The Stacks project

Lemma 38.16.6. Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$. Let $(S', s') \to (S, s)$ be a morphism of pointed schemes. Assume $S' \to S$ is flat at $s'$.

  1. If $\mathcal{F}_{S'}$ is pure along $X_{s'}$, then $\mathcal{F}$ is pure along $X_ s$.

  2. If $\mathcal{F}_{S'}$ is universally pure along $X_{s'}$, then $\mathcal{F}$ is universally pure along $X_ s$.

Proof. Let $(T \to S, t' \leadsto t, \xi )$ be an impurity of $\mathcal{F}$ above $s$. Set $T_1 = T \times _ S S'$, and let $t_1$ be the unique point of $T_1$ mapping to $t$ and $s'$. Since $T_1 \to T$ is flat at $t_1$, see Morphisms, Lemma 29.25.8, there exists a specialization $t'_1 \leadsto t_1$ lying over $t' \leadsto t$, see Algebra, Section 10.41. Choose a point $\xi _1 \in X_{t'_1}$ which corresponds to a generic point of $\mathop{\mathrm{Spec}}(\kappa (t'_1) \otimes _{\kappa (t')} \kappa (\xi ))$, see Schemes, Lemma 26.17.5. By Divisors, Lemma 31.7.3 we see that $\xi _1 \in \text{Ass}_{X_{T_1}/T_1}(\mathcal{F}_{T_1})$. As the Zariski closure of $\{ \xi _1\} $ in $X_{T_1}$ maps into the Zariski closure of $\{ \xi \} $ in $X_ T$ we conclude that this closure is disjoint from $X_{t_1}$. Hence $(T_1 \to S', t'_1 \leadsto t_1, \xi _1)$ is an impurity of $\mathcal{F}_{S'}$ above $s'$. In other words we have proved the contrapositive to part (2) of the lemma. Finally, if $(T, t) \to (S, s)$ is an elementary ├ętale neighbourhood, then $(T_1, t_1) \to (S', s')$ is an elementary ├ętale neighbourhood too, and in this way we see that (1) holds. $\square$


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Your email address will not be published.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 05J9. Beware of the difference between the letter 'O' and the digit '0'.