The Stacks project

Lemma 57.10.7. Let $k$ be a field. Let $X$ be a scheme proper and smooth over $k$. Then $D_{perf}(\mathcal{O}_ X)$ has a strong generator.

Proof. Using Lemma 57.10.6 choose finite locally free $\mathcal{O}_ X$-modules $\mathcal{E}$ and $\mathcal{G}$ such that $\mathcal{O}_\Delta \in \langle \mathcal{E} \boxtimes \mathcal{G} \rangle $ in $D(\mathcal{O}_{X \times X})$. We claim that $\mathcal{G}$ is a strong generator for $D_{perf}(\mathcal{O}_ X)$. With notation as in Derived Categories, Section 13.35 choose $m, n \geq 1$ such that

\[ \mathcal{O}_\Delta \in smd(add(\mathcal{E} \boxtimes \mathcal{G}[-m, m])^{\star n}) \]

This is possible by Derived Categories, Lemma 13.36.2. Let $K$ be an object of $D_{perf}(\mathcal{O}_ X)$. Since $L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} -$ is an exact functor and since

\[ L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} (\mathcal{E} \boxtimes \mathcal{G}) = (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{E}) \boxtimes \mathcal{G} \]

we conclude from Derived Categories, Remark 13.35.5 that

\[ L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} \mathcal{O}_\Delta \in smd(add( (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{E}) \boxtimes \mathcal{G}[-m, m])^{\star n}) \]

Applying the exact functor $R\text{pr}_{2, *}$ and observing that

\[ R\text{pr}_{2, *} \left((K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{E}) \boxtimes \mathcal{G}\right) = R\Gamma (X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{E}) \otimes _ k \mathcal{G} \]

by Derived Categories of Schemes, Lemma 36.22.1 we conclude that

\[ K = R\text{pr}_{2, *}(L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} \mathcal{O}_\Delta ) \in smd(add(R\Gamma (X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{E}) \otimes _ k \mathcal{G}[-m, m])^{\star n}) \]

The equality follows from the discussion in Example 57.9.6. Since $K$ is perfect, there exist $a \leq b$ such that $H^ i(X, K)$ is nonzero only for $i \in [a, b]$. Since $X$ is proper, each $H^ i(X, K)$ is finite dimensional. We conclude that the right hand side is contained in $smd(add(\mathcal{G}[-m + a, m + b])^{\star n})$ which is itself contained in $\langle \mathcal{G} \rangle _ n$ by one of the references given above. This finishes the proof. $\square$


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