Workshop

"Support Varieties in Representation Theory"

Paderborn (February 22 - 23, 2007)

Organisation: Paderborn Representation Theory Group

Speakers: Petter Bergh (Oxford), Karin Erdmann (Oxford), Rolf Farnsteiner (Bielefeld), Henning Krause (Paderborn), Øyvind Solberg (Trondheim)

About the workshop: Support varieties have been introduced into representation theory some 25 years ago and provide now an interesting bridge between representation theory and commutative algebra. This will be an informal workshop about support varieties with introductory lectures on modern aspects of this theory. A conference Representations, Cohomology and Support Spaces devoted to this subject will be held at Bielefeld (April 29 - May 1, 2007).

Registration: There is no formal registration. However, please send a short message to  Ms. K. Bornhorst if you intend to participate.

Accomodation: The recommended place is Gästehaus Meinwerk. Please contact Ms. K. Bornhorst for assistance.

Programme

All lectures will be in room D1.303 of the main university building.

Petter Bergh: Support varieties over complete intersections

Abstract: Support varieties for modules over commutative local complete intersections were defined by Avramov in the early 1990s. A decade later, Avramov and Buchweitz showed that these varieties to a large extent behave like the cohomological varieties for modules over group algebras, and they used these constructions to give surprising results on the vanishing of (co)homology. This talk is an introduction to the topic, with an overview of the most important results and some recent developments.

Karin Erdmann: Rank varieties and support varieties for truncated polynomial rings and quantum complete intersections

Abstract: This is a report on some results on rank varieties and support varieties for modules of truncated polynomial rings K[X₁,..., X_m]/(X_iⁿ) where K is some algebraically closed field. In joint work with D. Benson and M. Holloway, we constructed, via quantum complete intersections, rank varieties which satisfy Dade's Lemma. When n=2 we know that the rank varieties are isomorphic to support varieties, defined in terms of Hochschild  cohomology (and we think this should hold in general).

J. Pevtsova and S. Witherspoon independently introduced rank varieties for the truncated polynomial rings when the characteristic of the field does not divide n, by exploiting a related Hopf algebra; and show that they are isomorphic to the support varieties coming from the support varieties of this Hopf algebra. Miles Holloway has now shown that their rank varieties are isomorphic to to ours.

Rolf Farnsteiner: One-Parameter Subgroups, Support Spaces, and Applications

Abstract: Representation-theoretic support spaces for finite goup schemes unify and extend rank varieties that were previously defined for finite groups, restricted Lie algebras and infinitesimal groups. In my talks, which are mainly a review of recent work by Friedlander-Pevtsova, I will explain the salient features of p-points and π-points as well as their connection with the infinitesimal one-parameter subgroups defined by Suslin-Friedlander-Bendel.

Henning Krause: Local cohomology and support for triangulated categories

Abstract: This is a report on joint work with Dave Benson and Srikanth Iyengar. The notion of support is a fundamental concept, first introduced in algebraic geometry for modules, sheaves, and complexes, but now widely used in various areas of modern mathematics, in particular in representation theory. For instance, Benson, Carlson, and Rickard used it (following Hopkins and Neeman) to classify thick subcategories. To define the support of an object, one usually requires an abelian or triangulated category with a commutative tensor product. In this talk, I present an approach to define the support for objects in any triangulated category, which has small coproducts and is compactly generated. This approach covers the usual examples but has the potential to provide new insight, for instance in non-commuative situations. It is somewhat surprising, how little is needed to develop a satisfactory theory of support.

Øyvind Solberg: Support varieties - an axiomatic approach (jt. with A. B. Buan, H. Krause, N. Snashall)

Abstract: The main aim of the talk is to present a common framework where most of the existing occurrences of support varieties (for "finite objects") fit in. In addition we discuss examples of these constructions through support varieties of bounded complexes and support varieties for complete intersections.

Schedule

Thursday, February 22, 2007 (room D1.303)

13.30 - 14.30  Farnsteiner I

14.30 - 15.15  Coffee (D2.314)

15.15 - 16.15  Krause I

16.30 - 17.30  Bergh

18.30 Dinner at Restaurant "Ratskeller" (Rathausplatz 1)

Friday, February 23, 2007 (room D1.303)

09.30 - 10.30  Farnsteiner II

10.30 - 11.00  Coffee (D2.314)

11.00 - 12.00  Solberg

12.00 - 13.30  Lunch

13.30 - 14.30  Krause II

14.30 - 15.00  Coffee (D2.314)

15.00 - 16.00  Erdmann