The Stacks project

Lemma 38.2.9. Let $S$ be a scheme and $s \in S$ a point. Denote $\mathcal{O}_{S, s}^ h$ (resp. $\mathcal{O}_{S, s}^{sh}$) the henselization (resp. strict henselization), see Algebra, Definition 10.155.3. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module. The following are equivalent

  1. $s$ is a weakly associated point of $\mathcal{F}$,

  2. $\mathfrak m_ s$ is a weakly associated prime of $\mathcal{F}_ s$,

  3. $\mathfrak m_ s^ h$ is a weakly associated prime of $\mathcal{F}_ s \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^ h$, and

  4. $\mathfrak m_ s^{sh}$ is a weakly associated prime of $\mathcal{F}_ s \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^{sh}$.

Proof. The equivalence of (1) and (2) is the definition, see Divisors, Definition 31.5.1. The implications (2) $\Rightarrow $ (3) $\Rightarrow $ (4) follows from Divisors, Lemma 31.6.4 applied to the flat (More on Algebra, Lemma 15.45.1) morphisms

\[ \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^{sh}) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \]

and the closed points. To prove (4) $\Rightarrow $ (2) we may replace $S$ by an affine neighbourhood. Suppose that $x \in \mathcal{F}_ s \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^{sh}$ is an element whose annihilator has radical equal to $\mathfrak m_ s^{sh}$. (See Algebra, Lemma 10.66.2.) Since $\mathcal{O}_{S, s}^{sh}$ is equal to the limit of $\mathcal{O}_{U, u}$ over étale neighbourhoods $f : (U, u) \to (S, s)$ by Algebra, Lemma 10.155.11 we may assume that $x$ is the image of some $x' \in \mathcal{F}_ s \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{U, u}$. The local ring map $\mathcal{O}_{U, u} \to \mathcal{O}_{S, s}^{sh}$ is faithfully flat (as it is the strict henselization), hence universally injective (Algebra, Lemma 10.82.11). It follows that the annihilator of $x'$ is the inverse image of the annihilator of $x$. Hence the radical of this annihilator is equal to $\mathfrak m_ u$. Thus $u$ is a weakly associated point of $f^*\mathcal{F}$. By Lemma 38.2.8 we see that $s$ is a weakly associated point of $\mathcal{F}$. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CTU. Beware of the difference between the letter 'O' and the digit '0'.