Lemma 38.16.5. 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 $s \in S$. If $\mathcal{O}_{S, s}$ is Noetherian then $\mathcal{F}$ is pure along $X_ s$ if and only if $\mathcal{F}$ is universally pure along $X_ s$.

**Proof.**
First we may replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$, i.e., we may assume that $S$ is Noetherian. Next, use Lemma 38.15.6 and characterization (2) of purity given in discussion following Definition 38.16.1 to conclude.
$\square$

## 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Your email address will not be published.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.

## Comments (0)