Lemma 17.21.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F} \to 0$ be an exact sequence of sheaves of $\mathcal{O}_ X$-modules. For each $n \geq 1$ there is an exact sequence

$\mathcal{F}_2 \otimes _{\mathcal{O}_ X} \text{Sym}^{n - 1}(\mathcal{F}_1) \to \text{Sym}^ n(\mathcal{F}_1) \to \text{Sym}^ n(\mathcal{F}) \to 0$

and similarly an exact sequence

$\mathcal{F}_2 \otimes _{\mathcal{O}_ X} \wedge ^{n - 1}(\mathcal{F}_1) \to \wedge ^ n(\mathcal{F}_1) \to \wedge ^ n(\mathcal{F}) \to 0$

Proof. See Algebra, Lemma 10.13.2. $\square$

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