The Stacks project

Proof. Suppose that we have a diagram as in (1). Then the equivalence of the conditions in More on Flatness, Lemma 38.20.9 shows that (1) and (3) are equivalent. But condition (3) is inherited by $\varphi ^*\mathcal{F}$ for any $U \to V$ as in (2). Whence we see that (3) implies (2) by the result for schemes again. The result for schemes also implies the statement on base change. $\square$

Proof. Proof of (1). Suppose that $\{ T_ i \to T\} $ is an fpqc covering of a scheme $T$ over $Y$. We have to show that if $F_ n(T_ i)$ is nonempty for all $i$, then $F_ n(T)$ is nonempty. Choose a diagram as in part (1) of Lemma 77.11.2. Denote $F'_ n$ the corresponding functor for $\varphi ^*\mathcal{F}$ and the morphism $U \to V$. By More on Flatness, Lemma 38.20.12 we have the sheaf property for $F'_ n$. Thus we get the sheaf property for $F_ n$ because for $T \to Y$ we have $F_ n(T) = F'_ n(V \times _ Y T)$ by Lemma 77.11.2 and because $\{ V \times _ Y T_ i \to V \times _ Y T\} $ is an fpqc covering.

Proof of (2). Suppose that $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ is a filtered limit of affine schemes $T_ i$ over $Y$ and assume that $F_ n(T)$ is nonempty. We have to show that $F_ n(T_ i)$ is nonempty for some $i$. Choose a diagram as in part (1) of Lemma 77.11.2. Fix $i \in I$ and choose an affine open $W_ i \subset V \times _ Y T_ i$ mapping surjectively onto $T_ i$. For $i' \geq i$ let $W_{i'}$ be the inverse image of $W_ i$ in $V \times _ Y T_{i'}$ and let $W \subset V \times _ Y T$ be the inverse image of $W_ i$. Then $W = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} W_ i$ is a filtered limit of affine schemes over $V$. By Lemma 77.11.2 again it suffices to show that $F'_ n(W_{i'})$ is nonempty for some $i' \geq i$. But we know that $F'_ n(W)$ is nonempty because of our assumption that $F_ n(T) = F'_ n(V \times _ Y T)$ is nonempty. Thus we can apply More on Flatness, Lemma 38.20.12 to conclude. $\square$

Proof. Choose $U \to X$, $V \to Y$, $U \to V$ as in part (1) of Lemma 77.11.2. Choose a surjective étale morphism $U' \to U \times _ X X'$ where $U'$ is a scheme. Then we have the lemma for the two functors $F_{U, n}$ and $F_{U', n}$ determined by $U' \to U$ and $\mathcal{F}|_ U$ over $V$, see More on Flatness, Lemma 38.27.2. On the other hand, Lemma 77.11.2 tells us that given $T \to Y$ we have $F_ n(T) = F_{U, n}(V \times _ Y T)$ and $F'_ n(T) = F_{U', n}(V \times _ Y T)$. This proves the lemma. $\square$

Proof. The functor $F_ n$ is a sheaf for the fppf topology by Lemma 77.11.5. Since $F_ n \to Y$ is a monomorphism of sheaves on $(\mathit{Sch}/S)_{fppf}$ we see that $\Delta : F_ n \to F_ n \times F_ n$ is the pullback of the diagonal $\Delta _ Y : Y \to Y \times _ S Y$. Hence the representability of $\Delta _ Y$ implies the same thing for $F_ n$. Therefore it suffices to prove that there exists a scheme $W$ over $S$ and a surjective étale morphism $W \to F_ n$.

To construct $W \to F_ n$ choose an étale covering $\{ Y_ i \to Y\} $ with $Y_ i$ a scheme. Let $X_ i = X \times _ Y Y_ i$ and let $\mathcal{F}_ i$ be the pullback of $\mathcal{F}$ to $X_ i$. Then $\mathcal{F}_ i$ is pure relative to $Y_ i$ either by definition or by Lemma 77.3.3. The other assumptions of the theorem are preserved as well. Finally, the restriction of $F_ n$ to $Y_ i$ is the functor $F_ n$ corresponding to $X_ i \to Y_ i$ and $\mathcal{F}_ i$. Hence it suffices to show: Given $\mathcal{F}$ and $f : X \to Y$ as in the statement of the theorem where $Y$ is a scheme, the functor $F_ n$ is representable by a scheme $Z_ n$ and $Z_ n \to Y$ is a monomorphism of finite presentation.

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 $Y$.

In particular the situation is local for the Zariski topology on $Y$ and we may assume that $Y$ 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 $Z_{n + 1} \to Y$ of finite presentation. Consider the base change $X_{n + 1} = Z_{n + 1} \times _ Y X$ and the pullback $\mathcal{F}_{n + 1}$ of $\mathcal{F}$ to $X_{n + 1}$. The morphism $Z_{n + 1} \to Y$ is quasi-finite as it is a monomorphism of finite presentation, hence Lemma 77.3.3 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 $Y_{n + 1} = Y$, i.e., we may assume that $\mathcal{F}$ is flat in dimensions $\geq n + 1$ over $Y$.

Fix $n$ and assume $\mathcal{F}$ is flat in dimensions $\geq n + 1$ over the affine scheme $Y$. 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 $Y$ it suffices to show that for every $y \in Y$ there exists an étale neighbourhood $(Y', y') \to (Y, y)$ such that the result holds after base change to $Y'$. Thus by Lemma 77.4.1 we may assume there exist étale morphisms $h_ j : W_ j \to X$, $j = 1, \ldots , m$ such that for each $j$ there exists a complete dévissage of $\mathcal{F}_ j/W_ j/Y$ over $y$, where $\mathcal{F}_ j$ is the pullback of $\mathcal{F}$ to $W_ j$ and such that $|X_ y| \subset \bigcup h_ j(W_ j)$. Since $h_ j$ is étale, by Lemma 77.11.2 the sheaves $\mathcal{F}_ j$ are still flat over in dimensions $\geq n + 1$ over $Y$. Set $W = \bigcup h_ j(W_ j)$, which is a quasi-compact open of $X$. As $\mathcal{F}$ is pure along $X_ y$ we see that

contains all generalizations of $y$. By Divisors on Spaces, Lemma 71.4.10 $E$ is a constructible subset of $Y$. We have seen that $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \subset E$. By Morphisms, Lemma 29.22.4 we see that $E$ contains an open neighbourhood of $y$. Hence after shrinking $Y$ we may assume that $E = Y$. It follows from Lemma 77.11.6 that it suffices to prove the lemma for the functor $F_ n$ associated to $X = \coprod W_ j$ and $\mathcal{F} = \coprod \mathcal{F}_ j$. If $F_{j, n}$ denotes the functor for $W_ j \to Y$ and the sheaf $\mathcal{F}_ j$ we see that $F_ n = \prod F_{j, n}$. Hence it suffices to prove each $F_{j, n}$ is representable by some monomorphism $Z_{j, n} \to Y$ of finite presentation, since then

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

Proof. These statements follow immediately from Theorem 77.11.7 applied to $F_0 = F_{flat}$ and the fact that if $f$ is proper then $\mathcal{F}$ is automatically pure over the base, see Lemma 77.3.6. $\square$