## 38.20 Flattening functors

Let $S$ be a scheme. Recall that a functor $F : (\mathit{Sch}/S)^{opp} \to \textit{Sets}$ is called limit preserving if for every directed inverse system $\{ T_ i\} _{i \in I}$ of affine schemes with limit $T$ we have $F(T) = \mathop{\mathrm{colim}}\nolimits _ i F(T_ i)$.

Situation 38.20.1. Let $f : X \to S$ be a morphism of schemes. Let $u : \mathcal{F} \to \mathcal{G}$ be a homomorphism of quasi-coherent $\mathcal{O}_ X$-modules. For any scheme $T$ over $S$ we will denote $u_ T : \mathcal{F}_ T \to \mathcal{G}_ T$ the base change of $u$ to $T$, in other words, $u_ T$ is the pullback of $u$ via the projection morphism $X_ T = X \times _ S T \to X$. In this situation we can consider the functor

38.20.1.1
\begin{equation} \label{flat-equation-iso} F_{iso} : (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\}
& \text{if}
& u_ T \text{ is an isomorphism},
\\ \emptyset
& \text{else.}
& \end{matrix} \right. \end{equation}

There are variants $F_{inj}$, $F_{surj}$, $F_{zero}$ where we ask that $u_ T$ is injective, surjective, or zero.

Lemma 38.20.2. In Situation 38.20.1.

Each of the functors $F_{iso}$, $F_{inj}$, $F_{surj}$, $F_{zero}$ satisfies the sheaf property for the fpqc topology.

If $f$ is quasi-compact and $\mathcal{G}$ is of finite type, then $F_{surj}$ is limit preserving.

If $f$ is quasi-compact and $\mathcal{F}$ of finite type, then $F_{zero}$ is limit preserving.

If $f$ is quasi-compact, $\mathcal{F}$ is of finite type, and $\mathcal{G}$ is of finite presentation, then $F_{iso}$ is limit preserving.

**Proof.**
Let $\{ T_ i \to T\} _{i \in I}$ be an fpqc covering of schemes over $S$. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and $u_ i = u_{T_ i}$. Note that $\{ X_ i \to X_ T\} _{i \in I}$ is an fpqc covering of $X_ T$, see Topologies, Lemma 34.9.7. In particular, for every $x \in X_ T$ there exists an $i \in I$ and an $x_ i \in X_ i$ mapping to $x$. Since $\mathcal{O}_{X_ T, x} \to \mathcal{O}_{X_ i, x_ i}$ is flat, hence faithfully flat (see Algebra, Lemma 10.39.17) we conclude that $(u_ i)_{x_ i}$ is injective, surjective, bijective, or zero if and only if $(u_ T)_ x$ is injective, surjective, bijective, or zero. Whence part (1) of the lemma.

Proof of (2). Assume $f$ quasi-compact and $\mathcal{G}$ of finite type. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ be a directed limit of affine $S$-schemes and assume that $u_ T$ is surjective. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and $u_ i = u_{T_ i} : \mathcal{F}_ i = \mathcal{F}_{T_ i} \to \mathcal{G}_ i = \mathcal{G}_{T_ i}$. To prove part (2) we have to show that $u_ i$ is surjective for some $i$. Pick $i_0 \in I$ and replace $I$ by $\{ i \mid i \geq i_0\} $. Since $f$ is quasi-compact the scheme $X_{i_0}$ is quasi-compact. Hence we may choose affine opens $W_1, \ldots , W_ m \subset X$ and an affine open covering $X_{i_0} = U_{1, i_0} \cup \ldots \cup U_{m, i_0}$ such that $U_{j, i_0}$ maps into $W_ j$ under the projection morphism $X_{i_0} \to X$. For any $i \in I$ let $U_{j, i}$ be the inverse image of $U_{j, i_0}$. Setting $U_ j = \mathop{\mathrm{lim}}\nolimits _ i U_{j, i}$ we see that $X_ T = U_1 \cup \ldots \cup U_ m$ is an affine open covering of $X_ T$. Now it suffices to show, for a given $j \in \{ 1, \ldots , m\} $ that $u_ i|_{U_{j, i}}$ is surjective for some $i = i(j) \in I$. Using Properties, Lemma 28.16.1 this translates into the following algebra problem: Let $A$ be a ring and let $u : M \to N$ be an $A$-module map. Suppose that $R = \mathop{\mathrm{colim}}\nolimits _{i \in I} R_ i$ is a directed colimit of $A$-algebras. If $N$ is a finite $A$-module and if $u \otimes 1 : M \otimes _ A R \to N \otimes _ A R$ is surjective, then for some $i$ the map $u \otimes 1 : M \otimes _ A R_ i \to N \otimes _ A R_ i$ is surjective. This is Algebra, Lemma 10.127.5 part (2).

