# Seminars in Algebra

## Fall 2011

## December 16th 2011, 13:00 - 14:00 in 656

**Title:** When do dimensions of syzygies stabilize?

**Speaker:** Kristen Beck

**Abstract:** In 1991, Okiyama proved that the depths of the syzygies of a finitely generated module of infinite projective dimension over a commutative local Noetherian ring eventually stabilize to the depth of the ring. In this talk, we give necessary conditions under which a similar statement can be made concerning dimension. In particular, we will show that all high syzygies of a module with eventually non-decreasing Betti numbers over an equidimensional ring have dimension equal to the dimension of the ring.

## December 8th 2011, 11:00 - 12:00 in 656

**Title:** Nearly Morita equivalences and mutations of rigid objects

**Speaker:** Yann Palu

**Abstract:** Cluster algebras can be categorified by means of certain triangulated categories called cluster categories. In that context, clusters correspond to some specific objects: the maximal rigid objects (also called cluster tilting objects). Reflecting the mutation of clusters, a mutation theory for maximal rigid objects has been studied extensively. It has in partiular a nice application to representation theory, due to Buan–Marsh–Reiten: If two maximal rigid objects are related by a mutation, then their endomorphism algebras are nearly-Morita equivalent.

Based on the example of the \(A_n\) case, I will recall this result with more details. Then I will talk about some results obtained jointly with Robert Marsh as an attempt to generalise this phenomenon for (non-necessarily maximal) rigid objects.

## December 1st 2011, 11:00 - 12:30 in KJL24

**Title:** Øyvind's birthday - mini symposium

**Abstract:** An algebra seminar dedicated to Øyvind Solberg on his 50th birthday.

Speakers:

Aslak Bakke Buan - Øyvind and big modules.

Dag Madsen - Øyvind and Koszul.

Petter Andreas Bergh - Øyvind and Hochschild.

## November 24th 2011, 13:00 - 14:00 in 656

**Title:** Using QPA

**Speaker:** Øyvind Solberg

**Abstract:** We will demonstrate how to use the software package QPA, (Quiver and Path Algebras), in particular how to, enter algebras, modules and homomorphisms; do computations with them: tensor product of algebras, center of algebras, action of the algebra on modules, dimension vectors, basis, indecomposable projectives/injectives, simples, radical/socle series, common direct summands, kernel, image, cokernel of homomorphisms, minimal versions of homomorphisms, homomorphism sets, dual, add M-approximations, projective covers, DTr, Ext^1, etc.

## November 17th 2011, 13:00 - 14:00 in 656

**Title:** Wreath products, blocks of Hecke algebras and representation dimension

**Speaker:** Petter Andreas Bergh

**Abstract:** This is ongoing joint work with Will Turner (Aberdeen). The representation theory of blocks of Hecke algebras (of type A) is intimately connected with the theory of partitions of numbers. By results of Chuang-Rouquier and Chuang Miyachi, two blocks are derived equivalent if and only if they have the same weight, and Rouquier blocks are derived equivalent to certain wreath products. We shall take a look at all these concepts and results, and explain some implications for the representation dimension of the algebras involved.

## November 10th 2011, 13:00 - 14:00 in 656

**Title:** Quotient closed subcategories of quiver representations

**Speaker:** Idun Reiten

**Abstract:** This talk is based on joint work with Steffen Oppermann and Hugh Thomas. We recall the defininition of a Coxeter group associated with a finite acyclic quiver Q, and discuss a natural bijection between the elements of the Coxeter group and the cofinite quotient closed subcategories of mod kQ.

## November 3rd 2011, 13:00 - 14:00 in 656

**Title:** Products in negative Hochschild cohomology

**Speaker:** David Jorgensen

**Abstract:** Recently P. A. Bergh and the speaker introduced Tate-Hochschild cohomology over two-sided Noetherian Gorenstein k-algebras. In this talk we'll recall the construction of Tate-Hochschild cohomology and study the products in negative degree. A key tool will be a certainly duality of Tate-Hochschild cohomology, which we may entitle in the talk. This is joint work with P. A. Bergh and S. Oppermann.

