Remarks on complexities and entropies for singularity categories

Ryo Takahashi

doi:10.5802/crmath.482

Algèbre

Remarks on complexities and entropies for singularity categories

Ryo Takahashi ¹

¹ Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1611-1623.

Résumé

Let $R$ be a commutative noetherian local ring which is singular and has an isolated singularity. Let $D_{sg} (R)$ be the singularity category of $R$ in the sense of Buchweitz and Orlov. In this paper, we find real numbers $t$ such that the complexity $δ_{t} (G, X)$ in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich vanishes for any split generator $G$ of $D_{sg} (R)$ and any object $X$ of $D_{sg} (R)$ . In particular, the entropy $h_{t} (F)$ of an exact endofunctor $F$ of $D_{sg} (R)$ is not defined for such numbers $t$ .

Reçu le : 2022-07-28
Révisé le : 2023-02-02
Accepté le : 2023-02-22
Publié le : 2023-11-17

DOI : 10.5802/crmath.482

Classification : 13D09, 13H10, 18G80

Affiliations des auteurs :

Ryo Takahashi ¹

¹ Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G10_1611_0,
     author = {Ryo Takahashi},
     title = {Remarks on complexities and entropies for singularity categories},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1611--1623},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.482},
     language = {en},
}

TY  - JOUR
AU  - Ryo Takahashi
TI  - Remarks on complexities and entropies for singularity categories
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1611
EP  - 1623
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.482
LA  - en
ID  - CRMATH_2023__361_G10_1611_0
ER  -

%0 Journal Article
%A Ryo Takahashi
%T Remarks on complexities and entropies for singularity categories
%J Comptes Rendus. Mathématique
%D 2023
%P 1611-1623
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.482
%G en
%F CRMATH_2023__361_G10_1611_0

Ryo Takahashi. Remarks on complexities and entropies for singularity categories. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1611-1623. doi : 10.5802/crmath.482. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.482/

Bibliographie
Cité par

[1] Luchezar L. Avramov Infinite free resolutions, Six lectures on commutative algebra (Progress in Mathematics), Volume 166, Birkhäuser, 2010, pp. 1-118 | Zbl

[2] Luchezar L. Avramov; Ragnar-Olaf Buchweitz; Srikanth B. Iyengar; Claudia Miller Homology of perfect complexes, Adv. Math., Volume 223 (2010) no. 5, pp. 1731-1781 | DOI | MR | Zbl

[3] Winfried Bruns; Jürgen Herzog Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, 1998 | DOI | Zbl

[4] 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

[5] Hailong Dao; Toshinori Kobayashi; Ryo Takahashi Burch ideals and Burch rings, Algebra Number Theory, Volume 14 (2020) no. 8, pp. 2121-2150 | MR | Zbl

[6] George Dimitrov; Fabian Haiden; Ludmil Katzarkov; Maxim Kontsevich Dynamical systems and categories, The influence of Solomon Lefschetz in geometry and topology: 50 years of mathematics at CINVESTAV (Contemporary Mathematics), Volume 621, American Mathematical Society, 2014, pp. 133-170 | MR | Zbl

[7] Tobias Dyckerhoff Compact generators in categories of matrix factorizations, Duke Math. J., Volume 159 (2011) no. 2, pp. 223-274 | MR | Zbl

[8] David Eisenbud Homological algebra of a complete intersection, with an application to group representations, Trans. Am. Math. Soc., Volume 260 (1980) no. 1, pp. 35-64 | DOI | MR | Zbl

[9] Alexey Elagin; Valery A. Lunts Three notions of dimension for triangulated categories, J. Algebra, Volume 569 (2021), pp. 334-376 | DOI | MR | Zbl

[10] Yu-Wei Fan Entropy of an autoequivalence on Calabi-Yau manifolds, Math. Res. Lett., Volume 25 (2018) no. 2, pp. 509-519 | DOI | MR | Zbl