Proof of (3). Exactly the same arguments as given in the proof of (2) reduces this to the following algebra problem: Let $A$ be a ring and let $u : M \to N$ be an $A$-module map. Suppose that $R = \mathop{\mathrm{colim}}\nolimits _{i \in I} R_ i$ is a directed colimit of $A$-algebras. If $M$ is a finite $A$-module and if $u \otimes 1 : M \otimes _ A R \to N \otimes _ A R$ is zero, then for some $i$ the map $u \otimes 1 : M \otimes _ A R_ i \to N \otimes _ A R_ i$ is zero. This is Algebra, Lemma 10.127.5 part (1).

Proof of (4). Assume $f$ quasi-compact and $\mathcal{F}, \mathcal{G}$ of finite presentation. Arguing in exactly the same manner as in the previous paragraph (using in addition also Properties, Lemma 28.16.2) part (3) translates into the following algebra statement: Let $A$ be a ring and let $u : M \to N$ be an $A$-module map. Suppose that $R = \mathop{\mathrm{colim}}\nolimits _{i \in I} R_ i$ is a directed colimit of $A$-algebras. Assume $M$ is a finite $A$-module, $N$ is a finitely presented $A$-module, and $u \otimes 1 : M \otimes _ A R \to N \otimes _ A R$ is an isomorphism. Then for some $i$ the map $u \otimes 1 : M \otimes _ A R_ i \to N \otimes _ A R_ i$ is an isomorphism. This is Algebra, Lemma 10.127.5 part (3).
$\square$

Situation 38.20.3. Let $(A, \mathfrak m_ A)$ be a local ring. Denote $\mathcal{C}$ the category whose objects are $A$-algebras $A'$ which are local rings such that the algebra structure $A \to A'$ is a local homomorphism of local rings. A morphism between objects $A', A''$ of $\mathcal{C}$ is a local homomorphism $A' \to A''$ of $A$-algebras. Let $A \to B$ be a local ring map of local rings and let $M$ be a $B$-module. If $A'$ is an object of $\mathcal{C}$ we set $B' = B \otimes _ A A'$ and we set $M' = M \otimes _ A A'$ as a $B'$-module. Given $A' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, consider the condition

38.20.3.1
\begin{equation} \label{flat-equation-flat-at-primes} \forall \mathfrak q \in V(\mathfrak m_{A'}B' + \mathfrak m_ B B') \subset \mathop{\mathrm{Spec}}(B') : M'_{\mathfrak q}\text{ is flat over }A'. \end{equation}

Note the similarity with More on Algebra, Equation (15.19.1.1). In particular, if $A' \to A''$ is a morphism of $\mathcal{C}$ and (38.20.3.1) holds for $A'$, then it holds for $A''$, see More on Algebra, Lemma 15.19.2. Hence we obtain a functor

38.20.3.2
\begin{equation} \label{flat-equation-flat-at-point} F_{lf} : \mathcal{C} \longrightarrow \textit{Sets}, \quad A' \longrightarrow \left\{ \begin{matrix} \{ *\}
& \text{if }(05ML)\text{ holds},
\\ \emptyset
& \text{else.}
& \end{matrix} \right. \end{equation}

Lemma 38.20.4. In Situation 38.20.3.