## October 27th 2011, 13:00 - 14:00 in 656

**Title:** Quivers and admissible relations of tensor products and trivial extensions

**Speaker:** Øystein Ingmar Skartsæterhagen

**Abstract:** This talk is based on my master's thesis (2011). I will describe how one can algorithmically find quivers and sets of relations for tensor products of two algebras, triangular matrix algebras, and trivial extensions of algebras, when the original algebras are given by quivers and relations. The methods I present for triangular matrix algebras and trivial extensions produce quivers that may contain more arrows than necessary, so that there are relations containing paths of length one. I will describe a method to convert such quivers and relation sets into ones without this undesired property.

## October 20th 2011, 13:00 - 14:00 in 656

**Title:** On the algebra structure of Tor for trivariate monomial ideals

**Speaker:** Jared Painter

**Abstract:** Recently, L. Avramov classiﬁed the behavior of Bass numbers of embedding codepth 3 commutative local rings. His classification relied on a corresponding classification of their respecive Tor algebras, which is comprised of 5 categories. In this talk we explore the algebra structure of \(A = Tor^R(k,R/I)\), where \(R = k[x,y,z]\) is a trivariate polynomial ring over a ﬁeld k and I is a monomial ideal primary to the homogeneous maximal ideal m of R. We will determine which of the 5 categories of Tor algebras can be realized by monomial ideals. In addition we will identify classes of ideals I so that A will have the desired algebra structure.

## October 13th 2011, 13:00 - 14:00 in 656

**Title:** Non-commutative Gröbner bases and how to find them

**Speaker:** Kristin Krogh Arnesen

**Abstract:** I will give a short introduction to non-commutative Gröbner bases and the non-commutative version of Buchberger's algorithm, together with some time-saving modifications. This talk is based on parts of my master's thesis (2010). Gröbner bases can be seen as a tool for solving the ideal membership problem in a free k-algebra R = k<x1,x2,…,xn> where the variables x1,…,xn do not commute: Given a polynomial f in R and a two-sided ideal I in R, decide whether f is an element of I.

## October 6th 2011, 13:00 - 14:00 in 656

**Title:** A Novel Framework for Protocol Analysis

**Speaker:** George Petrides

**Abstract:** Many modern cryptographic protocols are described and analysed using the universal composability (UC) framework, whose purpose is to allow the preservation of their security when several of them are composed into larger ones in arbitrary environments.

I will begin this talk by briefly presenting the UC framework and pointing out a couple of weak-points that motivated our work, which is a reformulation of UC. I will then show how our framework remedies these weaknesses and also presents the main results, such as the composability theorem, in a clearer and easier to understand way.

With Kristian Gjøsteen and Asgeir Steine

## October 5th 2011, 13:15 - 14:15 in 656

**Title:** On the Category of Totally Reflexive Modules

**Speaker:** Denise Rangel

**Abstract:** Totally reflexive modules, also know as modules of Gorenstein dimension zero, were introduced by Auslander and Bridger in 1969 as a generalization of Maximal Cohen-Macaulay modules over a Gorenstein ring. They are defined in terms of certain vanishing Ext, and have many nice properties, such as admitting a complete cohomology theory, known as Tate cohomology. However, their existence over non-Gorenstein rings is rather mysterious. In this talk we give some background on totally reflexive modules, and then discuss some preliminaries of their representations over non-Gorenstein rings.

## September 29th 2011, 13:15 - 14:15 in 656

**Title:** Cohomologically Complete Complexes

**Speaker:** Amnon Yekutieli

**Abstract:** Let A be a noetherian commutative ring, and a an ideal in it. In this lecture I will talk about several properties of the derived a-adic completion functor and the derived a-torsion functor.