[11] Yu-Wei Fan; Simion Filip; Fabian Haiden; Ludmil Katzarkov; Yijia Liu On pseudo-Anosov autoequivalences, Adv. Math., Volume 384 (2021), 107732 | MR | Zbl

[12] Yu-Wei Fan; Lie Fu; Genki Ouchi Categorical polynomial entropy, Adv. Math., Volume 383 (2021), 107655 | MR | Zbl

[13] Akishi Ikeda Mass growth of objects and categorical entropy, Nagoya Math. J., Volume 244 (2021), pp. 136-157 | DOI | MR | Zbl

[14] Bernhard Keller; Dieter Vossieck Sous les catégories dérivées, . R. Math. Acad. Sci. Paris, Volume 305 (1987) no. 6, pp. 225-228 | Zbl

[15] Kohei Kikuta On entropy for autoequivalences of the derived category of curves, Adv. Math., Volume 308 (2017), pp. 699-712 | DOI | MR | Zbl

[16] Kohei Kikuta; Genki Ouchi; Atsushi Takahashi Serre dimension and stability conditions, Math. Z., Volume 299 (2021) no. 1, pp. 997-1013 | DOI | MR | Zbl

[17] Kohei Kikuta; Yuuki Shiraishi; Atsushi Takahashi A note on entropy of auto-equivalences: lower bound and the case of orbifold projective lines, Nagoya Math. J., Volume 238 (2020), pp. 86-103 | DOI | MR | Zbl

[18] Kohei Kikuta; Atsushi Takahashi On the categorical entropy and the topological entropy, Int. Math. Res. Not., Volume 2019 (2019) no. 2, pp. 457-469 | DOI | MR | Zbl

[19] Mahdi Majidi-Zolbanin; Nikita Miasnikov Entropy in the category of perfect complexes with cohomology of finite length, J. Pure Appl. Algebra, Volume 223 (2019) no. 6, pp. 2585-2597 | DOI | MR | Zbl

[20] Hideyuki Matsumura Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1989 (translated from the Japanese by M. Reid) | Zbl

[21] Dominique Mattei Categorical vs topological entropy of autoequivalences of surfaces, Mosc. Math. J., Volume 21 (2021) no. 2, pp. 401-412 | DOI | MR | Zbl

[22] Amnon Neeman Triangulated categories, Annals of Mathematics Studies, 148, Princeton University Press, 2001 | DOI | Zbl

[23] Dmitri O. Orlov Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova, Volume 246 (2004) no. 3, pp. 240-262 translation in Proc. Steklov Inst. Math. 2004, no. 3(246), 227–248 | MR | Zbl

[24] Genki Ouchi Automorphisms of positive entropy on some hyperKähler manifolds via derived automorphisms of $K 3$ surfaces, Adv. Math., Volume 335 (2018), pp. 1-26 | DOI | MR | Zbl

[25] Genki Ouchi On entropy of spherical twists, Entropy, Volume 148 (2020) no. 3, pp. 1003-1014 (with an appendix by Arend Bayer) | MR | Zbl

[26] Jeremy Rickard Derived categories and stable equivalence, J. Pure Appl. Algebra, Volume 61 (1989) no. 3, pp. 303-317 | DOI | MR | Zbl

[27] Judith D. Sally The Poincaré series of stretched Cohen-Macaulay rings, Can. J. Math., Volume 32 (1980) no. 5, pp. 1261-1265 | DOI | MR | Zbl

[28] Ryo Takahashi Reconstruction from Koszul homology and applications to module and derived categories, Pac. J. Math., Volume 268 (2014) no. 1, pp. 231-248 | DOI | MR | Zbl

[29] Yuji Yoshino Cohen–Macaulay modules over Cohen–Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, 1990 | DOI | Zbl

[30] Kōta Yoshioka Categorical entropy for Fourier–Mukai transforms on generic abelian surfaces, J. Algebra, Volume 556 (2020), pp. 448-466 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique