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.
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.
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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
@article{AIF_2020__70_6_2677_0, author = {Chen, Jianmin and Chen, Xiao-Wu and Ruan, Shiquan}, title = {The dual actions, equivariant autoequivalences and stable tilting objects}, journal = {Annales de l'Institut Fourier}, pages = {2677--2736}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {6}, year = {2020}, doi = {10.5802/aif.3361}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3361/} }
TY - JOUR AU - Chen, Jianmin AU - Chen, Xiao-Wu AU - Ruan, Shiquan TI - The dual actions, equivariant autoequivalences and stable tilting objects JO - Annales de l'Institut Fourier PY - 2020 SP - 2677 EP - 2736 VL - 70 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3361/ DO - 10.5802/aif.3361 LA - en ID - AIF_2020__70_6_2677_0 ER -
%0 Journal Article %A Chen, Jianmin %A Chen, Xiao-Wu %A Ruan, Shiquan %T The dual actions, equivariant autoequivalences and stable tilting objects %J Annales de l'Institut Fourier %D 2020 %P 2677-2736 %V 70 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3361/ %R 10.5802/aif.3361 %G en %F AIF_2020__70_6_2677_0
Chen, Jianmin; Chen, Xiao-Wu; Ruan, Shiquan. The dual actions, equivariant autoequivalences and stable tilting objects. Annales de l'Institut Fourier, Volume 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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