Vanishing of DHKK complexities for singularity categories and generation of syzygy modules

Let R be a commutative noetherian ring. In this paper, we study, for the singularity category of R, the vanishing of the complexity ${\delta }_t(X,Y)$ in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich. We prove that the set of real numbers t such that ${\delta }_t(X,Y)$ does not vanish is bounded in various cases. We do it by building the high syzygy modules and maximal Cohen-Macaulay modules out of a single module only by taking direct summands and extensions.