The Stacks project

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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