If $A' \to A''$ is a flat morphism in $\mathcal{C}$ then $F_{lf}(A') = F_{lf}(A'')$.

If $A \to B$ is essentially of finite presentation and $M$ is a $B$-module of finite presentation, then $F_{lf}$ is limit preserving: If $\{ A_ i\} _{i \in I}$ is a directed system of objects of $\mathcal{C}$, then $F_{lf}(\mathop{\mathrm{colim}}\nolimits _ i A_ i) = \mathop{\mathrm{colim}}\nolimits _ i F_{lf}(A_ i)$.

**Proof.**
Part (1) is a special case of More on Algebra, Lemma 15.19.3. Part (2) is a special case of More on Algebra, Lemma 15.19.4.
$\square$

Lemma 38.20.5. In Situation 38.20.3. Let $B \to C$ is a local map of local $A$-algebras and $N$ a $C$-module. Denote $F'_{lf} : \mathcal{C} \to \textit{Sets}$ the functor associated to the pair $(C, N)$. If $M \cong N$ as $B$-modules and $B \to C$ is finite, then $F_{lf} = F'_{lf}$.

**Proof.**
Let $A'$ be an object of $\mathcal{C}$. Set $C' = C \otimes _ A A'$ and $N' = N \otimes _ A A'$ similarly to the definitions of $B'$, $M'$ in Situation 38.20.3. Note that $M' \cong N'$ as $B'$-modules. The assumption that $B \to C$ is finite has two consequences: (a) $\mathfrak m_ C = \sqrt{\mathfrak m_ B C}$ and (b) $B' \to C'$ is finite. Consequence (a) implies that

\[ V(\mathfrak m_{A'}C' + \mathfrak m_ C C') = \left( \mathop{\mathrm{Spec}}(C') \to \mathop{\mathrm{Spec}}(B') \right)^{-1}V(\mathfrak m_{A'}B' + \mathfrak m_ B B'). \]

Suppose $\mathfrak q \subset V(\mathfrak m_{A'}B' + \mathfrak m_ B B')$. Then $M'_{\mathfrak q}$ is flat over $A'$ if and only if the $C'_{\mathfrak q}$-module $N'_{\mathfrak q}$ is flat over $A'$ (because these are isomorphic as $A'$-modules) if and only if for every maximal ideal $\mathfrak r$ of $C'_{\mathfrak q}$ the module $N'_{\mathfrak r}$ is flat over $A'$ (see Algebra, Lemma 10.39.18). As $B'_{\mathfrak q} \to C'_{\mathfrak q}$ is finite by (b), the maximal ideals of $C'_{\mathfrak q}$ correspond exactly to the primes of $C'$ lying over $\mathfrak q$ (see Algebra, Lemma 10.36.22) and these primes are all contained in $V(\mathfrak m_{A'}C' + \mathfrak m_ C C')$ by the displayed equation above. Thus the result of the lemma holds.
$\square$

Lemma 38.20.6. In Situation 38.20.3 suppose that $B \to C$ is a flat local homomorphism of local rings. Set $N = M \otimes _ B C$. Denote $F'_{lf} : \mathcal{C} \to \textit{Sets}$ the functor associated to the pair $(C, N)$. Then $F_{lf} = F'_{lf}$.

