Lemma 75.4.1. Let $S$ be a scheme. Let $X \to Y$ be a finite type morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $y \in |Y|$ be a point. There exists an étale morphism $(Y', y') \to (Y, y)$ with $Y'$ an affine scheme and étale morphisms $h_ i : W_ i \to X_{Y'}$, $i = 1, \ldots , n$ such that for each $i$ there exists a complete dévissage of $\mathcal{F}_ i/W_ i/Y'$ over $y'$, where $\mathcal{F}_ i$ is the pullback of $\mathcal{F}$ to $W_ i$ and such that $|(X_{Y'})_{y'}| \subset \bigcup h_ i(W_ i)$.

## 75.4 Flat finite type modules

Please compare with More on Flatness, Sections 38.10, 38.13, and 38.26. Most of these results have immediate consequences of algebraic spaces by étale localization.

**Proof.**
The question is étale local on $Y$ hence we may assume $Y$ is an affine scheme. Then $X$ is quasi-compact, hence we can choose an affine scheme $X'$ and a surjective étale morphism $X' \to X$. Then we may apply More on Flatness, Lemma 38.5.8 to $X' \to Y$, $(X' \to Y)^*\mathcal{F}$, and $y$ to get what we want.
$\square$

Lemma 75.4.2. 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 $y \in |Y|$ and $F = f^{-1}(\{ y\} ) \subset |X|$. Then the set

is open in $F$.

**Proof.**
Choose a scheme $V$, a point $v \in V$, and an étale morphism $V \to Y$ mapping $v$ to $y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Then $|U_ v| \to F$ is an open continuous map of topological spaces as $|U| \to |X|$ is continuous and open. Hence the result follows from the case of schemes which is More on Flatness, Lemma 38.10.4.
$\square$

Lemma 75.4.3. 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 $x \in |X|$ with image $y \in |Y|$. Let $\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. If $\mathcal{F}$ is flat at $x$ over $Y$, then

**Proof.**
Choose a commutative diagram

where $U$ and $V$ are schemes and the vertical arrows are surjective étale. Choose $u \in U$ mapping to $x$. Let $\mathcal{E} = \mathcal{F}|_ U$ and $\mathcal{H} = \mathcal{G}|_ V$. Let $v \in V$ be the image of $u$. Then $x \in \text{WeakAss}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G})$ if and only if $u \in \text{WeakAss}_ X(\mathcal{E} \otimes _{\mathcal{O}_ X} g^*\mathcal{H})$ by Divisors on Spaces, Definition 69.2.2. Similarly, $y \in \text{WeakAss}_ Y(\mathcal{G})$ if and only if $v \in \text{WeakAss}_ V(\mathcal{H})$. Finally, we have $x \in \text{Ass}_{X/Y}(\mathcal{F})$ if and only if $u \in \text{Ass}_{U_ v}(\mathcal{E}|_{U_ v})$ by Divisors on Spaces, Definition 69.4.5. Observe that flatness of $\mathcal{F}$ at $x$ is equivalent to flatness of $\mathcal{E}$ at $u$, see Morphisms of Spaces, Definition 65.31.2. The equivalence for $g : U \to V$, $\mathcal{E}$, $\mathcal{H}$, $u$, and $v$ is More on Flatness, Lemma 38.13.3. $\square$

Lemma 75.4.4. 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 finite type quasi-coherent sheaf on $X$ which is flat over $Y$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Then we have

**Proof.**
Immediate consequence of Lemma 75.4.3.
$\square$

Theorem 75.4.5. 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. Assume

$X \to Y$ is locally of finite presentation,

$\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type, and

the set of weakly associated points of $Y$ is locally finite in $Y$.

Then $U = \{ x \in |X| : \mathcal{F}\text{ flat at }x\text{ over }Y\} $ is open in $X$ and $\mathcal{F}|_ U$ is an $\mathcal{O}_ U$-module of finite presentation and flat over $Y$.

**Proof.**
Condition (3) means that if $V \to Y$ is a surjective étale morphism where $V$ is a scheme, then the weakly associated points of $V$ are locally finite on the scheme $V$. (Recall that the weakly associated points of $V$ are exactly the inverse image of the weakly associated points of $Y$ by Divisors on Spaces, Definition 69.2.2.) Having said this the question is étale local on $X$ and $Y$, hence we may assume $X$ and $Y$ are schemes. Thus the result follows from More on Flatness, Theorem 38.13.6.
$\square$

Lemma 75.4.6. 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 sheaf on $X$. Let $y \in |Y|$. Set $F = f^{-1}(\{ y\} ) \subset |X|$. Assume that

$f$ is of finite type,

$\mathcal{F}$ is of finite type, and

$\mathcal{F}$ is flat over $Y$ at all $x \in F$.