In the first half of the talk I will discuss a-adically projective modules, GM Duality (first proved by Alonso, Jeremias and Lipman), and the closely related MGM Equivalence. The latter is an equivalence between the category of cohomologically a-adically complete complexes and the category of cohomologically a-torsion complexes. These are triangulated subcategories of the derived category D(Mod A).

In the second half of the talk I will discuss new results:

(1) A characterization of the category of cohomologically a-adically complete complexes as the right perpendicular to the derived localization of A at a. This shows that our definition of cohomologically a-adically complete complexes coincides with the original definition of Kashiwara and Schapira.

(2) The Cohomologically Complete Nakayama Theorem.

(3) A characterization of cohomologically cofinite complexes.

(4) A theorem on completion by derived double centralizer.

This is joint work with Marco Porta and Liran Shaul.

## September 22nd 2011, 13:15 - 14:15 in 656

**Title:** n-angulated categories

**Speaker:** Steffen Oppermann

## August 24th 2011, 13:15 - 14:15 in 656

**Title:** The Integral Cluster Category

**Speaker:** Sarah Scherotzke

**Abstract:** In my talk, we will consider the question when orbit categories of triangulated categories are again triangulated. I will present some examples where this fails and give a sufficient condition proven by Bernhard Keller for the orbit category of a triangulated category to have a natural triangulated structure. Applying this result to the Cluster category associated to a finite acyclic quiver over a field shows that it is triangulated. In joint work with Bernhard Keller, we proved that the Cluster category defined over certain commutative rings are triangulated, we classify the Cluster-tilting objects and show that they are linked by mutation. The proof in the integral case does not use Keller's criteria and requires a different approach of which I will give sketch.

## Spring 2011

## June 28th 2011, 11:15 - 12:15 in 734

**Title:** Exceptional sequences and c-vectors

**Speaker:** Hugh Thomas

## June 28th 2011, 10:00 - 11:00 in 734

**Title:** Cohomology theories in the pure derived category of flats

**Speaker:** Javad Asadollahi

**Abstract:** Let \(A\) be an associative ring with identity. The quotient category \(D(Flat(A)):=K(Flat A)/K_p(Flat A)\), called the pure derived category of flats, was introduced and studied by Neeman. In this talk, we develop theories of Tate and complete cohomology in this category. These theories extend naturally to sheaves over semi-separated noetherian schemes, where there are not always enough projectives, but we do have enough flats. As application we characterize schemes which are locally Gorenstein. The talk is based on a joint work with D. Murfet and Sh. Salarian.

## June 15th 2011, 13:15 - 14:15 in 656

**Title:** Generalized Matrix Artin Algebras

**Speaker:** Edward L. Green

**Abstract:** I will report on some joint work with Chrysostomos Psauroudakis. When an artin algebra is in matrix form we call it a generalized matrix artin algebra. In my talk, I investigate the representation theory of such rings. In particular, under some restrictions, we find some covariantly, some contravariantly, and some functorially finite subcategories. We also give some global dimension bounds.

## June 8th 2011, 11:15 in F4, Gamle Fysikk

**Title:** Anonymous Credential Schemes with Encrypted Attributes

**Abstract:** In anonymous credential schemes, users obtain credentials on certain attributes from an issuer, and later show these credentials to a relying party anonymously and without fully disclosing the attributes. In this talk, we introduce the notion of (anonymous) credential schemes with encrypted attributes, in which issuers certify credentials on encrypted attributes to users, who later show these credentials to relying parties, such that possibly none of the involved parties—including the user—learns the values of the attributes.

