Talks in 2007


  • 17.12.2007, 17.15, Olaf Schnürer (Freiburg), im Rahmen des IRTG Seminars
    Perverse sheaves on flag varieties, DG modules, and formality
  • 06.12.2007, 16.45,  Jan Stovicek (Trondheim)
    The telescope conjecture in representation theory.
    Abstract: The telescope conjecture roughly says that every smashing localization of a compactly generated triangulated category is a finite localization. It is false in general, but it seems to hold in many interesting cases. After recalling some concepts, I will discuss a positive solution to the conjecture for stable module categories of standard domestic selfinjective algebras and for derived categories of right noetherian right hereditary rings.

  • 06.12.2007, 15.15,  Adam-Christiaan van Roosmalen (Hasselt)
    Classification of hereditary categories with Serre duality
    Abstract: Of of the goals of non-commutative geometry is to understand abelian categories with properties resembling those of the category of coherent sheaves over a non-singular proper scheme, for example being Ext-finite, having finite global dimension, and satisfying Serre duality.  We will present some recent results about the classification of such categories with global dimension 1, called hereditary categories.

  • 03.12.2007, 17.15, Geert van de Weyer (Antwerpen), im Rahmen des IRTG Seminars
    Noncommutative Poisson geometry and path algebras of quivers
  • 18.07.2007, 16.15, Lutz Hille (Bielefeld)
    Tilting modules, generic k[T]-module homomorphisms and mutations (joint with D. Vossieck)
    Abstract: We are mainly interested in a classification of generic homomorphisms between two finite dimensional $k[T]$--modules $M$ and $N$. We can, without loss of generality,  assume $T^t M = T^t N = 0$. Following previous work of Vossieck the indecomposable rigid homomorphisms $f: M ---> N$ are known, they correspond to certain pairs of subsets $T,S$ in $\{1,...,t\}$. So it remains to classify maximal rigid homomorphisms. We compare the classification of maximal rigid homomorphisms as above   with the classification of tilting modules over two algebras $A_t$ and $B_t$. The first one is the endomorphism algebra of $\oplus_i k[T]/T^i$ as a module over the polynomial ring. The classification was done in a joint work with Br\"ustle, Ringel and R\"ohrle. The algebra $B_t$ is a representation finite string algebra of infinite global dimension.  Finally we show, that the classification of maximal rigid homomorphisms is equivalent to the classification of tilting modules over $B_{t+2}$ and also to the classification of tilting modules (containing one particular indecomposable projective module) over $A_{t+2}$. This classification is compatible with mutations, but does not have good properties with respect to the corresponding Grothendieck groups. Moreover, we use the fan associated to tilting modules, respectively to maximal rigid homomorphisms, to prove several properties.

  • 12.07.2007, 14.45, Andrew Hubery (Paderborn)
    Hall algebras of triangulated categories
    Abstract: Toen showed how to construct from each DG category (under some mild homological conditions) an associative algebra, called the derived Hall algebra and emulating the Ringel-Hall algebra constructed from an exact category. Recently Xiao and Xu have simplified the proof considerably and shown that Toen's derived Hall algebra can be constructed starting from a triangulated category (under the same homological conditions). In the talk I will present Xiao and Xu's proof.

  • 12.07.2007, 13.15, Jan Saroch (Prag)
    Deconstruction of Cotorsion Pairs and Telescope Conjecture
    Abstract: Recently developed methods of deconstruction of cotorsion pairs are used to prove a weaker (countable) version of so-called Telescope Conjecture for Module Categories. Obstacles on the way to a proof of the full statement are discussed. This is a joint work with Jan Stovicek.
  • 05.07.2007, 15.15, Changchang Xi (Beijing/Köln)
    Finitistic dimensions and relatively projective modules
    Abstract: Over a field k, the finitistic dimension conjecture is equivalent to the following statement: If B is a subalgebra of a finite-dimensional k-algebra A such that the radical of B is a left ideal in A, and if A has finite finitistic dimension, then B has finite finitistic dimension. In this talk, we show that the statement is true if the category of the relatively projective A-modules is closed under kernels of surjective homomorphisms. In particular, the statement is true if A and B have the same radical. This is a joint work with Dengming Xu.

