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Automorphismes, graduations et catégories triangulées

Part of: Abelian categories Representation theory of rings and algebras

Published online by Cambridge University Press:  25 May 2011

Raphaël Rouquier
Raphaël Rouquier
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK (rouquier@maths.ox.ac.uk)
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Abstract

We give a moduli interpretation of the outer automorphism group Out of a finite-dimensional algebra similar to that of the Picard group of a scheme. We deduce that the connected component of Out is invariant under derived and stable equivalences. This allows us to transfer gradings between algebras and gives rise to conjectural homological constructions of interesting gradings on blocks of finite groups with abelian defect. We give applications to the lifting of stable equivalences to derived equivalences. We give a counterpart of the invariance result for smooth projective varieties: the product Pic0 × Aut0 is invariant under derived equivalence.

Résumé

Nous donnons une construction comme espace de modules du groupe d'automorphisme d'une algèbre de dimension finie analogue à celle du groupe de Picard d'un schéma. Nous en déduisons que la composante connexe de l'identité du groupe des automorphismes extérieurs est invariante par équivalences stables et dérivées. Ceci permet de transférer des graduations entre algèbres et fournit conjecturalement une construction homologique de graduations sur les blocs à défaut abé;lien de groupes finis. Nous donnons des applications au relèvement d'équivalences stables en équivalences dérivées. Nous donnons une version du résultat d'invariance pour les variétés projectives lisses : le produit Pic0 × Aut0 est invariant par équivalences dérivées.

Keywords

automorphismesgradingsreprésentationscatégories dérivéescatégories stablesautomorphismsgraduationsrepresentationsderived categoriesstable categories

MSC classification

Secondary: 16G10: Representations of Artinian rings 18E30: Derived categories, triangulated categories
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Special Issue 3: A Special Issue in Honour of Louis Boutet de Monvel and Pierre Schapira , July 2011 , pp. 713 - 751
Copyright
Copyright © Cambridge University Press 2011

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