**Proof.**
Let $A'$ be an object of $\mathcal{C}$. Set $C' = C \otimes _ A A'$ and $N' = N \otimes _ A A' = M' \otimes _{B'} C'$ similarly to the definitions of $B'$, $M'$ in Situation 38.20.3. Note that

\[ V(\mathfrak m_{A'}B' + \mathfrak m_ B B') = \mathop{\mathrm{Spec}}( \kappa (\mathfrak m_ B) \otimes _ A \kappa (\mathfrak m_{A'}) ) \]

and similarly for $V(\mathfrak m_{A'}C' + \mathfrak m_ C C')$. The ring map

\[ \kappa (\mathfrak m_ B) \otimes _ A \kappa (\mathfrak m_{A'}) \longrightarrow \kappa (\mathfrak m_ C) \otimes _ A \kappa (\mathfrak m_{A'}) \]

is faithfully flat, hence $V(\mathfrak m_{A'}C' + \mathfrak m_ C C') \to V(\mathfrak m_{A'}B' + \mathfrak m_ B B')$ is surjective. Finally, if $\mathfrak r \in V(\mathfrak m_{A'}C' + \mathfrak m_ C C')$ maps to $\mathfrak q \in V(\mathfrak m_{A'}B' + \mathfrak m_ B B')$, then $M'_{\mathfrak q}$ is flat over $A'$ if and only if $N'_{\mathfrak r}$ is flat over $A'$ because $B' \to C'$ is flat, see Algebra, Lemma 10.39.9. The lemma follows formally from these remarks.
$\square$

Situation 38.20.7. Let $f : X \to S$ be a smooth morphism with geometrically irreducible fibres. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. For any scheme $T$ over $S$ we will denote $\mathcal{F}_ T$ the base change of $\mathcal{F}$ to $T$, in other words, $\mathcal{F}_ T$ is the pullback of $\mathcal{F}$ via the projection morphism $X_ T = X \times _ S T \to X$. Note that $X_ T \to T$ is smooth with geometrically irreducible fibres, see Morphisms, Lemma 29.34.5 and More on Morphisms, Lemma 37.27.2. Let $p \geq 0$ be an integer. Given a point $t \in T$ consider the condition

38.20.7.1
\begin{equation} \label{flat-equation-free-at-generic-point-fibre} \mathcal{F}_ T \text{ is free of rank }p\text{ in a neighbourhood of }\xi _ t \end{equation}

where $\xi _ t$ is the generic point of the fibre $X_ t$. This condition for all $t \in T$ is stable under base change, and hence we obtain a functor

38.20.7.2
\begin{equation} \label{flat-equation-free-at-generic-points} H_ p : (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\}
& \text{if }\mathcal{F}_ T\text{ satisfies (05MQ) }\forall t\in T,
\\ \emptyset
& \text{else.}
\end{matrix} \right. \end{equation}

Lemma 38.20.8. In Situation 38.20.7.

The functor $H_ p$ satisfies the sheaf property for the fpqc topology.

If $\mathcal{F}$ is of finite presentation, then functor $H_ p$ is limit preserving.

**Proof.**
Let $\{ T_ i \to T\} _{i \in I}$ be an fpqc^{1} covering of schemes over $S$. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and denote $\mathcal{F}_ i$ the pullback of $\mathcal{F}$ to $X_ i$. Assume that $\mathcal{F}_ i$ satisfies (38.20.7.1) for all $i$. Pick $t \in T$ and let $\xi _ t \in X_ T$ denote the generic point of $X_ t$. We have to show that $\mathcal{F}$ is free in a neighbourhood of $\xi _ t$. For some $i \in I$ we can find a $t_ i \in T_ i$ mapping to $t$. Let $\xi _ i \in X_ i$ denote the generic point of $X_{t_ i}$, so that $\xi _ i$ maps to $\xi _ t$. The fact that $\mathcal{F}_ i$ is free of rank $p$ in a neighbourhood of $\xi _ i$ implies that $(\mathcal{F}_ i)_{x_ i} \cong \mathcal{O}_{X_ i, x_ i}^{\oplus p}$ which implies that $\mathcal{F}_{T, \xi _ t} \cong \mathcal{O}_{X_ T, \xi _ t}^{\oplus p}$ as $\mathcal{O}_{X_ T, \xi _ t} \to \mathcal{O}_{X_ i, x_ i}$ is flat, see for example Algebra, Lemma 10.78.6. Thus there exists an affine neighbourhood $U$ of $\xi _ t$ in $X_ T$ and a surjection $\mathcal{O}_ U^{\oplus p} \to \mathcal{F}_ U = \mathcal{F}_ T|_ U$, see Modules, Lemma 17.9.4. After shrinking $T$ we may assume that $U \to T$ is surjective. Hence $U \to T$ is a smooth morphism of affines with geometrically irreducible fibres. Moreover, for every $t' \in T$ we see that the induced map

