The Stacks project

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.

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

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

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

  4. 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 10.38.17) 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 10.126.5 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 10.126.5 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 10.126.5 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.

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

  2. 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 10.38.18). 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 10.35.22) 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 10.38.9. 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 29.33.5 and More on Morphisms, Lemma 37.25.2. 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.

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

  2. 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 fpqc1 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 10.77.6. 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

  1. 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

  2. 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.

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

  2. 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 32.16.4 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 32.16.1 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.

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

  2. 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). $\square$

[1] It is quite easy to show that $H_ p$ is a sheaf for the fppf topology using that flat morphisms of finite presentation are open. This is all we really need later on. But it is kind of fun to prove directly that it also satisfies the sheaf condition for the fpqc topology.

Comments (2)

Comment #5483 by James Borger on

You have "(05MQ)" in the text above. I guess this is a tag which got entered as plain text?

Comment #5484 by on

That seems to be a failure of the tag vs. number system, which doesn't deal with references within equations appropriately. I'll file it as a bug and take a look later. Thanks for noticing!


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