Lemma 38.2.8 (05FP)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.2: Lemmas on étale localization
Lemma 38.2.8 (cite)

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Lemma 38.2.8. Let $h : U \to S$ be an étale morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ S$ -module. Let $u \in U$ be a point with image $s \in S$ . Then

$u \in \text{WeakAss}(h^*\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}(\mathcal{G})$

Proof. After replacing $S$ and $U$ by affine neighbourhoods of $s$ and $u$ we may assume that $g$ is a standard étale morphism of affines, see Morphisms, Lemma 29.36.14. Thus we may assume $S = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(A[x, 1/g]/(f))$ , where $f$ is monic and $f'$ is invertible in $A[x, 1/g]$ . Note that $A[x, 1/g]/(f) = (A[x]/(f))_ g$ is also the localization of the finite free $A$ -algebra $A[x]/(f)$ . Hence we may think of $U$ as an open subscheme of the scheme $T = \mathop{\mathrm{Spec}}(A[x]/(f))$ which is finite locally free over $S$ . This reduces us to Lemma 38.2.7 above. $\square$

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