Lemma 38.18.5. Let $I$ be a directed set. Let $(S_ i, g_{ii'})$ be an inverse system of affine schemes over $I$. Set $S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$ and $s \in S$. Denote $g_ i : S \to S_ i$ the projections and set $s_ i = g_ i(s)$. Suppose that $f : X \to S$ is a morphism of finite presentation, $\mathcal{F}$ a quasi-coherent $\mathcal{O}_ X$-module of finite presentation which is pure along $X_ s$ and flat over $S$ at all points of $X_ s$. Then there exists an $i \in I$, a morphism of finite presentation $X_ i \to S_ i$, a quasi-coherent $\mathcal{O}_{X_ i}$-module $\mathcal{F}_ i$ of finite presentation which is pure along $(X_ i)_{s_ i}$ and flat over $S_ i$ at all points of $(X_ i)_{s_ i}$ such that $X \cong X_ i \times _{S_ i} S$ and such that the pullback of $\mathcal{F}_ i$ to $X$ is isomorphic to $\mathcal{F}$.

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

$\xymatrix{ X_ i \ar[d] & X_ i' \ar[l]^{g_ i} \ar[d] \\ S_ i & S_ i' \ar[l] }$

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

38.18.5.1
$$\label{flat-equation-inclusion} f^{-1}(\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})) \cap \text{Ass}_{X \times _ S S'/S'}(\mathcal{F} \times _ S S') \subset W'$$

By More on Morphisms, Lemma 37.25.5 we see that

$E = \{ t \in S' \mid \text{Ass}_{X_ t}(\mathcal{F}_ t) \subset W' \} \quad \text{and}\quad E_{i'} = \{ t \in S'_{i'} \mid \text{Ass}_{X_ t}(\mathcal{F}_{i', t}) \subset W'_{i'} \}$

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 $(X_{i'})_{s_{i'}}$. $\square$

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