Then there exists an étale morphism $(Y', y') \to (Y, y)$ where $Y'$ is a scheme and a commutative diagram of algebraic spaces

such that $X' \to X \times _ Y \mathop{\mathrm{Spec}}(\mathcal{O}_{Y', y'})$ is étale, $|X'_{y'}| \to F$ is surjective, $X'$ is affine, and $\Gamma (X', g^*\mathcal{F})$ is a free $\mathcal{O}_{Y', y'}$-module.

**Proof.**
Choose an étale morphism $(Y', y') \to (Y, y)$ where $Y'$ is an affine scheme. Then $X \times _ Y Y'$ is quasi-compact. Choose an affine scheme $X'$ and a surjective étale morphism $X' \to X \times _ Y Y'$. Picture

Then $\mathcal{F}' = g^*\mathcal{F}$ is flat over $Y'$ at all points of $X'_{y'}$, see Morphisms of Spaces, Lemma 65.31.3. Hence we can apply the lemma in the case of schemes (More on Flatness, Lemma 38.12.11) to the morphism $X' \to Y'$, the quasi-coherent sheaf $g^*\mathcal{F}$, and the point $y'$. This gives an étale morphism $(Y'', y'') \to (Y', y')$ and a commutative diagram

To get what we want we take $(Y'', y'') \to (Y, y)$ and $g \circ g' : X'' \to X$. $\square$

Theorem 75.4.7. 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 $x \in |X|$ with image $y \in |Y|$. Set $F = f^{-1}(\{ y\} ) \subset |X|$. Consider the conditions

$\mathcal{F}$ is flat at $x$ over $Y$, and

for every $x' \in F \cap \text{Ass}_{X/Y}(\mathcal{F})$ which specializes to $x$ we have that $\mathcal{F}$ is flat at $x'$ over $Y$.

Then we always have (2) $\Rightarrow $ (1). If $X$ and $Y$ are decent, then (1) $\Rightarrow $ (2).

**Proof.**
Assume (2). Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Choose a point $u \in U$ mapping to $x$. Let $v \in V$ be the image of $u$. We will deduce the result from the corresponding result for $\mathcal{F}|_ U = (U \to X)^*\mathcal{F}$ and the point $u$. $U_ v$. This works because $\text{Ass}_{U/V}(\mathcal{F}|_ U) \cap |U_ v|$ is equal to $\text{Ass}_{U_ v}(\mathcal{F}|_{U_ v})$ and equal to the inverse image of $F \cap \text{Ass}_{X/Y}(\mathcal{F})$. Since the map $|U_ v| \to F$ is continuous we see that specializations in $|U_ v|$ map to specializations in $F$, hence condition (2) is inherited by $U \to V$, $\mathcal{F}|_ U$, and the point $u$. Thus More on Flatness, Theorem 38.26.1 applies and we conclude that (1) holds.

If $Y$ is decent, then we can represent $y$ by a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k) \to Y$ (by definition of decent spaces, see Decent Spaces, Definition 66.6.1). Then $F = |X_ k|$, see Decent Spaces, Lemma 66.18.6. If in addition $X$ is decent (or more generally if $f$ is decent, see Decent Spaces, Definition 66.17.1 and Decent Spaces, Lemma 66.17.3), then $X_ y$ is a decent space too. Furthermore, specializations in $F$ can be lifted to specializations in $U_ v \to X_ y$, see Decent Spaces, Lemma 66.12.2. Having said this it is clear that the reverse implication holds, because it holds in the case of schemes. $\square$

Lemma 75.4.8. Let $S$ be a local scheme with closed point $s$. Let $f : X \to S$ be a morphism from an algebraic space $X$ to $S$ which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Assume that

every point of $\text{Ass}_{X/S}(\mathcal{F})$ specializes to a point of the closed fibre $X_ s$

^{1},$\mathcal{F}$ is flat over $S$ at every point of $X_ s$.

Then $\mathcal{F}$ is flat over $S$.

**Proof.**
This is immediate from the fact that it suffices to check for flatness at points of the relative assassin of $\mathcal{F}$ over $S$ by Theorem 75.4.7.
$\square$

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