The Stacks project

Lemma 70.2.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in |X|$. The following are equivalent

  1. for some étale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is weakly associated to $f^*\mathcal{F}$,

  2. for every étale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is weakly associated to $f^*\mathcal{F}$,

  3. the maximal ideal of $\mathcal{O}_{X, \overline{x}}$ is a weakly associated prime of the stalk $\mathcal{F}_{\overline{x}}$.

If $X$ is locally Noetherian, then these are also equivalent to

  1. for some étale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is associated to $f^*\mathcal{F}$,

  2. for every étale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is associated to $f^*\mathcal{F}$,

  3. the maximal ideal of $\mathcal{O}_{X, \overline{x}}$ is an associated prime of the stalk $\mathcal{F}_{\overline{x}}$.

Proof. Choose a scheme $U$ with a point $u$ and an étale morphism $f : U \to X$ mapping $u$ to $x$. Lift $\overline{x}$ to a geometric point of $U$ over $u$. Recall that $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}$ where the strict henselization is with respect to our chosen lift of $\overline{x}$, see Properties of Spaces, Lemma 65.22.1. Finally, we have

\[ \mathcal{F}_{\overline{x}} = (f^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} = (f^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{U, u}^{sh} \]

by Properties of Spaces, Lemma 65.29.4. Hence the equivalence of (1), (2), and (3) follows from More on Flatness, Lemma 38.2.9. If $X$ is locally Noetherian, then any $U$ as above is locally Noetherian, hence we see that (1), resp. (2) are equivalent to (4), resp. (5) by Divisors, Lemma 31.5.8. On the other hand, in the locally Noetherian case the local ring $\mathcal{O}_{X, \overline{x}}$ is Noetherian too (Properties of Spaces, Lemma 65.24.4). Hence the equivalence of (3) and (6) by the same lemma (or by Algebra, Lemma 10.66.9). $\square$


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