# Seminars in Algebra

## Fall 2006

## December 19th 2006, 13:00-14:00 in Rom 734

**Title:** Periodicity for self-injective algebras

**Speaker:** Karin Erdmann

**Abstract:** For some self-injective algebras, all modules have periodic projective resolutions; these include modular group algebras of finite groups with cyclic or quaternion Sylow subgroups on one end of the spectrum, but also, apart from self-injective algebras of finite type, all finite-dimensional preprojective algebras that are of interest for quiver varieties and quantizations of singularities. We report on some recent results in which such algebras arise.

## December 19th 2006, 11:00-12:00 in Rom 734

**Title:** Homology of perfect complexes

**Speaker:** Luchezar Avramov

**Abstract:** It will be proved that the sum of the Loewy length of the homology modules of a finite free complex \(F\) over a local ring \(R\) is bounded below by an invariant that measures the singularity of \(R\). In the special case of the group algebra of an elementary abelian group one recovers results of G.~Carlsson and of C.~Allday and V.~Puppe. The arguments use numerical invariants of objects in general triangulated categories, introduced in the talk and called levels. One such level models projective dimension; a lower bound for this level contains the New Intersection Theorem for commutative local rings containing fields. The lower bound on Loewy length of the homology of \(F\) is sharp in general. Under additional hypothesis on the ring \(R\) stronger estimates are proved for Loewy lengths of modules of finite projective dimension. This is joint work with Ragnar-Olaf Buchweitz, Srikanth Iyengar, and Claudia Miller.

## December 13th 2006, 11:00-12:00 in Rom 734

**Title:** From Coxeter functors to Tilting theory.

**Speaker:** Marco Angel Bertani-Økland

**Abstract:** During this talk we explain the connection between the Dynkin diagrams and the hereditary algebras of the form H = kQ , where Q is a connected quiver and k an algebraically closed field. This connection gives birth to the partial Coxeter functors and we see how these evolve into tilting modules.

## December 6th 2006, 14:45-15:45 in Rom 734

**Title:** A classification of jetbundles on projective spaces and grassmannians

**Speaker:** Helge Maakestad

**Abstract:** For a scheme X smooth over a basefield k of characteristic zero the generalized sheaf of differential operators D_X(L) where L is a linebundle, has a filtration D^a_X(L) by degree. Its dual J^a_X(L) the a'th sheaf of jets of the linebundle L is a sheaf of bimodules on X and I will in this talk discuss the structure of this sheaf as left and right module over the structure sheaf on the projective line, projective space and flag-variety SL(V)/P.

## November 22nd 2006, 14:45-15:45 in Rom 734

**Title:** The quiver of tilting modules II

**Speaker:** Dieter Happel

## November 15th 2006, 14:45-15:45 in Rom 734

**Title:** A vanishing theorem for sheaves of differential operators in positive characteristic and tilting equivalences

**Speaker:** Alexandre Samokhin

**Abstract:** A tilting object on a (complex) algebraic variety furnishes an equivalence between the derived category of coherent sheaves on this variety and the derived category of modules over an associative algebra. We will show that in some cases such a tilting object can be obtained using reduction to positive characteristic and considering the Frobenius morphism. We will also touch on the question of the D-affinity of homogeneous spaces in positive characteristic and discuss related vanishing theorems.

## November 8th 2006, 14:45-15:45 in Rom 734

**Title:** The quiver of tilting modules I

**Speaker:** Dieter Happel

## November 1st 2006, 14:45-15:45 in Rom 734

**Title:** Vanishing of Ext and Tor over complete intersections

**Speaker:** Petter Andreas Bergh

**Abstract:** We consider the vanishing of Ext and Tor for finitely generated modules over a commutative Noetherian local complete intersection, a topic which began with Maurice Auslander's study of torsion properties over reguar local rings in 1961. Using the notion of complexity of a module, and a method for reducing it, we present some vanishing results which generalize the results already known. All the necessary concepts from commutative algebra will be defined. The talk should therefore be accessible to those following the course in commutative algebra (and who are familiar with the basic properties of Ext and Tor).

