The dual actions, equivariant autoequivalences and stable tilting objects
[Actions duales, auto-équivalences équivariantes et objets basculants stables]
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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
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
@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, Tome 70 (2020) no. 6, pp. 2677-2736. doi : 10.5802/aif.3361. https://aif.centre-mersenne.org/articles/10.5802/aif.3361/

[1] Asashiba, Hideto A generalization of Gabriel’s Galois covering functors and derived equivalences, J. Algebra, Volume 334 (2011), pp. 109-149 | DOI | MR | Zbl

[2] Asashiba, Hideto Gluing derived equivalences together, Adv. Math., Volume 235 (2013), pp. 134-160 | DOI | MR | Zbl

[3] Asashiba, Hideto A generalization of Gabriel’s Galois covering functors II: 2-categorical Cohen-Montgomery duality, Appl. Categ. Struct., Volume 25 (2017) no. 2, pp. 155-186 | DOI | MR | Zbl

[4] Balmer, Paul Separability and triangulated categories, Adv. Math., Volume 226 (2011) no. 5, pp. 4352-4372 | DOI | MR | Zbl

[5] Barot, Michael; de la Peña, José Antonio Derived tubular strongly simply connected algebras, Proc. Am. Math. Soc., Volume 127 (1999) no. 3, pp. 647-655 | DOI | MR | Zbl

[6] Borceux, Francis Handbook of categorical algebra. Volume 1: Basic category theory., Encyclopedia of Mathematics and Its Applications, 50, Cambridge University Press, 1994 | MR | Zbl

[7] Chen, Jianmin; Chen, Xiao-Wu Weighted projective lines of tubular type and equivariantization, J. Algebra, Volume 470 (2017), pp. 77-90 | DOI | MR | Zbl

[8] Chen, Jianmin; Chen, Xiao-Wu; Zhou, Zhenqiang Monadicity theorem and weighted projective lines of tubular type, Int. Math. Res. Not. (2015) no. 24, pp. 13324-13359 | DOI | MR | Zbl

[9] Chen, Xiao-Wu The stable monomorphism category of a Frobenius category, Math. Res. Lett., Volume 18 (2011) no. 1, pp. 125-137 | DOI | MR | Zbl

[10] Chen, Xiao-Wu A note on separable functors and monads with an application to equivariant derived categories, Abh. Math. Semin. Univ. Hamb., Volume 85 (2015) no. 1, pp. 43-52 | DOI | MR | Zbl

[11] Chen, Xiao-Wu Equivariantization and Serre duality I, Appl. Categ. Struct., Volume 25 (2017) no. 4, pp. 539-568 | DOI | MR | Zbl

[12] Chen, Xiao-Wu; Ye, Yu The D-standard and K-standard categories, Adv. Math., Volume 333 (2018), pp. 159-193 | DOI | MR | Zbl

[13] Deligne, Pierre Action du groupe des tresses sur une catégorie, Invent. Math., Volume 128 (1997) no. 1, pp. 159-175 | DOI | MR | Zbl

[14] Dionne, Julie; Lanzilotta, Marcelo; Smith, David Skew group algebras of piecewise hereditary algebras are piecewise hereditary, J. Pure Appl. Algebra, Volume 213 (2009) no. 2, pp. 241-249 | DOI | MR | Zbl

[15] Drinfeld, Vladimir; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor On braided fusion categories. I, Sel. Math., New Ser., Volume 16 (2010) no. 1, pp. 1-119 | DOI | MR | Zbl

[16] Elagin, Alexey On equivariant triangulated categories (2014) (https://arxiv.org/abs/1403.7027v2)

[17] Epstein, David B. A. Functors between tensored categories, Invent. Math., Volume 1 (1966), pp. 221-228 | DOI | MR | Zbl

[18] Geigle, Werner; Lenzing, Helmut A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) (Lecture Notes in Mathematics), Volume 1273, Springer, 1987, pp. 265-297 | DOI | MR | Zbl

[19] Hubery, Andrew Quiver representations respecting a quiver automorphism: a generalisation of a theorem of Kac, J. Lond. Math. Soc., Volume 69 (2004) no. 1, pp. 79-96 | DOI | MR | Zbl

[20] Jasso, Gustavo τ 2 -stable tilting complexes over weighted projective lines, Adv. Math., Volume 273 (2015), pp. 1-31 | DOI | MR | Zbl

[21] Keller, Bernhard Derived categories and tilting, Handbook of tilting theory (London Mathematical Society Lecture Note Series), Volume 332, Cambridge University Press, 2007, pp. 49-104 | DOI | MR

[22] Lenzing, Helmut Weighted projective lines and Riemann surfaces, Proceedings of the 49th Symposium on Ring Theory and Representation Theory (2017), pp. 67-79 | MR | Zbl

[23] Lenzing, Helmut; Meltzer, Hagen The automorphism group of the derived category for a weighted projective line, Commun. Algebra, Volume 28 (2000) no. 4, pp. 1685-1700 | DOI | MR | Zbl

[24] Mac Lane, Saunders Categories for the working mathematician, Graduate Texts in Mathematics, 5, Springer, 1998 | MR | Zbl

[25] Meltzer, Hagen Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines, Memoirs of the American Mathematical Society, 808, American Mathematical Society, 2004 | DOI | MR | Zbl

[26] Năstăsescu, Constantin; Van Oystaeyen, Freddy Methods of graded rings, Lecture Notes in Mathematics, 1836, Springer, 2004 | DOI | MR | Zbl

[27] Novaković, Saša Tilting objects on some global quotient stacks, J. Commut. Algebra, Volume 10 (2018) no. 1, pp. 107-137 | DOI | MR | Zbl

[28] Ploog, David Equivariant autoequivalences for finite group actions, Adv. Math., Volume 216 (2007) no. 1, pp. 62-74 | DOI | MR | Zbl

[29] Polishchuk, Alexander Holomorphic bundles on 2-dimensional noncommutative toric orbifolds, Noncommutative geometry and number theory (Aspects Math.), Volume 37, Springer, 2006, pp. 341-359 | DOI | MR | Zbl

[30] Power, A. John A general coherence result, J. Pure Appl. Algebra, Volume 57 (1989) no. 2, pp. 165-173 | DOI | MR | Zbl

[31] Reiten, Idun; Riedtmann, Christine Skew group algebras in the representation theory of Artin algebras, J. Algebra, Volume 92 (1985) no. 1, pp. 224-282 | DOI | MR | Zbl

[32] Rickard, Jeremy Derived equivalences as derived functors, J. Lond. Math. Soc., Volume 43 (1991) no. 1, pp. 37-48 | DOI | MR | Zbl

[33] Ringel, Claus Michael Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, 1099, Springer, 1984 | DOI | MR | Zbl

[34] Schiffmann, Olivier Lectures on Hall algebras (2009) (https://arxiv.org/abs/math/0611617) | Zbl

[35] Shinder, Evgeny Group actions on categories and Elagin’s theorem revisited, Eur. J. Math., Volume 4 (2018) no. 1, pp. 413-422 | DOI | MR | Zbl

[36] Skowroński, Andrzej Selfinjective algebras of polynomial growth, Math. Ann., Volume 285 (1989) no. 2, pp. 177-199 | DOI | MR | Zbl

[37] Zhang, Yingying; Huang, Zhaoyong G-stable support τ-tilting modules, Front. Math. China, Volume 11 (2016) no. 4, pp. 1057-1077 | DOI | MR | Zbl

Cité par Sources :