Remarks on complexities and entropies for singularity categories

Takahashi, Ryo

Comptes Rendus. Mathématique, Tome 361 (2023) no. G10, pp. 1611-1623

Let $R$ be a commutative noetherian local ring which is singular and has an isolated singularity. Let ${\mathsf{D}}_{\mathsf{sg}}\left(R\right)$ 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 ${\mathsf{D}}_{\mathsf{sg}}\left(R\right)$ and any object $X$ of ${\mathsf{D}}_{\mathsf{sg}}\left(R\right)$. In particular, the entropy ${\mathrm{h}}_t\left(F\right)$ of an exact endofunctor $F$ of ${\mathsf{D}}_{\mathsf{sg}}\left(R\right)$ is not defined for such numbers $t$.

Classification : 13D09, 13H10, 18G80

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