Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T11:43:54.482Z Has data issue: false hasContentIssue false
  • English
  • Français

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 ([email protected])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

1.Asashiba, H., A covering technique for derived equivalence, J. Alg. 191 (1997), 382–415.CrossRefGoogle Scholar
2.Asashiba, H., On a lift of an individual stable equivalence to a standard derived equivalence for representation-finite self-injective algebras, Alg. Representat. Theory 6 (2003), 427–447.CrossRefGoogle Scholar
3.Bogdanic, D., Graded Brauer tree algebras, J. Pure Appl. Alg. 214 (2010), 1534–1552.CrossRefGoogle Scholar
4.Broué, M., Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181–182 (1990), 61–92.Google Scholar
5.Brundan, J. and Kleshchev, A., Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras, Invent. Math. 178 (2009), 451–484.CrossRefGoogle Scholar
6.Chuang, J. et Rouquier, R., Derived equivalences for symmetric groups and 2-categorification, Annals Math. 167 (2008), 245–298CrossRefGoogle Scholar
7.Dade, E., Algebraically rigid modules, in Representation Theory II, Proceedings of the Second International Conference, Carleton University, Ottawa, Ontario, 1979, Lecture Notes in Mathematics, Volume 832, pp. 195–215 (Springer, 1980).Google Scholar
8.Donald, J. D. et Flanigan, F. J., Deformations of algebra modules, J. Alg. 31 (1974), 245–256.CrossRefGoogle Scholar
9.Guil-Asensio, F. et Saorín, M., On automorphism groups induced by bimodules, Arch. Math. 76 (2001), 12–19.CrossRefGoogle Scholar
10.Happel, D., On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), 339–389.CrossRefGoogle Scholar
11.Huisgen-Zimmermann, B. et Saorín, M., Geometry of chain complexes and outer automorphisms under derived equivalence, Trans. Am. Math. Soc. 353 (2001), 4757–4777.CrossRefGoogle Scholar
12.Huybrechts, D., Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs (Oxford University Press, 2006).Google Scholar
13.Külshammer, B., Crossed products and blocks with normal defect groups, Commun. Alg. 13 (1985), 147–168.CrossRefGoogle Scholar
14.Orlov, D., Equivalences of derived categories and K3 surfaces, J. Math. Sci. 84 (1997), 1361–1381.CrossRefGoogle Scholar
15.Orlov, D., Derived categories of coherent sheaves on abelian varieties and equivalences between them, Izv. Math. 66 (2002), 569–594.CrossRefGoogle Scholar
16.Pollack, R. D., Algebras and their automorphism groups, Commun. Alg. 17 (1989), 1843–1866.CrossRefGoogle Scholar
17.Popa, M. and Schnell, C., Derived invariance of the number of holomorphic 1-forms and vector fields, preprint (arXiv:0912.4040).Google Scholar
18.Rickard, J., Derived equivalences as derived functors, J. Lond. Math. Soc. 43 (1991), 37–48.CrossRefGoogle Scholar
19.Rouquier, R., Block theory via stable and Rickard equivalences, in Modular representation theory of finite groups, pp. 101–146 (Walter de Gruyter, Berlin, 2001).Google Scholar
20.Rouquier, R., Catégories dérivées et géométrie birationnelle, Séminaire Bourbaki 946, Mars 2005, Astérisque 307 (2006), 283–307.Google Scholar
21.Rouquier, R., Derived equivalences and finite dimensional algebras, International Congress of Mathematicians, Volume II, pp. 191–221 (European Mathematical Society, 2006).Google Scholar
22.Rouquier, R., Local constructions in block theory, en preparation.Google Scholar
23.Rouquier, R., 2-Kac-Moody algebras, preprint (arXiv:0812.5023).Google Scholar
24.Rouquier, R. et Zimmermann, A., Picard groups for derived module categories, J. Lond. Math. Soc. 87 (2003), 197–225.CrossRefGoogle Scholar
25.Serre, J. P., Groupes algébriques et corps de classes (Hermann, Paris, 1959).Google Scholar
26.Soergel, W., Kategorie , perverse Garben und Moduln über den Koinvarienten zur Weylgruppe, J. Am. Math. Soc. 3 (1990), 421–445.Google Scholar
27.Thévenaz, J., G-algebras and modular representation theory (Oxford University Press, 1995).Google Scholar
28.Yekutieli, A., Dualizing complexes, Morita equivalence and the derived Picard group of a ring, J. Lond. Math. Soc. 60 (1999), 723–746.CrossRefGoogle Scholar
29.Yekutieli, A., The derived Picard group is a locally algebraic group, Alg. Representat. Theory 7 (2004), 53–57.CrossRefGoogle Scholar