Lemma 17.21.1. In the situation described above. The sheaf $\wedge ^ n\mathcal{F}$ is the sheafification of the presheaf

$U \longmapsto \wedge ^ n_{\mathcal{O}_ X(U)}(\mathcal{F}(U)).$

See Algebra, Section 10.13. Similarly, the sheaf $\text{Sym}^ n\mathcal{F}$ is the sheafification of the presheaf

$U \longmapsto \text{Sym}^ n_{\mathcal{O}_ X(U)}(\mathcal{F}(U)).$

Proof. Omitted. It may be more efficient to define $\text{Sym}(\mathcal{F})$ and $\wedge (\mathcal{F})$ in this way instead of the method given above. $\square$

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