  • 28.06.2007, 16.45, Idun Reiten (Trondheim)
    Cluster tilting for hypersufaces
    Abstract: The talk is based upon work with Burban, Iyama and Keller. We investigate cluster tilting objects and maximal rigid objects in 2-Calabi-Yau categories which are stable categories of Cohen-Macaulay modules over hypersurfaces. Under some conditions we give the number of cluster tilting objects and a description of them. We also discuss the associated endomorphism algebras. There are close connections with the noncommutative crepant resolutions of Van den Bergh.
  • 28.06.2007, 14.15, Yu Ye (Hefei/Paderborn)
    On centers of triangulated categories
    Abstract: In this talk, I will talk about the graded center of triangulated categories. Let $T$ be a triangulated category and $S$ a triangulated subcategory. We show that there exists a natural graded algebra map from $Z(T)$ to $Z(S)$ and also a graded algebra map from $Z(T)$ to $Z(T/S)$, where $T/S$ is the quotient category. In the case that $T$ is the bounded derived category of an abelian category which has enough projectives and $S$ the full subcategory of perfect complexes, the above map gives an isomorphism between $Z(T)$ and $Z(S)$. I will also discuss the center of the bounded derived category of the path algebra of finite and affine type, as well as the algebra $k[x]/<x^2>$.

  • 21.06.2007, 16.00, Srikanth Iyengar (Lincoln)
    Modules with prescribed cohomological support
    Abstract: In this talk, I will discuss a cohomological support, Supp_A(M), defined for finitely generated modules M over an left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of R-modules. The main result is that if the A-module Ext^R(M,M) is noetherian and Ext_i^R(M,R)=0 for i>>0, then every closed subset of Supp_A(M) is the support of some finitely generated R-module. This theorem specializes to realizability results for varieties of modules over group algebras, over local complete intersections, and over finite dimensional algebras over a field. It can also used to produce large families of finitely generated modules of finite projective dimension over commutative local noetherian rings. This is based on an on-going collaboration with Lucho Avramov.

  • 18.06.2007, 17.15, Nikolay Dichev (Paderborn), im Rahmen des IRTG Seminars
    Thick subcategories of quiver representations
  • 21.05.2007, 17.15, Stefan Wolf (Paderborn), im Rahmen des IRTG Seminars
    The composition monoid over different fields
  • 03.05.2007, 16.15, Julia Pevtsova (Seattle)
    Modules of  constant Jordan type
    Abstract: The support variety of a module of a finite group scheme is a much studied geometric invariant which has its roots in the fundamental work of  Quillen on cohomology of finite groups. Approaching the study of support varieties through the "rank variety" prospective introduced by J. Carlson we define a refinement  of the  usual support variety, called the "non-maximal" support variety.  This leads naturally to the study of a special class of modules, "modules of constant Jordan types", for which the non-maximal support variety is trivial. I shall discuss general properties and particular examples of modules of constant Jordan type and finish with some open questions. This is joint work with J. Carlson and E. Friedlander.
  • 19.04.2007, 14.15, Hagen Meltzer (Stettin)
    An algorithm for exceptional modules
    Abstract: This is a report on joint work with Piotr Dowbor and Andrzej Mroz. We describe an algorithm and develop a computer program for exceptional modules over tubular  canonical algebras. The input for the algorithm is a quadruple consisting of the slope, the number of the tube, the quasi-socle and the quasi-length. The output are explicit matrices for the module with the data above.
  • 12.04.2007, 15.45, David J. Green (Jena)
    Testing Benson's regularity conjecture
    Abstract: Dave Benson conjectures that the cohomology ring of a finite group with coefficients in a finite field always has Castelnuovo-Mumford regularity zero. A slight strengthening of the conjecture states that the cohomology ring always has a system of parameters which only fails to be a regular sequence in very specific low degrees. Benson calls such parameter systems very strongly quasi-regular (VSQR). I shall discuss where one might look for counterexamples and report that Benson's conjecture has successfully withstood the first serious test: all groups of order less than 256 satisfy the conjecture. In particular the conjecture holds for the 14 groups of order 128 where the difference between Krull dimension and depth of the cohomology ring is as large as 3.
  • 12.04.2007, 14.15, Grigory Garkusha (Swansea)
    Classifying finite localizing subcategories of modules
    Abstract: Given a commutative ring R (resp. graded commutaive ring A), we classify finite localizing subcategories of R-modules (resp. graded A-modules) in terms of topology for Spec R (resp. Proj A). The classification applies to reconstructing Spec R (resp. Proj A) from the category of R-modules (resp. graded A-modules).
  • 05.04.2007, 16.15, Nikolay Dichev (Paderborn)
    Thick subcategories for hereditary algebras
    This is a report on the recent paper "Noncrossing partitions and representations of quivers" by Colin Ingalls and Hugh Thomas. For a finite and acyclic quiver denote by repQ the category of finite dimensional kQ-modules. The proof of the following theorem will be presented :In repQ there are bijections between support tilting modules, finitely generated torsion classes and finitely generated thick subcategories.