\[ \alpha : \mathcal{O}_{U, \xi _{t'}}^{\oplus p} \longrightarrow \mathcal{F}_{U, \xi _{t'}} \]

is an isomorphism (since by the same argument as before the module on the right is free of rank $p$). It follows from Lemma 38.10.1 that

\[ \Gamma (U, \mathcal{O}_ U^{\oplus p}) \otimes _{\Gamma (T, \mathcal{O}_ T)} \mathcal{O}_{T, t'} \longrightarrow \Gamma (U, \mathcal{F}_ U) \otimes _{\Gamma (T, \mathcal{O}_ T)} \mathcal{O}_{T, t'} \]

is injective for every $t' \in T$. Hence we see the surjection $\alpha $ is an isomorphism. This finishes the proof of (1).

Assume that $\mathcal{F}$ is of finite presentation. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ be a directed limit of affine $S$-schemes and assume that $\mathcal{F}_ T$ satisfies (38.20.7.1). Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and denote $\mathcal{F}_ i$ the pullback of $\mathcal{F}$ to $X_ i$. Let $U \subset X_ T$ denote the open subscheme of points where $\mathcal{F}_ T$ is flat over $T$, see More on Morphisms, Theorem 37.15.1. By assumption every generic point of every fibre is a point of $U$, i.e., $U \to T$ is a smooth surjective morphism with geometrically irreducible fibres. We may shrink $U$ a bit and assume that $U$ is quasi-compact. Using Limits, Lemma 32.4.11 we can find an $i \in I$ and a quasi-compact open $U_ i \subset X_ i$ whose inverse image in $X_ T$ is $U$. After increasing $i$ we may assume that $\mathcal{F}_ i|_{U_ i}$ is flat over $T_ i$, see Limits, Lemma 32.10.4. In particular, $\mathcal{F}_ i|_{U_ i}$ is finite locally free hence defines a locally constant rank function $\rho : U_ i \to \{ 0, 1, 2, \ldots \} $. Let $(U_ i)_ p \subset U_ i$ denote the open and closed subset where $\rho $ has value $p$. Let $V_ i \subset T_ i$ be the image of $(U_ i)_ p$; note that $V_ i$ is open and quasi-compact. By assumption the image of $T \to T_ i$ is contained in $V_ i$. Hence there exists an $i' \geq i$ such that $T_{i'} \to T_ i$ factors through $V_ i$ by Limits, Lemma 32.4.11. Then $\mathcal{F}_{i'}$ satisfies (38.20.7.1) as desired. Some details omitted.
$\square$

Lemma 38.20.9. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $n \geq 0$. The following are equivalent

for $s \in S$ the closed subset $Z \subset X_ s$ of points where $\mathcal{F}$ is not flat over $S$ (see Lemma 38.10.4) satisfies $\dim (Z) < n$, and

for $x \in X$ such that $\mathcal{F}$ is not flat at $x$ over $S$ we have $\text{trdeg}_{\kappa (f(x))}(\kappa (x)) < n$.

If this is true, then it remains true after any base change.

