Remarks on complexities and entropies for singularity categories
Let $R$ be a commutative noetherian local ring which is singular and has an isolated singularity. Let $\mathsf {D_{sg}}(R)$ 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 $\delta _t(G,X)$ in the sense of Dimitr...
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Académie des sciences
2023-11-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.482/ |
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| author | Takahashi, Ryo |
| author_facet | Takahashi, Ryo |
| author_sort | Takahashi, Ryo |
| collection | DOAJ |
| description | Let $R$ be a commutative noetherian local ring which is singular and has an isolated singularity. Let $\mathsf {D_{sg}}(R)$ 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 $\delta _t(G,X)$ in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich vanishes for any split generator $G$ of $\mathsf {D_{sg}}(R)$ and any object $X$ of $\mathsf {D_{sg}}(R)$. In particular, the entropy $\mathrm{h}_t(F)$ of an exact endofunctor $F$ of $\mathsf {D_{sg}}(R)$ is not defined for such numbers $t$. |
| format | Article |
| id | doaj-art-5d4295bc7e5740fba811743e12ad3841 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-5d4295bc7e5740fba811743e12ad38412025-02-07T11:11:47ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-11-01361G101611162310.5802/crmath.48210.5802/crmath.482Remarks on complexities and entropies for singularity categoriesTakahashi, Ryo0Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, JapanLet $R$ be a commutative noetherian local ring which is singular and has an isolated singularity. Let $\mathsf {D_{sg}}(R)$ 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 $\delta _t(G,X)$ in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich vanishes for any split generator $G$ of $\mathsf {D_{sg}}(R)$ and any object $X$ of $\mathsf {D_{sg}}(R)$. In particular, the entropy $\mathrm{h}_t(F)$ of an exact endofunctor $F$ of $\mathsf {D_{sg}}(R)$ is not defined for such numbers $t$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.482/ |
| spellingShingle | Takahashi, Ryo Remarks on complexities and entropies for singularity categories Comptes Rendus. Mathématique |
| title | Remarks on complexities and entropies for singularity categories |
| title_full | Remarks on complexities and entropies for singularity categories |
| title_fullStr | Remarks on complexities and entropies for singularity categories |
| title_full_unstemmed | Remarks on complexities and entropies for singularity categories |
| title_short | Remarks on complexities and entropies for singularity categories |
| title_sort | remarks on complexities and entropies for singularity categories |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.482/ |
| work_keys_str_mv | AT takahashiryo remarksoncomplexitiesandentropiesforsingularitycategories |