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 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 .