  • 29.03.2007, 15.45, Jon Carlson (Athens, Georgia)
    Gluing representations via idempotent modules and constructing endotrivial modules
    Abstract: This is joint work with Paul Balmer and Dave Benson. Suppose that G is a finite group and k is a field of characteristic p > 0. The endotrivial kG-modules are the elements of the Picard group of invertible objects in the stable category of kG-modules.  They form an important subgroup of the group of all self equivalences of the stable category. In addition, endotrivial modules play a significant role in the block theory and modular representation theory of G. In this paper we investigate a new construction of endotrivial modules, which is not limited to p-groups. The method is a gluing construction developed by Balmer and Favi applied to the idempotent modules of Jeremy Rickard.

  • 29.03.2007, 14.15, Carles Bivià Ausina (Valencia)
    Invariants of singularities and integrally closed monomial ideals
    Abstract: We study integral closures of submodules and its relation with Buchsbaum-Rim multiplicity and mixed multiplicities of a set of ideals, defined by Teissier and Risler. We apply these notions to compute known invariants of complex analytic singularities, like the Milnor number of complete intersections.
  • 15.03.2007, 16.15, Dave Benson (Aberdeen)
    An algebraic model for the loops on the p-completion of the classifying space of a finite group
    Abstract: I shall give a representation theoretic recipe which computes the homology of the p-completion of the classifying space of a finite group. I shall illustrate with an example, and talk about some questions concerning the boundary between polynomial and exponential growth.

  • 08.03.2007, 16.15, Øyvind Solberg (Trondheim)
    Artin-Schelter regular algebras.
    Abstract: We define and generalize the class of Artin-Schelter regular algebras. Some central properties of these algebras are recalled, and our concept of an AR-regular algebra is modelled on this behaviour. The generalization is motivated through considering the class of Auslander algebras for finite dimensional algebras.

  • 15.02.2007, 16.45, Yu Ye (USTC Heifei)
    Koszul stuctrures from generalized Koszul algebras
    Abstract: Given a generalized Koszul algebra, I will discuss the structure of its Yoneda algebra.  It is known that the even part of the Yoneda algebra is a Koszul algebra. Then we study the odd part and show that it is a Koszul module over the even part. Moreover, after regrading, the Yoneda algebra also becomes a Koszul algebra.
  • 15.02.2007, 15.15, Martin Hamm (Paderborn)
    Freyd's generating hypothesis, III
    Abstract: This is a report on the paper "The generating hypothesis in the derived category of R-modules" by Keir H. Lockridge.
  • 01.02.2007, 16.15, Birgit Huber (Paderborn)
    Freyd's generating hypothesis, II
    Abstract: This is a report on the recent paper "The generating hypothesis for the stable module category of a p-group" by Benson, Chebolu, Christensen and Minac.
  • 11.01.2007, 16.15, Kristian Brüning (Paderborn)
    Freyd's generating hypothesis, I
    Abstract: This is the first talk in series of three talks on Freyd's generating hypothesis (GH). We will formulate the GH and introduce briefly the required notions from stable homotopy theory. Furthermore, some consequences of the GH will be described.

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