Strong generators in $\mathbf{D}^{\mathrm{perf}}(X)$ and $\mathbf{D}^b_{\mathrm{coh}}(X)$

Abstract

We solve two open problems: first we prove a conjecture of Bondal and Van den Bergh, showing that the category $\mathbf{D}^{\mathrm{perf}}(X)$ is strongly generated whenever $X$ is a quasicompact, separated scheme, admitting a cover by open affine subsets $\mathrm{Spec}(R_i)$ 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 $\le 2$, the derived category $\mathbf{D}^b_{\mathrm{coh}}(X)$ 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 $f:X \longrightarrow Y$ is a separated morphism of quasicompact, quasiseparated schemes such that $\mathbf{R} f_*: \mathbf{D}_{\mathbf{qc}}(X)\longrightarrow \mathbf{D}_{\mathbf{qc}}(Y)$ takes perfect complexes to complexes of bounded-below Tor-amplitude, then $f$ must be of finite Tor-dimension.