The Stacks project

Lemma 57.9.6. Let $k$ be a field. Let $X$ be a quasi-compact separated smooth scheme over $k$. There exist 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 $D(\mathcal{O}_{X \times X})$ where the notation is as in Derived Categories, Section 13.36.

Proof. Recall that $X$ is regular by Varieties, Lemma 33.25.3. Hence $X$ has the resolution property by Derived Categories of Schemes, Lemma 36.36.8. Hence we may choose a resolution as in Lemma 57.9.3. Say $\dim (X) = d$. Since $X \times X$ is smooth over $k$ it is regular. Hence $X \times X$ is a regular Noetherian scheme with $\dim (X \times X) = 2d$. The object

\[ K = (\mathcal{E}_{2d} \boxtimes \mathcal{G}_{2d} \to \ldots \to \mathcal{E}_0 \boxtimes \mathcal{G}_0) \]

of $D_{perf}(\mathcal{O}_{X \times X})$ has cohomology sheaves $\mathcal{O}_\Delta $ in degree $0$ and $\mathop{\mathrm{Ker}}(\mathcal{E}_{2d} \boxtimes \mathcal{G}_{2d} \to \mathcal{E}_{2d-1} \boxtimes \mathcal{G}_{2d-1})$ in degree $-2d$ and zero in all other degrees. Hence by Lemma 57.9.5 we see that $\mathcal{O}_\Delta $ is a summand of $K$ in $D_{perf}(\mathcal{O}_{X \times X})$. Clearly, the object $K$ is in

\[ \left\langle \bigoplus \nolimits _{i = 0, \ldots , 2d} \mathcal{E}_ i \boxtimes \mathcal{G}_ i \right\rangle \subset \left\langle \left(\bigoplus \nolimits _{i = 0, \ldots , 2d} \mathcal{E}_ i\right) \boxtimes \left(\bigoplus \nolimits _{i = 0, \ldots , 2d} \mathcal{G}_ i\right) \right\rangle \]

which finishes the proof. (The reader may consult Derived Categories, Lemmas 13.36.1 and 13.35.7 to see that our object is contained in this category.) $\square$

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