## May 25th 2011, 13:15 - 14:15 in 656

**Title:** A finite set of equations that determine projective resolutions

**Speaker:** Magdalini Lada

**Abstract:** Let \(L=kQ/I\), where \(kQ\) is the path algebra of a finite quiver \(Q\), over a field \(k\), and \(I\) an admissible ideal. Using the right Gröbner basis of \(I\), we provide a finite set of formulas that are enough for computing a projective resolution of any \(L\)-module. Using these formulas we give a proof of the finitistic dimension conjecture for finite dimensional monomial algebras.

## May 12th 2011, 10:15 - 11:15 in 734

**Title:** Cluster-additive functions on stable translation quivers

**Speaker:** Claus M. Ringel

**Abstract:** TBA

## May 4th 2011, 13:15 - 14:15 in 656

**Title:** Derived equivalence and graded mutation

**Speaker:** Claire Amiot

**Abstract:** Understand when two algebras are derived equivalent is a fundamental question in representation theory. In the 80's Happel gave a complete answer to this problem for finite dimensional hereditary algebras. In a joint work with Steffen Oppermann, we generalised this result for algebras of global dimension 2 using cluster-tilting theory and the mutation of quiver with potential of Derksen Weyman and Zelevinsky.

## April 6th 2011, 13:15 - 14:15 in 656

**Title:** Mittag-Leffler conditions on modules

**Speaker:** Dolors Herbera

**Abstract:** A right module M is said to be Mittag-Leffler if the canonical map \(M \otimes \prod _{i \in I}Q_i \to \prod _{i \in I}M \otimes Q_i\) for any family of left modules \(Q_i\). Mittag-Lefler modules were introduced and deeply studied by Raynaud and Gruson in the 70's. In this talk I want to explain how this type of modules appears in relation to the vanishing of derived functors. I will also explain some new characterization of this class of modules that shows a connection to the ideas developed by Shelah to "solve" the Whitehead problem (i.e. determining whether an abelian group such that \( Ext_Z(A,Z)=0 \) must be free).

## March 30th 2011, 13:15 - 14:15 in 656

**Title:** Filtrations determined by tilting objects of projective dimension two

**Speaker:** Dag Madsen

**Abstract:** This is joint work with Bernt Tore Jensen and Xiuping Su. It is well known that a tilting object of projective dimension one in an abelian category determines a torsion pair \((\mathcal T, \mathcal F)\). In this talk I will discuss the corresponding statement for tilting objects of projective dimension two. A titling object of dimension two determines a triple \((\mathcal E_0, \mathcal E_1, \mathcal E_2)\) of disjoint extension closed subcategories such that every object \(X\) has a unique filtration \(0 = X_0 subseteq X_1 subseteq X_2 subseteq X_3 = X\) with \(X_{i+1}/X_i in \mathcal E_i\) for \(i=0, 1, 2\). Dag Madsen, Høgskolen i Østfold

## March 24th 2011, 14:15 - 15:15 in 656

**Title:** Stable categories of Cohen-Macaulay modules and cluster categories

**Speaker:** Osamu Iyama

**Abstract:** It is well-known that the stable categories of Cohen-Macaulay modules over Kleinian singularities are equivalent to the orbit categories of the derived categories of the path algebras by \\tau. We give a family of similar triangle equivalences between the stable categories of Cohen-Macaulay modules over certain Gorenstein rings and generalized cluster categories of certain finite dimensional algebras. A special case was given by Keller-Reiten. We also give a graded version as a byproduct, which generalizes a result by Kajiura-Saito-Takahashi and Lenzing-de la Pena.

## March 23rd 2011, 13:15-14:15 in 656

**Title:** Deformations of crossed products and graded Hecke algebras

**Speaker:** Sarah Witherspoon

**Abstract:** A crossed product of an algebra with a group of automorphisms encodes the group action in a larger algebra. In case the group acts on a polynomial ring, deformations of the crossed product include graded Hecke algebras, symplectic reflection algebras, and rational Cherednik algebras. In this talk, we will introduce these algebras, put them in the wider context of deformations of algebras, and discuss the graded Lie bracket on Hochschild cohomology which encodes obstructions to deforming algebras.

