Neeman, Amnon2023-03-060003-486Xhttp://hdl.handle.net/1885/286664We 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.The research was partly supported by the Australian Research Council, and partly carried out while visiting the CRM in Barcelonaapplication/pdfen-AU© 2021 Department of Mathematics, Princeton UniversityDerived categoriesschemescompact generators202110.4007/annals.2021.193.3.12021-12-26