The Stacks project

Proof. Say $x' \leadsto t$ with $t \in X_ s$. Then we can find specializations $x' \leadsto x \leadsto t$ with $x$ corresponding to a generic point of an irreducible component of $\overline{\{ x'\} } \cap f^{-1}(\{ s\} )$. By assumption $\mathcal{F}$ is flat over $S$ at $x$. By More on Morphisms, Lemma 37.18.3 we see that $x \in \text{Ass}_{X/S}(\mathcal{F})$ as desired. $\square$

Proof. Since the morphism $S' \to S$ is étale, we see that if $\mathcal{F}$ is pure along $X_ s$, then $\mathcal{F} \times _ S S'$ is pure along $X_ s$, see Lemma 38.16.4. Since purity satisfies flat descent, see Lemma 38.16.6, we see that if $\mathcal{F} \times _ S S'$ is pure along $X_{s'}$, then $\mathcal{F}$ is pure along $X_ s$. Hence we may replace $S$ by $S'$ and assume that $S = S'$ so that $g : X' \to X$ is an étale morphism between schemes of finite type over $S$. Moreover, we may replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ and assume that $S$ is local.

First, assume that $\mathcal{F}$ is pure along $X_ s$. In this case every point of $\text{Ass}_{X/S}(\mathcal{F})$ specializes to a point of $X_ s$ by purity. Hence by Lemma 38.18.1 we see that every point of $\text{Ass}_{X/S}(\mathcal{F})$ specializes to a point of $\text{Ass}_{X_ s}(\mathcal{F}_ s)$. Thus every point of $\text{Ass}_{X/S}(\mathcal{F})$ is in the image of $g$ (as the image is open and contains $\text{Ass}_{X_ s}(\mathcal{F}_ s)$).

Conversely, assume that $g(X')$ contains $\text{Ass}_{X/S}(\mathcal{F})$. Let $S^ h = \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h)$ be the henselization of $S$ at $s$. Denote $g^ h : (X')^ h \to X^ h$ the base change of $g$ by $S^ h \to S$, and denote $\mathcal{F}^ h$ the pullback of $\mathcal{F}$ to $X^ h$. By Divisors, Lemma 31.7.3 and Remark 31.7.4 the relative assassin $\text{Ass}_{X^ h/S^ h}(\mathcal{F}^ h)$ is the inverse image of $\text{Ass}_{X/S}(\mathcal{F})$ via the projection $X^ h \to X$. As we have assumed that $g(X')$ contains $\text{Ass}_{X/S}(\mathcal{F})$ we conclude that the base change $g^ h((X')^ h) = g(X') \times _ S S^ h$ contains $\text{Ass}_{X^ h/S^ h}(\mathcal{F}^ h)$. In this way we reduce to the case where $S$ is the spectrum of a henselian local ring. Let $x \in \text{Ass}_{X/S}(\mathcal{F})$. To finish the proof of the lemma we have to show that $x$ specializes to a point of $X_ s$, see criterion (4) for purity in discussion following Definition 38.16.1. By assumption there exists a $x' \in X'$ such that $g(x') = x$. As $g : X' \to X$ is étale, we see that $x' \in \text{Ass}_{X'/S}(g^*\mathcal{F})$, see Lemma 38.2.8 (applied to the morphism of fibres $X'_ w \to X_ w$ where $w \in S$ is the image of $x'$). Since $g^*\mathcal{F}$ is pure along $X'_ s$ we see that $x' \leadsto y$ for some $y \in X'_ s$. Hence $x = g(x') \leadsto g(y)$ and $g(y) \in X_ s$ as desired. $\square$

Proof. We first make a preliminary remark. Suppose that $(S', s') \to (S, s)$ is an elementary étale neighbourhood. Denote $\mathcal{F}'$ the pullback of $\mathcal{F}$ to $X' = X \times _ S S'$. By the discussion following Definition 38.16.1 we see that $\mathcal{F}'$ is pure along $X'_{s'}$. Moreover, $\mathcal{F}'$ is flat over $S'$ along $X'_{s'}$. Then it suffices to prove that $\mathcal{F}'$ is universally pure along $X'_{s'}$. Namely, given any morphism $(T, t) \to (S, s)$ of pointed schemes the fibre product $(T', t') = (T \times _ S S', (t, s'))$ is flat over $(T, t)$ and hence if $\mathcal{F}_{T'}$ is pure along $X_{t'}$ then $\mathcal{F}_ T$ is pure along $X_ t$ by Lemma 38.16.6. Thus during the proof we may always replace $(s, S)$ by an elementary étale neighbourhood. We may also replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ due to the local nature of the problem.

Choose an elementary étale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram

such that $X' \to X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is étale, $X_ s = g((X')_{s'})$, the scheme $X'$ is affine, and such that $\Gamma (X', g^*\mathcal{F})$ is a free $\mathcal{O}_{S', s'}$-module, see Lemma 38.12.11. Note that $X' \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is of finite type (as a quasi-compact morphism which is the composition of an étale morphism and the base change of a finite type morphism). By our preliminary remarks in the first paragraph of the proof we may replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$. Hence we may assume there exists a commutative diagram

of schemes of finite type over $S$, where $g$ is étale, $X_ s \subset g(X')$, with $S$ local with closed point $s$, with $X'$ affine, and with $\Gamma (X', g^*\mathcal{F})$ a free $\Gamma (S, \mathcal{O}_ S)$-module. Note that in this case $g^*\mathcal{F}$ is universally pure over $S$, see Lemma 38.17.4.

