JournalsowrVol. 3 / No. 1DOI 10.4171/owr/2006/08

Mini-Workshop: Thick Subcategories - Classifications and Applications

  • Henning Krause

    Universität Bielefeld, Germany
  • Stefan Schwede

    Universität Bonn, Germany
  • Ragnar-Olaf Buchweitz

    University of Toronto at Scarborough, Canada
Mini-Workshop: Thick Subcategories - Classifications and Applications cover

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Abstract

Thick subcategories of triangulated categories have been the main topic of this workshop. Triangulated categories arise in many areas of modern mathematics, for instance in algebraic geometry, in representation theory of groups and algebras, or in stable homotopy theory. We give three typical examples of such triangulated categories: \begin{itemize} \item the category of perfect complexes of -modules over a scheme , \item the stable category of finite dimensional representations of a finite group, \item the stable homotopy category of finite spectra. \end{itemize} In each case, there is a classification of thick subcategories under some appropriate conditions. Recall that a subcategory of a triangulated category is {\em thick}, if it is a triangulated subcategory and closed under taking direct factors. Historically, the first classification has been established by Hopkins and Smith for the stable homotopy category, using the nilpotence theorem. A similar idea was then applied by Hopkins and Neeman to categories of perfect complexes over commutative noetherian rings. Later, Thomason extended this classification to schemes. For stable categories of finite group representations, the classification of thick subcategories is due to Benson, Carlson, and Rickard.

The format of the workshop has been a combination of introductory survey lectures and more specialized talks on recent progress and open problems. The mix of participants from different mathematical areas and the relatively small size of the workshop provided an ideal atmosphere for fruitful interaction and exchange of ideas. It is a pleasure to thank the administration and the staff of the Oberwolfach Institute for their efficient support and hospitality.