Remarks on complexities and entropies for singularity categories

Abstract

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 $t$.