Lemma 38.17.2. Let $f : X \to S$ be a separated, finite type morphism of schemes. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module. Assume that $\text{Supp}(\mathcal{F}_ s)$ is finite for every $s \in S$. Then the following are equivalent

1. $\mathcal{F}$ is pure relative to $S$,

2. the scheme theoretic support of $\mathcal{F}$ is finite over $S$, and

3. $\mathcal{F}$ is universally pure relative to $S$.

In particular, given a quasi-finite separated morphism $X \to S$ we see that $X$ is pure relative to $S$ if and only if $X \to S$ is finite.

Proof. Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$, see Morphisms, Definition 29.5.5. Then $Z \to S$ is a separated, finite type morphism of schemes with finite fibres. Hence it is separated and quasi-finite, see Morphisms, Lemma 29.20.10. By Lemma 38.16.7 it suffices to prove the lemma for $Z \to S$ and the sheaf $\mathcal{F}$ viewed as a finite type quasi-coherent module on $Z$. Hence we may assume that $X \to S$ is separated and quasi-finite and that $\text{Supp}(\mathcal{F}) = X$.

It follows from Lemma 38.17.1 and Morphisms, Lemma 29.44.11 that (2) implies (3). Trivially (3) implies (1). Assume (1) holds. We will prove that (2) holds. It is clear that we may assume $S$ is affine. By More on Morphisms, Lemma 37.43.3 we can find a diagram

$\xymatrix{ X \ar[rd]_ f \ar[rr]_ j & & T \ar[ld]^\pi \\ & S & }$

with $\pi$ finite and $j$ a quasi-compact open immersion. If we show that $j$ is closed, then $j$ is a closed immersion and we conclude that $f = \pi \circ j$ is finite. To show that $j$ is closed it suffices to show that specializations lift along $j$, see Schemes, Lemma 26.19.8. Let $x \in X$, set $t' = j(x)$ and let $t' \leadsto t$ be a specialization. We have to show $t \in j(X)$. Set $s' = f(x)$ and $s = \pi (t)$ so $s' \leadsto s$. By More on Morphisms, Lemma 37.41.4 we can find an elementary étale neighbourhood $(U, u) \to (S, s)$ and a decomposition

$T_ U = T \times _ S U = V \amalg W$

into open and closed subschemes, such that $V \to U$ is finite and there exists a unique point $v$ of $V$ mapping to $u$, and such that $v$ maps to $t$ in $T$. As $V \to T$ is étale, we can lift generalizations, see Morphisms, Lemmas 29.25.9 and 29.36.12. Hence there exists a specialization $v' \leadsto v$ such that $v'$ maps to $t' \in T$. In particular we see that $v' \in X_ U \subset T_ U$. Denote $u' \in U$ the image of $t'$. Note that $v' \in \text{Ass}_{X_ U/U}(\mathcal{F})$ because $X_{u'}$ is a finite discrete set and $X_{u'} = \text{Supp}(\mathcal{F}_{u'})$. As $\mathcal{F}$ is pure relative to $S$ we see that $v'$ must specialize to a point in $X_ u$. Since $v$ is the only point of $V$ lying over $u$ (and since no point of $W$ can be a specialization of $v'$) we see that $v \in X_ u$. Hence $t \in X$. $\square$

## 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.

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 05K4. Beware of the difference between the letter 'O' and the digit '0'.