Mini-Workshop: Localising and Tilting in Categories

Abstract

The workshop brought together experts on localisation theory and tilting theory from different parts of mathematics with the aim of fully exploiting the power of some recent developments in so far rather independent contexts. The intensive exchange during the workshop is expected to lead to new and strengthened synergies and to new applications. Mathematics Subject Classification (2010): primary 16xx, 18xx, secondary 13xx, 55xx, 79xx. Introduction by the Organisers The workshop Localising and tilting in abelian and triangulated categories, organised by Lidia Angeleri Hugel (Verona), Steffen Koenig (Stuttgart) and Changchang Xi (Beijing) was attended by 17 participants with a broad mathematical and geographical background. The aim was to bring together researchers from different branches of mathematics to discuss and compare recent results on localisation theory and tilting theory. In order to optimise interaction between different areas and techniques, the organisers gave the programme a precise structure and ordering to emphasize connections and point out applicability of the new results. Historically, localisation theory has its origin in commutative algebra with the classical notions of a quotient field and of localisation of rings at prime ideals. Later on, these ring theoretic constructions were generalised to non-commutative rings and studied from a more categorical point of view, leading to a theory of localisation, first for abelian, and, more recently, for triangulated categories. Here the relevant notions are t-structures and recollements. 1534 Oberwolfach Report 26/2013 Localisation in abelian and triangulated categories plays an important role in many parts of mathematics. Commutative and non-commutative algebra, algebraic topology and homotopy theory, homological algebra and category theory have contributed in different and so far rather unrelated ways to better understanding localisations and to extending the scope of these methods. There are also close relations to microlocal analysis and to the study of D-modules, which in turn relates to representation theory through Kazhdan-Lusztig conjectures/theorems. Recent exciting developments link localisation theory with tilting theory a fundamental branch of representation theory which allows to compare different categories of representations both in abstract and in combinatorial ways. In fact, tilting theory is undergoing a major change due to new and unexpected applications of localisation techniques. For example, localisation methods have turned out to be very useful in establishing classification results for tilting modules, both in commutative and in non-commutative situations. The rapid progress in the theory during the last few years makes an exchange between researchers from different areas more necessary than ever. In fact, we believe that the impact of some of the recent developments has not been completely understood yet and can be strengthened much by combining and applying different techniques. A first step in this direction was done during the week in Oberwolfach. A lot of attention has been devoted to the interplay between the topological and the algebraic perspective on localisation. New results in commutative algebra and algebraic geometry were intensively discussed also from the point of view of non-commutative ring theory. Recent developments in a more categorical algebraic setting were explored together with explicit and often combinatorial applications. The active and promising exchange during the workshop lays the foundation for new projects. The positive outcome of the workshop is also due to the pleasant and well organised environment at the institute in Oberwolfach. In particular, the concept of bringing together such a small number of researchers in a mini-workshop creates an intense and almost private atmosphere, which turned out to be very productive. Finally, the active and helpful interaction between the different mini-workshops during the week should not remain unmentioned. Having parallel mini-workshops with a potential common interest proved to be a good idea. Mini-Workshop: Localising and Tilting in Categories 1535 Mini-Workshop: Localising and Tilting in Categories