## October 25th 2006, 14:45-15:45 in Rom 734

**Title:** Approximations and Auslander-Reiten theory in subcategories

**Speaker:** Dag Madsen

**Abstract:** Let A be a finite dimensional algebra and let X \\subset mod A be the subcategory of modules orthogonal to a tilting A-module. Many "naturally" occurring subcategories of mod A can be described in this way. As a relatively easy application of derived categories, we show how to compute right X-approximations. In other words, given M in mod A, we find the "best" possible approximation X_M→M with X_M in X. Knowing how to deal with approximations, we have a practical way of doing Auslander-Reiten theory in X. This is joint work with K.Erdmann and V. Miemietz.

## October 11th 2006, 14:45-15:45 in Rom 734

**Title:** Algebraic Geometric Secret Sharing Schemes

**Speaker:** Sasa Radomirovic

**Abstract:** Secret sharing schemes are an essential tool for secure multiparty computations. Furthermore, they can be used to keep data unconditionally secret while maintaining a certain degree of fault tolerance. I will present a secret sharing scheme introduced by Cramer and Chen at CRYPTO 2006. The scheme is constructed using rational functions of a smooth projective algebraic curve over a finite field. It is a generalization of the very first secret sharing scheme invented by Shamir in 1979 who used polynomials over a finite field. The talk will be mostly self-contained.

## October 4th 2006, 14:45-15:45 in Rom 734

**Title:** Support varieties for triangulated categories

**Speaker:** Henning Krause

## September 27th 2006, 14:45-15:45 in Rom 734

**Title:** Real roots of a hyperbolic quiver

**Speaker:** Bernt Tore Jensen

## September 20th 2006, 14:45-15:45 in Rom 734

**Title:** Symmetries of tensors and Young diagrams II

**Speaker:** Alexei Rudakov

## September 13th 2006, 14:45-15:45 in Rom 734

**Title:** Symmetries of tensors and Young diagrams

**Speaker:** Alexei Rudakov

**Abstract:** The simplest example of a tensor would be a bilinear form on a vector space. We all know that there are symmetric and skew-symmetric (or anti-symmetric) bilinear forms. In the case of bilinear forms these two are the only symmetry types, but in general there are more, and the symmetry types of tensors are described by Young diagrams. In the talk we shall explain what is a Young diagram, then consider the case of tri-linear forms that is the first case when there are not only symmetric and skew-symmetric forms, but one more symmetry type shows itself. The talk is the first of two and meant to be an introduction in the subject and accessible to the students who know Linear Algebra.

## September 6th 2006, 14:45-15:45 in Rom 734

**Title:** Homological properties of determinantal varieties

**Speaker:** Jon Eivind Vatne

**Abstract:** A determinantal variety is a variety whose ideal in projective space is given by the minors of a matrix. As a module over the symmetric algebra, it has a well understood resolution (in the cases we will consider), the Eagon-Northcott complex. The simplest non-trivial example of a determinantal variety is a rational normal curve. The homogeneous coordinate ring of a rational normal curve is a Koszul algebra, and its Hochschild cohomology therefore has a simple expression via the Koszul dual. We will see a nice connection between the numerical data of the Koszul dual and a construction based on the Eagon-Northcott complex. In particular, this determines precise information about a certain change of rings spectral sequence. We conjecture that analogous properties hold for a more general class of determinantal varieties. The definition and basic properties of several classical methods in algebraic geometry and homological algebra (e.g. Eagon-Northcott complex, Koszul duality, change of rings spectral sequence) will given in the talk.

## August 24th 2006, 16:00-17:00 in Rom 734

**Title:** A geometric description of m-cluster categories

**Speaker:** Robert Marsh

**Abstract:** Joint work with Karin Baur. The m-cluster category is a category recently appearing in the theory of cluster-tilting. We give a geometric definition of the m-cluster category of type A_{n-1} in terms of diagonals of a regular polygon. This generalises a result of Caldero, Chapoton and Schiffler for m=1. The approach uses the theory of translation quivers and their corresponding mesh categories. We also introduce the notion of the m-th power of a translation quiver and show how it can be used to realise the m-cluster category in terms of the cluster category.