## March 9th 2011, 13:15 - 14:15 in 656

**Title:** Factors and localizations of triangulated categories

**Speaker:** Aslak Buan

**Abstract:** Let A be the endomorphism ring of a rigid object T in a triangulated category C (with some finiteness conditions). It is known that if T is a cluster-tilting object, meaning that Ker Hom(T,-) = add T[1], then mod A can be obtained from C by factoring out Ker Hom(T, -). In our more general set-up, this does not hold. However, we obtain a generalization: mod A can be obtained as a certain Gabriel-Zisman localisation of C.

## March 2nd 2011, 13:15 - 14:15 in 656

**Title:** Quivers and intersection of parabolic Lie algebras in type A

**Speaker:** Bernt Tore Jensen

## February 16th 2011, 13:15 - 14:15 in 656

**Title:** Infinitely generated projective modules over semilocal rings

**Speaker:** Dolors Herbera

**Abstract:** A ring R is said to be semilocal if it is semisimple modulo the Jacobson radical. In this talk I would like to explain what we know about the behavior of projective modules over such rings, emphasizing on the methods to construct examples with infinitely generated projective modules that are not direct sum of finitely generated ones. Since each projective module is a direct sum of countably generated projective modules we restrict our study to the countably generated case. Let V^*(R) to denote the set of isomorphism classes of countably generated right projective modules. With the sum induced by the direct sum of modules, V^*(R) becomes an additive monoid. We will see which monoids can be realized as V^*(R) for R semilocal and noetherian. We still do not know a characterization for the general case but we will give examples to illustrate the additional difficulties that appear: a projective module that is finitely generated modulo the Jacobson radical need not be finitely generated, the monoids V^* for right and left projective modules need not be isomorphic. The talk is based on past and ongoing joint work with Pavel Prihoda from Charles University, Prague.

## January 26th 2011, 13:15-14:15 in 656

**Title:** Construction of 2-representation-finite algebras

**Speaker:** Martin Herschend

**Abstract:** This talk concerns joint work with Osamu Iyama (see arXiv: 0908.3510,1006.1917). Let n be a positive integer. A finite dimensional algebra A is called n-representation finite if it has global dimension at most n and and there exists an n-cluster tilting A-module M. This concept is a natural analogue of representation finiteness from the view point of higher Auslander-Reiten theory. In particular 1-representation-finite algebras are precisely hereditary and representation-finite, and thus by Gabriel's Theorem, they are given by Dynkin quivers. In my talk I will treat the next natural case, i.e., 2-representation-finite algebras. I will focus on constructing 2-representation-finite algebras from pairs of Dynkin diagrams using various methods including tensor products, tilting and mutation.

## January 19th 2011, 13:15-14:15 in F4

**Title:** Trees of relations, clusters and RNA secondary structures

**Speaker:** Robert Marsh

**Abstract:** Joint work with Sibylle Schroll (Leicester). We exhibit a sequence of combinatorial sets, consisting of certain labelled trees, whose cardinality is given by the m-fold convolution of the sequence of Fuss-Catalan numbers of degree m-1. The proof involves showing that a certain subset of the kth set in the sequence is in bijection with the set of m-angulations of an (m-2)k+2-sided polygon, and thus to the set of (m-2)-clusters of type A_{k-1}. We characterize the sets in terms of an operation on the trees, known as induction, which generalises an induction defined by Cassaigne, Ferenczi and Zamboni in the context of interval exchange transformations.

## January 14th 2011, 13:15 - 14:15 in 734

**Title:** A geometric model of tube categories

**Speaker:** Robert Marsh

**Abstract:** Joint work with Karin Baur (ETH, Zurich, Switzerland). We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative crossing numbers between corresponding arcs, giving a geometric interpretation of the description of an extension group in the cluster category of a tube as a symmetrized version of the extension group in the tube.