Strong generators in D-Perf(X) and D-coh(b)(X)

Abstract

We solve two open problems: first we prove a conjecture of Bondal and Van den Bergh, showing that the category D-Perf (X) is strongly generated whenever X is a quasicompact, separated scheme, admitting a cover by open affine subsets Spec(R-i) with each R-i of finite global dimension. We also prove that, for a noetherian scheme X of finite type over an excellent scheme of dimension <= 2, the derived category D-coh(b)(X) is strongly generated. The known results in this direction all assumed equal characteristic; we have no such restriction. The method is interesting in other contexts: our key lemmas turn out to give a simple proof that, if f : X -> Y is a separated morphism of quasicompact, quasiseparated schemes such that Rf(*) : D-qc(X) -> D-qc(Y) takes perfect complexes to complexes of bounded-below Tor-amplitude, then f must be of finite Tor-dimension.