Ranks of Verdier quotients of derived categories over local rings

Abstract

Let $R$ be a commutative noetherian local ring with residue field $k$. Denote by $\operatorname{\mathsf{D^b}}(R)$ the bounded derived category of finitely generated $R$-modules. In this paper, we introduce the notion of ranks of triangulated categories and study Verdier quotients of $\operatorname{\mathsf{D^b}}(R)$ from the viewpoint of this invariant. Our main result determines the rank of the Verdier quotient $\operatorname{\mathsf{D^b}}(R)/\operatorname{\mathsf{thick}}(R\oplus k)$ in terms of the ranks of the categories of maximal Cohen-Macaulay modules on the punctured spectrum.