The Stacks project

Lemma 76.2.5. In Situation 76.2.1. Let $\overline{y}$ be a geometric point lying over $y$. Let $\mathcal{O} = \mathcal{O}_{Y, \overline{y}}$ be the étale local ring of $Y$ at $\overline{y}$. Denote $Y^{sh} = \mathop{\mathrm{Spec}}(\mathcal{O})$, $X^{sh} = X \times _ Y Y^{sh}$, and $\mathcal{F}^{sh}$ the pullback of $\mathcal{F}$ to $X^{sh}$. The following are equivalent

  1. there exists an impurity $(Y^{sh} \to Y, y' \leadsto \overline{y}, \xi )$ of $\mathcal{F}$ above $y$,

  2. every point of $\text{Ass}_{X^{sh}/Y^{sh}}(\mathcal{F}^{sh})$ specializes to a point of the closed fibre $X_{\overline{y}}$,

  3. there exists an impurity $(T \to Y, t' \leadsto t, \xi )$ of $\mathcal{F}$ above $y$ such that $(T, t) \to (Y, y)$ is an étale neighbourhood, and

  4. there exists an impurity $(T \to Y, t' \leadsto t, \xi )$ of $\mathcal{F}$ above $y$ such that $T \to Y$ is quasi-finite at $t$.

Proof. That parts (1) and (2) are equivalent is immediate from the definition.

Recall that $\mathcal{O} = \mathcal{O}_{Y, \overline{y}}$ is the filtered colimit of $\mathcal{O}(V)$ over the category of étale neighbourhoods $(V, \overline{v}) \to (Y, \overline{y})$ (Properties of Spaces, Lemma 65.19.3). Moreover, it suffices to consider affine étale neighbourhoods $V$. Hence $Y^{sh} = \mathop{\mathrm{Spec}}(\mathcal{O}) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Spec}}(\mathcal{O}(V)) = \mathop{\mathrm{lim}}\nolimits V$. Thus we see that (1) implies (3) by Lemma 76.2.3.

Since an étale morphism is locally quasi-finite (Morphisms of Spaces, Lemma 66.39.5) we see that (3) implies (4).

Finally, assume (4). After replacing $T$ by an open neighbourhood of $t$ we may assume $T \to Y$ is locally quasi-finite. By Lemma 76.2.4 we find an impurity $(T_1 \to Y^{sh}, t_1' \leadsto t_1, \xi _1)$ with $T_1 \to T \times _ Y Y^{sh}$ étale. Since an étale morphism is locally quasi-finite and using Morphisms of Spaces, Lemma 66.27.4 and Morphisms, Lemma 29.20.12 we see that $T_1 \to Y^{sh}$ is locally quasi-finite. As $\mathcal{O}$ is strictly henselian, we can apply More on Morphisms, Lemma 37.41.1 to see that after replacing $T_1$ by an open and closed neighbourhood of $t_1$ we may assume that $T_1 \to Y^{sh} = \mathop{\mathrm{Spec}}(\mathcal{O})$ is finite. Let $\theta \in |X^{sh}|$ be the image of $\xi _1$ and let $y' \in \mathop{\mathrm{Spec}}(\mathcal{O})$ be the image of $t_1'$. By Divisors on Spaces, Lemma 70.4.7 we see that $\theta \in \text{Ass}_{X^{sh}/Y^{sh}}(\mathcal{F}^{sh})$. Since $\pi : X_{T_1} \to X^{sh}$ is finite, it induces a closed map $|X_{T_1}| \to |X^{sh}|$. Hence the image of $\overline{\{ \xi _1\} }$ is $\overline{\{ \theta \} }$. It follows that $(Y^{sh} \to Y, y' \leadsto \overline{y}, \theta )$ is an impurity of $\mathcal{F}$ above $y$ and the proof is complete. $\square$

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