The Stacks project

Lemma 56.10.3. Let $R$ be a Noetherian ring. Let $X$ be a separated finite type scheme over $R$ which has the resolution property. Set $\mathcal{O}_\Delta = \Delta _*(\mathcal{O}_ X)$ where $\Delta : X \to X \times _ R X$ is the diagonal of $X/k$. There exists a resolution

\[ \ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \to \mathcal{O}_\Delta \to 0 \]

where each $\mathcal{E}_ i$ and $\mathcal{G}_ i$ is a finite locally free $\mathcal{O}_ X$-module.

Proof. Since $X$ is separated, the diagonal morphism $\Delta $ is a closed immersion and hence $\mathcal{O}_\Delta $ is a coherent $\mathcal{O}_{X \times _ R X}$-module (Cohomology of Schemes, Lemma 30.9.8). Thus the lemma follows immediately from Lemma 56.10.1. $\square$


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