On the Hochschild homology of singularity categories

Yu Wang; Umamaheswaran Arunachalam; Bernhard Keller

doi:10.5802/crmath.318

Algèbre, Combinatoire

On the Hochschild homology of singularity categories

Yu Wang ^1,² ; Umamaheswaran Arunachalam ³ ; Bernhard Keller ⁴

¹ Department of Mathematics, Nanjing University, Nanjing 210093, PR China
² Université Paris Cité, UFR de mathématiques, CNRS IMJ–PRG, 8 place Aurélie Nemours, 75013 Paris, France
³ Department of Mathematics, National Institute of Technology (NIT) Warangal, Warangal, Telangana, 506004, India
⁴ Université Paris Cité, Sorbonne Université, CNRS IMJ–PRG 8 place Aurélie Nemours, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 491-496.

Résumé

Let $k$ be an algebraically closed field and $A$ a finite-dimensional $k$ -algebra. In this note, we determine complexes which compute the Hochschild homology of the canonical dg enhancement of the bounded derived category of $A$ and of the canonical dg enhancement of the singularity category of $A$ . As an application, we obtain a new approach to the computation of Hochschild homology of Leavitt path algebras.

Reçu le : 2021-10-09
Révisé le : 2021-12-21
Accepté le : 2021-12-21
Publié le : 2022-05-23

DOI : 10.5802/crmath.318

Classification : 16E35, 16E40, 16E45, 18G80

Affiliations des auteurs :

Yu Wang ^{1, 2} ; Umamaheswaran Arunachalam ³ ; Bernhard Keller ⁴

¹ Department of Mathematics, Nanjing University, Nanjing 210093, PR China
² Université Paris Cité, UFR de mathématiques, CNRS IMJ–PRG, 8 place Aurélie Nemours, 75013 Paris, France
³ Department of Mathematics, National Institute of Technology (NIT) Warangal, Warangal, Telangana, 506004, India
⁴ Université Paris Cité, Sorbonne Université, CNRS IMJ–PRG 8 place Aurélie Nemours, 75013 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Yu Wang and Umamaheswaran Arunachalam and Bernhard Keller},
     title = {On the {Hochschild} homology of singularity categories},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {491--496},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.318},
     language = {en},
}

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Yu Wang; Umamaheswaran Arunachalam; Bernhard Keller. On the Hochschild homology of singularity categories. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 491-496. doi : 10.5802/crmath.318. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.318/

Bibliographie
Cité par

[1] Pere Ara; Guillermo Cortiñas Tensor products of Leavitt path algebras, Proc. Am. Math. Soc., Volume 141 (2013) no. 8, pp. 2629-2639 | MR | Zbl

[2] Michel Van den Bergh Calabi–Yau algebras and superpotentials, Sel. Math., New Ser., Volume 21 (2015) no. 2, pp. 555-603 | DOI | MR | Zbl

[3] Ragnar-Olaf Buchweitz Maximal Cohen–Macaulay modules and Tate cohomology, Mathematical Surveys and Monographs, 262, American Mathematical Society, 2021 (with appendices by Luchezar L. Avramov, Benjamin Briggs, Srikanth B. Iyengar and Janina C. Letz.) | DOI | Zbl

[4] Xiao-Wu Chen The singularity category of a quadratic monomial algebra, Q. J. Math, Volume 69 (2018) no. 3, pp. 1015-1033 | DOI | MR | Zbl

[5] Xiao-Wu Chen; Zhengfang Wang The dg Leavitt algebra, singular Yoneda category and singularity category (2021) (https://arxiv.org/abs/2109.11278, with an appendix by B. Keller and Yu Wang)

[6] Xiao-Wu Chen; Dong Yang Homotopy categories, Leavitt path algebras, and Gorenstein projective modules, Int. Math. Res. Not., Volume 2015 (2015) no. 10, pp. 2597-2633 | DOI | MR | Zbl

[7] Estanislao Herscovich Hochschild (co)homology of Koszul dual pairs, J. Noncommut. Geom., Volume 13 (2019) no. 1, pp. 59-85 | DOI | MR | Zbl

[8] Bernhard Keller Deriving DG categories, Ann. Sci. Éc. Norm. Supér., Volume 27 (1994) no. 1, pp. 63-102 | DOI | Numdam | MR | Zbl

[9] Bernhard Keller Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra, Volume 123 (1998) no. 1-3, pp. 223-273 | DOI | MR | Zbl

[10] Bernhard Keller On differential graded categories, Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures, European Mathematical Society, 2006, pp. 151-190 | Zbl

[11] Dmitri O. Orlov Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova, Volume 246 (2004), pp. 240-262 | Zbl

[12] S. Paul Smith Category equivalences involving graded modules over path algebras of quivers, Adv. Math., Volume 230 (2012) no. 4-6, pp. 1780-1810 | DOI | MR | Zbl

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