**Proof.**
Let $x \in X$ be a point over $s \in S$. Then the dimension of the closure of $\{ x\} $ in $X_ s$ is $\text{trdeg}_{\kappa (s)}(\kappa (x))$ by Varieties, Lemma 33.20.3. Conversely, if $Z \subset X_ s$ is a closed subset of dimension $d$, then there exists a point $x \in Z$ with $\text{trdeg}_{\kappa (s)}(\kappa (x)) = d$ (same reference). Therefore the equivalence of (1) and (2) holds (even fibre by fibre). The statement on base change follows from Morphisms, Lemmas 29.25.7 and 29.28.3.
$\square$

Definition 38.20.10. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $n \geq 0$. We say *$\mathcal{F}$ is flat over $S$ in dimensions $\geq n$* if the equivalent conditions of Lemma 38.20.9 are satisfied.

Situation 38.20.11. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. For any scheme $T$ over $S$ we will denote $\mathcal{F}_ T$ the base change of $\mathcal{F}$ to $T$, in other words, $\mathcal{F}_ T$ is the pullback of $\mathcal{F}$ via the projection morphism $X_ T = X \times _ S T \to X$. Note that $X_ T \to T$ is of finite type and that $\mathcal{F}_ T$ is an $\mathcal{O}_{X_ T}$-module of finite type (Morphisms, Lemma 29.15.4 and Modules, Lemma 17.9.2). Let $n \geq 0$. By Definition 38.20.10 and Lemma 38.20.9 we obtain a functor

38.20.11.1
\begin{equation} \label{flat-equation-flat-dimension-n} F_ n : (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\}
& \text{if }\mathcal{F}_ T\text{ is flat over }T\text{ in }\dim \geq n,
\\ \emptyset
& \text{else.}
\end{matrix} \right. \end{equation}

Lemma 38.20.12. In Situation 38.20.11.

The functor $F_ n$ satisfies the sheaf property for the fpqc topology.

If $f$ is quasi-compact and locally of finite presentation and $\mathcal{F}$ is of finite presentation, then the functor $F_ n$ is limit preserving.

**Proof.**
Let $\{ T_ i \to T\} _{i \in I}$ be an fpqc covering of schemes over $S$. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and denote $\mathcal{F}_ i$ the pullback of $\mathcal{F}$ to $X_ i$. Assume that $\mathcal{F}_ i$ is flat over $T_ i$ in dimensions $\geq n$ for all $i$. Let $t \in T$. Choose an index $i$ and a point $t_ i \in T_ i$ mapping to $t$. Consider the cartesian diagram

\[ \xymatrix{ X_{\mathop{\mathrm{Spec}}(\mathcal{O}_{T, t})} \ar[d] & X_{\mathop{\mathrm{Spec}}(\mathcal{O}_{T_ i, t_ i})} \ar[d] \ar[l] \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{T, t}) & \mathop{\mathrm{Spec}}(\mathcal{O}_{T_ i, t_ i}) \ar[l] } \]

As the lower horizontal morphism is flat we see from More on Morphisms, Lemma 37.15.2 that the set $Z_ i \subset X_{t_ i}$ where $\mathcal{F}_ i$ is not flat over $T_ i$ and the set $Z \subset X_ t$ where $\mathcal{F}_ T$ is not flat over $T$ are related by the rule $Z_ i = Z_{\kappa (t_ i)}$. Hence we see that $\mathcal{F}_ T$ is flat over $T$ in dimensions $\geq n$ by Morphisms, Lemma 29.28.3.

