We solve two open problems: first we prove a conjecture of Bondal and Van den Bergh, showing that the category $\mathbf D^{\rm perf}( X)$ 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 $\leq 2$, 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 $f\colon X\rightarrow Y$ is a separated morphism of quasicompact, quasiseparated schemes such that $\mathbf{R} f_*\colon \mathbf{D}_{\mathrm{\mathbf{qc}}}(X) \rightarrow \mathbf{D}_{\mathrm{\mathbf{qc}}}(Y)$ takes perfect complexes to complexes of bounded-below Tor-amplitude, then $f$ must be of finite Tor-dimension.

我们解决了两个未解决的问题：首先，我们证明了Bondal和Van den Bergh的猜想，表明只要$ X $是拟紧凑的分离方案，就会强烈生成类别$ \ mathbf D ^ {\ rm perf}（X）$，接受开放仿射子集$ \ mathrm {Spec}（{R_i}）$的覆盖，每个$ R_i $具有有限的全局维度。我们还证明，对于维数为$ \ leq 2 $的优秀方案而言，有限类型的Noether方案$ X $，派生类别$ \ mathbf {D} ^ b _ {\ mathrm {coh}}（X）$为强烈产生。在这个方向上的已知结果都假定具有相同的特性；我们没有这样的限制。

该方法在其他情况下很有趣：我们的关键引理给出一个简单的证明，如果$ f \冒号X \ rightarrow Y $是准紧凑的分离形态，则采用拟分隔的格式，例如$ \ mathbf {R} f _ * \冒号\ mathbf {D} _ {\ mathrm {\ mathbf {qc}}}（X）\ rightarrow \ mathbf {D} _ {\ mathrm {\ mathbf {qc}}}（Y）$在低于Tor振幅的范围内，则$ f $必须具有有限的Tor维度。