## Spring 2006

## June 26th 2006, 09:00-10:00 in Rom 1329

**Title:** The McKay correspondence

**Speaker:** John McKay

**Abstract:** I give a personal perspective of the discovery of this simple bijection between G < SU2 and Dynkin data of type A-D-E. Many open questions arise from it. I shall illustrate the correspondence with the D4 singularity of Klein. We also discover some facts about the three sporadic finite simple groups, M, B, and F24'. Reference: McKay: Graphs, singularities and finite groups, AMS Proc Symp Pure Math vol 37 (1980) pp 183-186.

## June 23rd 2006, 13:30-14:30 in Rom 734

**Title:** Projectives in abelian hereditary categories with duality

**Speaker:** Adam-Christiaan Ruben van Roosmalen

## June 14th 2006, 14:30-15:15 in Rom 734

**Title:** Identity-based encryption schemes

**Speaker:** Sasa Radomirovic

**Abstract:** Identity-based encryption schemes are a hot topic in contemporary cryptography. They have been invented in 1984, but only in 2001 was a practical and secure scheme found. The scheme is based on the Weil pairing on Supersingular Elliptic Curves. Assuming the audience's familiarity with abstract algebra, this talk will explain the scheme and show how elliptic curves are being used to realise it.

## June 14th 2006, 13:15-14:15 in Rom 734

**Title:** A duality for generalized Koszul algebras

**Speaker:** Manolo Saorín

## May 10th 2006, 14:15-15:00 in KJL 21

**Title:** The existence of short exact sequences with some of the terms in prescribed subcategories

**Speaker:** Øyvind Bakke

## May 3rd 2006, 16:10-17:10 in Rom 1329

**Title:** Realizability and localization for cohomology modules

**Speaker:** Birgit Huber

**Abstract:** Let G be a finite group and k a field. Motivated by the question whether a module over the group cohomology ring H*(G,k) can be written as H*(G,X) for some kG-module X, we study differential graded algebras and modules over their cohomology rings: Let A be a dg algebra such that its cohomology ring H*(A) is graded commutative. A module over H*(A) is called realizable if it is (up to direct summands) of the form H*(M) for some dg A-module M. Benson, Krause and Schwede have given a local and a global obstruction for the realizability. We show that a module X is realizable if and only if X localized at p is realizable for all primes p of the spectrum of H*(A). The global obstruction is given by the secondary multiplication of the A-infinity algebra H*(A). We define a localization A_p of the dg algebra A at a prime p of H*(A) and show that the secondary multiplications of H*(A) and H*(A_p) are related in a nice way. Further we discuss whether the global obstruction being trivial is a local property.

## April 26th 2006, 16:10-17:10 in Rom 1329

**Title:** Generalized minors and the even orthogonal group

**Speaker:** Jeanne Scott

**Abstract:** In this talk I will explain the notion of 'generalized minors' — as introduced in 1998 by S.Fomin and A.Zelevinsky — for complex semi- simple algebraic groups and I will work out examples in the case of the even orthogonal group. If time permits I will discuss connections with the isotropic Grassmannians and their possible cluster algebra structure.

## April 19th 2006, 16:10-17:40 in Rom 546

**Title:** Cluster variables and exceptional objects

**Speaker:** Bernhard Keller

## April 5th 2006, 16:10-17:10 in Rom 546

**Title:** Structure theorems for basic finite dimensional algebras

**Abstract:** Abstract: P. Gabriel showed that every basic finite dimensional algebra over an algebraically closed field was isomorphic to a quotient of the path algebra. When the algebra in addition was hereditary, it was isomorphic to a path algebra. I want to give similar results for algebras over arbitrary fields. In this setting we will need the notion of species with relations.

## March 29th 2006, 16:10-17:10 in Rom 546

**Title:** A monoidal category of linear ODEs

**Speaker:** Cathrine Jensen

**Abstract:** Viewing linear homogenous ODEs as differential modules shows that these equations form a monoidal category. The category contains tensor products, direct sums, homomorphisms, symmetric - and alternating products etc. This may be used to study various structures on equations and solution spaces. We get an analogue of the representation ring of a semisimple Lie algebra, in terms of a structure theorem for equations.