Assume that $f$ is quasi-compact and locally of finite presentation and that $\mathcal{F}$ is of finite presentation. In this paragraph we first reduce the proof of (2) to the case where $f$ is of finite presentation. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ be a directed limit of affine $S$-schemes and assume that $\mathcal{F}_ T$ is flat in dimensions $\geq n$. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and denote $\mathcal{F}_ i$ the pullback of $\mathcal{F}$ to $X_ i$. We have to show that $\mathcal{F}_ i$ is flat in dimensions $\geq n$ for some $i$. Pick $i_0 \in I$ and replace $I$ by $\{ i \mid i \geq i_0\} $. Since $T_{i_0}$ is affine (hence quasi-compact) there exist finitely many affine opens $W_ j \subset S$, $j = 1, \ldots , m$ and an affine open overing $T_{i_0} = \bigcup _{j = 1, \ldots , m} V_{j, i_0}$ such that $T_{i_0} \to S$ maps $V_{j, i_0}$ into $W_ j$. For $i \geq i_0$ denote $V_{j, i}$ the inverse image of $V_{j, i_0}$ in $T_ i$. If we can show, for each $j$, that there exists an $i$ such that $\mathcal{F}_{V_{j, i_0}}$ is flat in dimensions $\geq n$, then we win. In this way we reduce to the case that $S$ is affine. In this case $X$ is quasi-compact and we can choose a finite affine open covering $X = W_1 \cup \ldots \cup W_ m$. In this case the result for $(X \to S, \mathcal{F})$ is equivalent to the result for $(\coprod W_ j, \coprod \mathcal{F}|_{W_ j})$. Hence we may assume that $f$ is of finite presentation.

Assume $f$ is of finite presentation and $\mathcal{F}$ is of finite presentation. Let $U \subset X_ T$ denote the open subscheme of points where $\mathcal{F}_ T$ is flat over $T$, see More on Morphisms, Theorem 37.15.1. By assumption the dimension of every fibre of $Z = X_ T \setminus U$ over $T$ has dimension $< n$. By Limits, Lemma 32.18.4 we can find a closed subscheme $Z \subset Z' \subset X_ T$ such that $\dim (Z'_ t) < n$ for all $t \in T$ and such that $Z' \to X_ T$ is of finite presentation. By Limits, Lemmas 32.10.1 and 32.8.5 there exists an $i \in I$ and a closed subscheme $Z'_ i \subset X_ i$ of finite presentation whose base change to $T$ is $Z'$. By Limits, Lemma 32.18.1 we may assume all fibres of $Z'_ i \to T_ i$ have dimension $< n$. By Limits, Lemma 32.10.4 we may assume that $\mathcal{F}_ i|_{X_ i \setminus T'_ i}$ is flat over $T_ i$. This implies that $\mathcal{F}_ i$ is flat in dimensions $\geq n$; here we use that $Z' \to X_ T$ is of finite presentation, and hence the complement $X_ T \setminus Z'$ is quasi-compact! Thus part (2) is proved and the proof of the lemma is complete.
$\square$

Situation 38.20.13. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. For any scheme $T$ over $S$ we will denote $\mathcal{F}_ T$ the base change of $\mathcal{F}$ to $T$, in other words, $\mathcal{F}_ T$ is the pullback of $\mathcal{F}$ via the projection morphism $X_ T = X \times _ S T \to X$. Since the base change of a flat module is flat we obtain a functor

38.20.13.1
\begin{equation} \label{flat-equation-flat} F_{flat} : (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\}
& \text{if } \mathcal{F}_ T \text{ is flat over }T,
\\ \emptyset
& \text{else.}
\end{matrix} \right. \end{equation}

Lemma 38.20.14. In Situation 38.20.13.

The functor $F_{flat}$ satisfies the sheaf property for the fpqc topology.

If $f$ is quasi-compact and locally of finite presentation and $\mathcal{F}$ is of finite presentation, then the functor $F_{flat}$ is limit preserving.

**Proof.**
Part (1) follows from the following statement: If $T' \to T$ is a surjective flat morphism of schemes over $S$, then $\mathcal{F}_{T'}$ is flat over $T'$ if and only if $\mathcal{F}_ T$ is flat over $T$, see More on Morphisms, Lemma 37.15.2. Part (2) follows from Limits, Lemma 32.10.4 after reducing to the case where $X$ and $S$ are affine (compare with the proof of Lemma 38.20.12).
$\square$

## Comments (2)

Comment #5483 by James Borger on

Comment #5484 by Pieter Belmans on