In this situation we apply Lemma 38.18.2 to deduce that $\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')$ from our assumption that $\mathcal{F}$ is pure along $X_ s$ and flat over $S$ along $X_ s$. By Divisors, Lemma 31.7.3 and Remark 31.7.4 we see that for any morphism of pointed schemes $(T, t) \to (S, s)$ we have

Hence by Lemma 38.18.2 applied to the base change of our displayed diagram to $(T, t)$ we conclude that $\mathcal{F}_ T$ is pure along $X_ t$ as desired. $\square$

Proof. Immediate consequence of Lemma 38.18.3 and the definitions. $\square$

Proof. Let $U \subset X$ be the set of points where $\mathcal{F}$ is flat over $S$. By More on Morphisms, Theorem 37.15.1 this is an open subscheme of $X$. By assumption $X_ s \subset U$. As $X_ s$ is quasi-compact, we can find a quasi-compact open $U' \subset U$ with $X_ s \subset U'$. By Limits, Lemma 32.10.1 we can find an $i \in I$ and a morphism of finite presentation $f_ i : X_ i \to S_ i$ whose base change to $S$ is isomorphic to $f_ i$. Fix such a choice and set $X_{i'} = X_ i \times _{S_ i} S_{i'}$. Then $X = \mathop{\mathrm{lim}}\nolimits _{i'} X_{i'}$ with affine transition morphisms. By Limits, Lemma 32.10.2 we can, after possible increasing $i$ assume there exists a quasi-coherent $\mathcal{O}_{X_ i}$-module $\mathcal{F}_ i$ of finite presentation whose base change to $S$ is isomorphic to $\mathcal{F}$. By Limits, Lemma 32.4.11 after possibly increasing $i$ we may assume there exists an open $U'_ i \subset X_ i$ whose inverse image in $X$ is $U'$. Note that in particular $(X_ i)_{s_ i} \subset U'_ i$. By Limits, Lemma 32.10.4 (after increasing $i$ once more) we may assume that $\mathcal{F}_ i$ is flat on $U'_ i$. In particular we see that $\mathcal{F}_ i$ is flat along $(X_ i)_{s_ i}$.

Next, we use Lemma 38.12.5 to choose an elementary étale neighbourhood $(S_ i', s_ i') \to (S_ i, s_ i)$ and a commutative diagram of schemes

such that $g_ i$ is étale, $(X_ i)_{s_ i} \subset g_ i(X_ i')$, the schemes $X_ i'$, $S_ i'$ are affine, and such that $\Gamma (X_ i', g_ i^*\mathcal{F}_ i)$ is a projective $\Gamma (S_ i', \mathcal{O}_{S_ i'})$-module. Note that $g_ i^*\mathcal{F}_ i$ is universally pure over $S'_ i$, see Lemma 38.17.4. We may base change the diagram above to a diagram with morphisms $(S'_{i'}, s'_{i'}) \to (S_{i'}, s_{i'})$ and $g_{i'} : X'_{i'} \to X_{i'}$ over $S_{i'}$ for any $i' \geq i$ and we may base change the diagram to a diagram with morphisms $(S', s') \to (S, s)$ and $g : X' \to X$ over $S$.

At this point we can use our criterion for purity. Set $W'_ i \subset X_ i \times _{S_ i} S'_ i$ equal to the image of the étale morphism $X'_ i \to X_ i \times _{S_ i} S'_ i$. For every $i' \geq i$ we have similarly the image $W'_{i'} \subset X_{i'} \times _{S_{i'}} S'_{i'}$ and we have the image $W' \subset X \times _ S S'$. Taking images commutes with base change, hence $W'_{i'} = W'_ i \times _{S'_ i} S'_{i'}$ and $W' = W_ i \times _{S'_ i} S'$. Because $\mathcal{F}$ is pure along $X_ s$ the Lemma 38.18.2 implies that

By More on Morphisms, Lemma 37.25.5 we see that

are locally constructible subsets of $S'$ and $S'_{i'}$. By More on Morphisms, Lemma 37.25.4 we see that $E_{i'}$ is the inverse image of $E_ i$ under the morphism $S'_{i'} \to S'_ i$ and that $E$ is the inverse image of $E_ i$ under the morphism $S' \to S'_ i$. Thus Equation (38.18.5.1) is equivalent to the assertion that $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ maps into $E_ i$. As $\mathcal{O}_{S', s'} = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \mathcal{O}_{S'_{i'}, s'_{i'}}$ we see that $\mathop{\mathrm{Spec}}(\mathcal{O}_{S'_{i'}, s'_{i'}})$ maps into $E_ i$ for some $i' \geq i$, see Limits, Lemma 32.4.10. Then, applying Lemma 38.18.2 to the situation over $S_{i'}$, we conclude that $\mathcal{F}_{i'}$ is pure along $(X_{i'})_{s_{i'}}$. $\square$

Proof. Let $s \in U$. Using Lemma 38.12.5 we can find an elementary étale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram

such that $g$ is étale, $X_ s \subset g(X')$, the schemes $X'$, $S'$ are affine, and such that $\Gamma (X', g^*\mathcal{F})$ is a projective $\Gamma (S', \mathcal{O}_{S'})$-module. Note that $g^*\mathcal{F}$ is universally pure over $S'$, see Lemma 38.17.4. Set $W' \subset X \times _ S S'$ equal to the image of the étale morphism $X' \to X \times _ S S'$. Note that $W$ is open and quasi-compact over $S'$. Set

By More on Morphisms, Lemma 37.25.5 $E$ is a constructible subset of $S'$. By Lemma 38.18.2 we see that $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \subset E$. By Morphisms, Lemma 29.22.4 we see that $E$ contains an open neighbourhood $V'$ of $s'$. Applying Lemma 38.18.2 once more we see that for any point $s_1$ in the image of $V'$ in $S$ the sheaf $\mathcal{F}$ is pure along $X_{s_1}$. Since $S' \to S$ is étale the image of $V'$ in $S$ is open and we win. $\square$