## March 22nd 2006, 16:10-17:10 in Rom 546

**Title:** SO(2n)-Grassmannians and the type D_n pre-projective algebra

**Speaker:** Jeanne Scott

**Abstract:** This talk will begin by reviewing the cluster algebra structure of Grassmannian \(Gr(k,n)\), especially in connetion with the type \(A_{n-1}\) pre-projective algebra. I will then discuss the \(SO(2n)\)-Grassmannians, the coordinates, and indicate what their cluster algebra structures may be.

## March 15th 2006, 16:10-17:10 in Rom 546

**Title:** Mutation in d-Calabi-Yau triangulated categories

**Speaker:** Osamu Iyama

**Abstract:** We study d-cluster subcategories \(C\) of a d-Calabi-Yau triangulated category \(T\). Under certain "no-loop" condition on \(C\), we can construct other \((d-1)\) d-cluster subcategories of \(T\). Moreover, any "almost d-cluster" subcategory of \(T\) is contained in precisely \(d\) d-cluster subcategories. This is shown by certain reduction of triangulated categories. We will show some applications.

## March 8th 2006, 16:10-17:10 in Rom 546

**Title:** Tilting modules over Calabi-Yau algebras II

**Speaker:** Osamu Iyama

## March 1st 2006, 16:10-17:10 in Rom 546

**Title:** Tilting modules over Calabi-Yau algebras

**Speaker:** Osamu Iyama

## February 22nd 2006, 16:10-17:10 in Rom 546

**Title:** Higher order relations of the Plücker embedding of Grassmanian (2,n)

**Speaker:** Anton Khoroshkin

**Abstract:** We apply the theory of Lie algebra cohomologies for the calculations of the spaces of syzygies of arbitrary projective quadratic embeddings ( i.e. the space of equations, the relations between equations, the relations between relations and so on). This method proves the existence of Massey operations on the syzygies. As an illustration we calculate syzygies of the plucker embedding of the grassmanian of 2-dimensional planes.

## February 15th 2006, 16:10 - 17:10 in Rom 546

**Title:** Infinitely generated tilting modules and finite type

**Speaker:** Jan Stovicek

**Abstract:** This talk will focus on infinitely generated tilting modules, and in particular on the fact that every such module is of finite type. That is, the Ext-orthogonal class of a tilting module is actually an Ext-orthogonal class of a set of finitely presented modules. A sketch of the proof will be given.

## February 8th 2006, 14:30-15:30 in Rom 546

**Title:** From a single filtration to a large family of submodules

**Speaker:** Jan Stovicek

**Abstract:** The aim is to present a rather general method how to obtain from a single filtration of a module a large family of filtered submodules. This is useful for instance when dealing with cotorsion pairs. Some applications are: a generalization of Kaplansky's structure theorem for projective modules, a characterization of strongly flat modules over valuation domains, or a short proof of the structure of Matlis localizations.

## February 1st 2006, 15:30-16:30 in Rom 546

**Title:** Stability data on triangulated categories

**Speaker:** A. L. Gorodentsev

**Abstract:** We will discuss some generalisation of the T. Bridgeland approach to the stability data for the triangulated categories. We treat stability data as a category of continous form rigt side finite-valued step funcors form a fixed category of "slopes" (say, an ordered set, considered as a category) to a given triangulated category \(T\) and show that this category admits canonical full faithful embedding in \(T\). This leads to some DG-enhancement for \(T\), which can be used for precise "\(A_\\infinity\)-computations" (higher Massey products, say) in \(T\). Depending on the time, we will try to discuss refinments of the stability data, how to construct the finest t-stabilities on (derived categories of) abelian categories satisfying some quite common finiteness conditions, and designing interplay between all this stuff with diophantine approximations.

## February 1st 2006, 14:15-15:15 in Rom 546

**Title:** Simplicial complexes, Stanley-Reisner rings and the Nakayama functor

**Speaker:** Gunnar Fløystad

## January 18th 2006, 14:30-15:30 in Rom 1329

**Title:** On syzygies of projective coordinate algebra of highest weight orbits

**Speaker:** A. L. Gorodentsev