## 76.11 Universal flattening

This section is the analogue of More on Flatness, Section 38.27. Our main aim is to prove Lemma 76.11.8. However, we do not see a way to deduce this result from the corresponding result for schemes directly. Hence we have to redevelop some of the material here. But first a definition.

Definition 76.11.1. Let $S$ be a scheme. Let $X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. We say that the universal flattening of $\mathcal{F}$ exists if the functor $F_{flat}$ defined in Situation 76.7.8 is an algebraic space. We say that the universal flattening of $X$ exists if the universal flattening of $\mathcal{O}_ X$ exists.

This is a bit unsatisfactory, because here the definition of universal flattening does not agree with the one used in the case of schemes, as we don't know whether every monomorphism of algebraic spaces is representable (More on Morphisms of Spaces, Section 75.4). Hopefully no confusion will ever result from this.

Lemma 76.11.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces 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 some commutative diagram

$\xymatrix{ U \ar[d]_\varphi \ar[r] & V \ar[d] \\ X \ar[r] & Y }$

with surjective, étale vertical arrows where $U$ and $V$ are schemes, the sheaf $\varphi ^*\mathcal{F}$ is flat over $V$ in dimensions $\geq n$ (More on Flatness, Definition 38.20.10),

2. for every commutative diagram

$\xymatrix{ U \ar[d]_\varphi \ar[r] & V \ar[d] \\ X \ar[r] & Y }$

with étale vertical arrows where $U$ and $V$ are schemes, the sheaf $\varphi ^*\mathcal{F}$ is flat over $V$ in dimensions $\geq n$, and

3. for $x \in |X|$ such that $\mathcal{F}$ is not flat at $x$ over $Y$ the transcendence degree of $x/f(x)$ is $< n$ (Morphisms of Spaces, Definition 66.33.1).

If this is true, then it remains true after any base change $Y' \to Y$.

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. $\square$

Definition 76.11.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ 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 $Y$ in dimensions $\geq n$ if the equivalent conditions of Lemma 76.11.2 are satisfied.

Situation 76.11.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ 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 $Y$ 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 $X_ T = X \times _ Y T \to X$. Note that $f_ T : X_ T \to T$ is of finite type and that $\mathcal{F}_ T$ is an $\mathcal{O}_{X_ T}$-module of finite type (Morphisms of Spaces, Lemma 66.23.3 and Modules on Sites, Lemma 18.23.4). Let $n \geq 0$. By Definition 76.11.3 and Lemma 76.11.2 we obtain a functor

76.11.4.1
$$\label{spaces-flat-equation-flat-dimension-n} F_ n : (\mathit{Sch}/Y)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\} & \text{if }\mathcal{F}_ T\text{ is flat over }T\text{ in }\dim \geq n, \\ \emptyset & \text{else.} \end{matrix} \right.$$

In Situation 76.11.4 we sometimes think of $F_ n$ as a functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$ endowed with a morphism $F_ n \to Y$. Namely, if $T$ is a scheme over $S$, then an element $h \in F_ n(T)$ is a morphism $h : T \to Y$ such that the base change of $\mathcal{F}$ via $h$ is flat over $T$ in $\dim \geq n$. In particular, when we say that $F_ n$ is an algebraic space, we mean that the corresponding functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$ is an algebraic space.

Lemma 76.11.5. In Situation 76.11.4.

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. 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 76.11.2. 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 76.11.2 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 76.11.2. 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 76.11.2 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. $\square$

Lemma 76.11.6. In Situation 76.11.4. Let $h : X' \to X$ be an étale morphism. Set $\mathcal{F}' = h^*\mathcal{F}$ and $f' = f \circ h$. Let $F_ n'$ be (76.11.4.1) associated to $(f' : X' \to Y, \mathcal{F}')$. Then $F_ n$ is a subfunctor of $F_ n'$ and if $h(X') \supset \text{Ass}_{X/Y}(\mathcal{F})$, then $F_ n = F'_ n$.

Proof. Choose $U \to X$, $V \to Y$, $U \to V$ as in part (1) of Lemma 76.11.2. 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 76.11.2 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. $\square$

Theorem 76.11.7. In Situation 76.11.4. 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 $Y$. Then $F_ n$ is an algebraic space and $F_ n \to Y$ is a monomorphism of finite presentation.

Proof. The functor $F_ n$ is a sheaf for the fppf topology by Lemma 76.11.5. 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 76.3.3. 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 37.55.1 , 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 76.3.3 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 76.4.1 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 76.11.2 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

$E = \{ t \in |Y| : \text{Ass}_{X_ t}(\mathcal{F}_ t) \subset W \} .$

contains all generalizations of $y$. By Divisors on Spaces, Lemma 70.4.10 $E$ 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 76.11.6 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

$Z_ n = Z_{1, n} \times _ Y \ldots \times _ Y Z_{m, n}$

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

Thus we finally obtain the desired result.

Lemma 76.11.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

1. If $f$ is of finite presentation, $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation, and $\mathcal{F}$ is pure relative to $Y$, then there exists a universal flattening $Y' \to Y$ of $\mathcal{F}$. Moreover $Y' \to Y$ is a monomorphism of finite presentation.

2. If $f$ is of finite presentation and $X$ is pure relative to $Y$, then there exists a universal flattening $Y' \to Y$ of $X$. Moreover $Y' \to Y$ is a monomorphism of finite presentation.

3. If $f$ is proper and of finite presentation and $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation, then there exists a universal flattening $Y' \to Y$ of $\mathcal{F}$. Moreover $Y' \to Y$ is a monomorphism of finite presentation.

4. If $f$ is proper and of finite presentation then there exists a universal flattening $Y' \to Y$ of $X$.

Proof. These statements follow immediately from Theorem 76.11.7 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 76.3.6. $\square$

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