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Stable equivalences of Morita type

Final Activity Report Summary - STABLEQ (Stable equivalences of Morita type)

This project is in the area of representation theory, whose aim is to describe and study abstract algebraic objects by representing them in explicit way; there are usually many different such representations. Representations are of interest not just by individual examples, but in particular as categories of representations which comprise and connect many different representations of the same object.

A central problem is to compare different categories of representations. There are two levels of such comparisons. The level of abelian categories is the easier one and has been intensively studied. The more difficult level is that of triangulated categories, which includes two main topics, namely derived categories and stable categories.

Stable categories are currently the most mysterious ones and were the object of this project. The main questions addressed by this initiative were to:

1. define and study invariants of stable categories that allowed to distinguish different categories, and
2. investigate properties of equivalences between stable categories.

The most important problem that was studied during research on the second question was the so-called Auslander-Reiten conjecture, which asserted that the number of non-projective simple representations was invariant under stable equivalences.

There was much progress on all main problems. In particular, a surprising new characterisation of the Auslander-Reiten conjecture was found in terms of Hochschild homology. This could be regarded as the most striking result achieved in this project.