The Stacks project

Theorem 38.27.5. In Situation 38.20.11. Assume moreover that $f$ is of finite presentation, that $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation, and that $\mathcal{F}$ is pure relative to $S$. Then $F_ n$ is representable by a monomorphism $Z_ n \to S$ of finite presentation.

Proof. The functor $F_ n$ is a sheaf for the fppf topology by Lemma 38.20.12. Observe that a monomorphism of finite presentation is separated and quasi-finite (Morphisms, Lemma 29.20.15). Hence combining Descent, Lemma 35.39.1, More on Morphisms, Lemma 37.57.1 , and Descent, Lemmas 35.23.31 and 35.23.13 we see that the question is local for the étale topology on $S$.

In particular the situation is local for the Zariski topology on $S$ and we may assume that $S$ is affine. In this case the dimension of the fibres of $f$ is bounded above, hence we see that $F_ n$ is representable for $n$ large enough. Thus we may use descending induction on $n$. Suppose that we know $F_{n + 1}$ is representable by a monomorphism $Z_{n + 1} \to S$ of finite presentation. Consider the base change $X_{n + 1} = Z_{n + 1} \times _ S X$ and the pullback $\mathcal{F}_{n + 1}$ of $\mathcal{F}$ to $X_{n + 1}$. The morphism $Z_{n + 1} \to S$ is quasi-finite as it is a monomorphism of finite presentation, hence Lemma 38.16.4 implies that $\mathcal{F}_{n + 1}$ is pure relative to $Z_{n + 1}$. Since $F_ n$ is a subfunctor of $F_{n + 1}$ we conclude that in order to prove the result for $F_ n$ it suffices to prove the result for the corresponding functor for the situation $\mathcal{F}_{n + 1}/X_{n + 1}/Z_{n + 1}$. In this way we reduce to proving the result for $F_ n$ in case $S_{n + 1} = S$, i.e., we may assume that $\mathcal{F}$ is flat in dimensions $\geq n + 1$ over $S$.

Fix $n$ and assume $\mathcal{F}$ is flat in dimensions $\geq n + 1$ over $S$. To finish the proof we have to show that $F_ n$ is representable by a monomorphism $Z_ n \to S$ of finite presentation. Since the question is local in the étale topology on $S$ it suffices to show that for every $s \in S$ there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ such that the result holds after base change to $S'$. Thus by Lemma 38.5.8 we may assume there exist étale morphisms $h_ j : Y_ j \to X$, $j = 1, \ldots , m$ such that for each $j$ there exists a complete dévissage of $\mathcal{F}_ j/Y_ j/S$ over $s$, where $\mathcal{F}_ j$ is the pullback of $\mathcal{F}$ to $Y_ j$ and such that $X_ s \subset \bigcup h_ j(Y_ j)$. Note that by Lemma 38.27.2 the sheaves $\mathcal{F}_ j$ are still flat over in dimensions $\geq n + 1$ over $S$. Set $W = \bigcup h_ j(Y_ j)$, which is a quasi-compact open of $X$. As $\mathcal{F}$ is pure along $X_ s$ we see that

\[ E = \{ t \in S \mid \text{Ass}_{X_ t}(\mathcal{F}_ t) \subset W \} . \]

contains all generalizations of $s$. By More on Morphisms, Lemma 37.25.5 $E$ is a constructible subset of $S$. We have seen that $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \subset E$. By Morphisms, Lemma 29.22.4 we see that $E$ contains an open neighbourhood of $s$. Hence after shrinking $S$ we may assume that $E = S$. It follows from Lemma 38.27.2 that it suffices to prove the lemma for the functor $F_ n$ associated to $X = \coprod Y_ j$ and $\mathcal{F} = \coprod \mathcal{F}_ j$. If $F_{j, n}$ denotes the functor for $Y_ j \to S$ and the sheaf $\mathcal{F}_ i$ we see that $F_ n = \prod F_{j, n}$. Hence it suffices to prove each $F_{j, n}$ is representable by some monomorphism $Z_{j, n} \to S$ of finite presentation, since then

\[ Z_ n = Z_{1, n} \times _ S \ldots \times _ S Z_{m, n} \]

Thus we have reduced the theorem to the special case handled in Lemma 38.27.4. $\square$

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