The Stacks project

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 ├ętale 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 ├ętale, 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 $t \in X$. $\square$

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