Lemma 17.21.2 (01CH)—The Stacks project

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Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.21: Symmetric and exterior powers
Lemma 17.21.2 (cite)

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Lemma 17.21.2. In the situation described above. Let $x \in X$ . There are canonical isomorphisms of $\mathcal{O}_{X, x}$ -modules $\text{T}(\mathcal{F})_ x = \text{T}(\mathcal{F}_ x)$ , $\text{Sym}(\mathcal{F})_ x = \text{Sym}(\mathcal{F}_ x)$ , and $\wedge (\mathcal{F})_ x = \wedge (\mathcal{F}_ x)$ .

Proof. Clear from Lemma 17.21.1 above, and Algebra, Lemma 10.13.5. $\square$

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