Lemma 77.4.1. 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 $|(X_{Y'})_{y'}| \subset \bigcup h_ i(W_ i)$.
77.4 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 77.4.2. 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 $F$.
\[ \{ x \in F \mid \mathcal{F} \text{ flat over }Y\text{ at }x\} \]
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 77.4.3. 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
\[ x \in \text{WeakAss}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}_ Y(\mathcal{G}) \text{ and } x \in \text{Ass}_{X/Y}(\mathcal{F}). \]
Proof. Choose a commutative diagram
\[ \xymatrix{ U \ar[d] \ar[r]_ g & V \ar[d] \\ X \ar[r]^ f & Y } \]
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 71.2.2. 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 71.4.5. Observe that flatness of $\mathcal{F}$ at $x$ is equivalent to flatness of $\mathcal{E}$ at $u$, see Morphisms of Spaces, Definition 67.31.2. The equivalence for $g : U \to V$, $\mathcal{E}$, $\mathcal{H}$, $u$, and $v$ is More on Flatness, Lemma 38.13.3. $\square$
Lemma 77.4.4. 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
\[ \text{WeakAss}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}) = \text{Ass}_{X/Y}(\mathcal{F}) \cap |f|^{-1}(\text{WeakAss}_ Y(\mathcal{G})) \]
Proof. Immediate consequence of Lemma 77.4.3. $\square$
Theorem 77.4.5. 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 71.2.2.) 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 77.4.6. 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
\[ \xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ Y & \mathop{\mathrm{Spec}}(\mathcal{O}_{Y', y'}) \ar[l] } \]
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
\[ \xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ Y & Y' \ar[l] } \]
Then $\mathcal{F}' = g^*\mathcal{F}$ is flat over $Y'$ at all points of $X'_{y'}$, see Morphisms of Spaces, Lemma 67.31.3. 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
\[ \xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] & X'' \ar[l]^{g'} \ar[d] \\ Y & Y' \ar[l] & \mathop{\mathrm{Spec}}(\mathcal{O}_{Y'', y''}) \ar[l] } \]
To get what we want we take $(Y'', y'') \to (Y, y)$ and $g \circ g' : X'' \to X$. $\square$
Theorem 77.4.7. 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 68.6.1). Then $F = |X_ k|$, see Decent Spaces, Lemma 68.18.6. If in addition $X$ is decent (or more generally if $f$ is decent, see Decent Spaces, Definition 68.17.1 and Decent Spaces, Lemma 68.17.3), 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 68.12.2. Having said this it is clear that the reverse implication holds, because it holds in the case of schemes. $\square$
Lemma 77.4.8. 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 77.4.7. $\square$
[1] For example this holds if $f$ is finite type and $\mathcal{F}$ is pure along $X_ s$, or if $f$ is proper.
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