no. 6
The dual actions, equivariant autoequivalences and stable tilting objects
[Actions duales, auto-équivalences équivariantes et objets basculants stables]
Chen, Jianmin1 ; Chen, Xiao-Wu2 ; Ruan, Shiquan1
1 School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, PR (China)
2 Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, No. 96 Jinzhai Road, Hefei, 230026, Anhui, PR (China)
Annales de l'Institut Fourier, Tome 70 (2020) no. 6, pp. 2677-2736.

Pour une catégorie linéaire munie d’une action d’un groupe abélien fini, on étudie l’action duale du groupe des caractères de ce groupe abélien sur la catégorie des objets équivariants. On montre que les groupes des auto-équivalences équivariantes de ces deux catégories sont isomorphes. Dans la situation où la catégorie linéaire est triangulée, les objets basculants stables dans ces deux catégories sont en bijection naturelle compatible avec l’isomorphisme ci-dessus. On applique ces résultats aux complexes basculants stables sur les droites projectives à poids de type tubulaire.

For a finite abelian group action on a linear category, we study the dual action given by the character group acting on the category of equivariant objects. We prove that the groups of equivariant autoequivalences on these two categories are isomorphic. In the triangulated situation, this isomorphism implies that the classifications of stable tilting objects for these two categories are in a natural bijection. We apply these results to stable tilting complexes on weighted projective lines of tubular type.

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Révisé le :
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DOI : 10.5802/aif.3361
Classification : 18E30, 16S35, 58E40, 16D99
Keywords: group action, equivariantization, autoequivalence, tilting object, weighted projective line
Mot clés : action de groupe, équivariantisation, auto-équivalence, objet basculant, droite projective à poids

Chen, Jianmin 1 ; Chen, Xiao-Wu 2 ; Ruan, Shiquan 1

1 School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, PR (China)
2 Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, No. 96 Jinzhai Road, Hefei, 230026, Anhui, PR (China)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Chen, Jianmin; Chen, Xiao-Wu; Ruan, Shiquan. The dual actions, equivariant autoequivalences and stable tilting objects. Annales de l'Institut Fourier, Tome 70 (2020) no. 6, pp. 2677-2736. doi : 10.5802/aif.3361. https://aif.centre-mersenne.org/articles/10.5802/aif.3361/

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