} D. McNamee, On the mathematical structure of topological defects in {L}andau-{G}inzburg models, MSc Thesis, Trinity College Dublin, 2009.
• {m0912.1629} D. Murfet, Residues and duality for singularity categories of isolated {G}orenstein singularities, arXiv:0912.1629.
• {ps0910.1306} K. Ponto and M. Shulman, Shadows and traces in bicategories, arXiv:0910.1306.
• {pv1002.2116} A. Polishchuk and A. Vaintrob, Chern characters and {H}irzebruch-{R}iemann-{R}och formula for matrix factorizations, arXiv:1002.2116.

Last update: 25 April 2013