Lemma 57.9.7. 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 57.9.6 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 57.8.6. 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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