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38.17 Examples of relatively pure sheaves

Here are some example cases where it is possible to see what purity means.

Lemma 38.17.1. 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.

  1. If the support of $\mathcal{F}$ is proper over $S$, then $\mathcal{F}$ is universally pure relative to $S$.

  2. If $f$ is proper, then $\mathcal{F}$ is universally pure relative to $S$.

  3. If $f$ is proper, then $X$ is universally pure relative to $S$.

Proof. First we reduce (1) to (2). Namely, let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$. Let $i : Z \to X$ be the corresponding closed immersion and write $\mathcal{F} = i_*\mathcal{G}$ for some finite type quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{G}$, see Morphisms, Section 29.5. In case (1) $Z \to S$ is proper by assumption. Thus by Lemma 38.16.7 case (1) reduces to case (2).

Assume $f$ is proper. Let $(g : T \to S, t' \leadsto t, \xi )$ be an impurity of $\mathcal{F}$ above $s \in S$. Since $f$ is proper, it is universally closed. Hence $f_ T : X_ T \to T$ is closed. Since $f_ T(\xi ) = t'$ this implies that $t \in f(\overline{\{ \xi \} })$ which is a contradiction. $\square$

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 $t \in X$. $\square$

Lemma 38.17.3. Let $f : X \to S$ be a finite type, flat morphism of schemes with geometrically integral fibres. Then $X$ is universally pure over $S$.

Proof. Let $\xi \in X$ with $s' = f(\xi )$ and $s' \leadsto s$ a specialization of $S$. If $\xi $ is an associated point of $X_{s'}$, then $\xi $ is the unique generic point because $X_{s'}$ is an integral scheme. Let $\xi _0$ be the unique generic point of $X_ s$. As $X \to S$ is flat we can lift $s' \leadsto s$ to a specialization $\xi ' \leadsto \xi _0$ in $X$, see Morphisms, Lemma 29.25.9. The $\xi \leadsto \xi '$ because $\xi $ is the generic point of $X_{s'}$ hence $\xi \leadsto \xi _0$. This means that $(\text{id}_ S, s' \to s, \xi )$ is not an impurity of $\mathcal{O}_ X$ above $s$. Since the assumption that $f$ is finite type, flat with geometrically integral fibres is preserved under base change, we see that there doesn't exist an impurity after any base change. In this way we see that $X$ is universally $S$-pure. $\square$

Lemma 38.17.4. Let $f : X \to S$ be a finite type, affine morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module such that $f_*\mathcal{F}$ is locally projective on $S$, see Properties, Definition 28.21.1. Then $\mathcal{F}$ is universally pure over $S$.

Proof. After reducing to the case where $S$ is the spectrum of a henselian local ring this follows from Lemma 38.14.1. $\square$

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