Strong generators in Dperf(X) and Dbcoh(X)

We solve two open problems: first we prove a conjecture of Bondal and Van den Bergh, showing that the category Dperf(X) is strongly generated whenever X is a quasicompact, separated scheme, admitting a cover by open affine subsets Spec(Ri) with each Ri 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 Dbcoh(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∗:Dqc(X)→Dqc(Y) takes perfect complexes to complexes of bounded-below Tor-amplitude, then f must be of finite Tor-dimension.