The Stacks project

67.30 Flat morphisms

The property “flat” of morphisms of schemes is étale local on the source-and-target, see Descent, Remark 35.32.7. It is also stable under base change and fpqc local on the target, see Morphisms, Lemma 29.25.8 and Descent, Lemma 35.23.15. Hence, by Lemma 67.22.1 above, we may define the notion of a flat morphism of algebraic spaces as follows and it agrees with the already existing notion defined in Section 67.3 when the morphism is representable.

Definition 67.30.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  1. We say $f$ is flat if the equivalent conditions of Lemma 67.22.1 with $\mathcal{P} =$“flat”.

  2. Let $x \in |X|$. We say $f$ is flat at $x$ if the equivalent conditions of Lemma 67.22.5 hold with $\mathcal{Q} =$“induced map local rings is flat”.

Note that the second part makes sense by Descent, Lemma 35.33.4.

We do a quick sanity check.

Lemma 67.30.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is flat if and only if $f$ is flat at all points of $|X|$.

Proof. Choose a commutative diagram

\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]

where $U$ and $V$ are schemes, the vertical arrows are étale, and $a$ is surjective. By definition $f$ is flat if and only if $h$ is flat (Definition 67.22.2). By definition $f$ is flat at $x \in |X|$ if and only if $h$ is flat at some (equivalently any) $u \in U$ which maps to $x$ (Definition 67.22.6). Thus the lemma follows from the fact that a morphism of schemes is flat if and only if it is flat at all points of the source (Morphisms, Definition 29.25.1). $\square$

Lemma 67.30.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. $f$ is flat,

  2. for every $x \in |X|$ the morphism $f$ is flat at $x$,

  3. for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is flat,

  4. for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is flat,

  5. there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times _ Y X \to V$ is flat,

  6. there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi $ is flat,

  7. for every commutative diagram

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

    where $U$, $V$ are schemes and the vertical arrows are étale the top horizontal arrow is flat,

  8. there exists a commutative diagram

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

    where $U$, $V$ are schemes, the vertical arrows are étale, and $U \to X$ is surjective such that the top horizontal arrow is flat, and

  9. there exists a Zariski coverings $Y = \bigcup Y_ i$ and $f^{-1}(Y_ i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_ i$ is flat.

Proof. Omitted. $\square$

Lemma 67.30.6. A flat morphism locally of finite presentation is universally open.

Proof. Let $f : X \to Y$ be a flat morphism locally of finite presentation of algebraic spaces over $S$. Choose a diagram

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

where $U$ and $V$ are schemes and the vertical arrows are surjective and étale, see Spaces, Lemma 65.11.6. By Lemmas 67.30.5 and 67.28.4 the morphism $\alpha $ is flat and locally of finite presentation. Hence by Morphisms, Lemma 29.25.10 we see that $\alpha $ is universally open. Hence $X \to Y$ is universally open according to Lemma 67.6.5. $\square$

Lemma 67.30.7. Let $S$ be a scheme. Let $f : X \to Y$ be a flat, quasi-compact, surjective morphism of algebraic spaces over $S$. A subset $T \subset |Y|$ is open (resp. closed) if and only $f^{-1}(|T|)$ is open (resp. closed) in $|X|$. In other words $f$ is submersive, and in fact universally submersive.

Proof. Choose affine schemes $V_ i$ and étale morphisms $V_ i \to Y$ such that $V = \coprod V_ i \to Y$ is surjective, see Properties of Spaces, Lemma 66.6.1. For each $i$ the algebraic space $V_ i \times _ Y X$ is quasi-compact. Hence we can find an affine scheme $U_ i$ and a surjective étale morphism $U_ i \to V_ i \times _ Y X$, see Properties of Spaces, Lemma 66.6.3. Then the composition $U_ i \to V_ i \times _ Y X \to V_ i$ is a surjective, flat morphism of affines. Of course then $U = \coprod U_ i \to X$ is surjective and étale and $U = V \times _ Y X$. Moreover, the morphism $U \to V$ is the disjoint union of the morphisms $U_ i \to V_ i$. Hence $U \to V$ is surjective, quasi-compact and flat. Consider the diagram

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

By definition of the topology on $|Y|$ the set $T$ is closed (resp. open) if and only if $g^{-1}(T) \subset |V|$ is closed (resp. open). The same holds for $f^{-1}(T)$ and its inverse image in $|U|$. Since $U \to V$ is quasi-compact, surjective, and flat we win by Morphisms, Lemma 29.25.12. $\square$

Lemma 67.30.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{x}$ be a geometric point of $X$ lying over the point $x \in |X|$. Let $\overline{y} = f \circ \overline{x}$. The following are equivalent

  1. $f$ is flat at $x$, and

  2. the map on étale local rings $\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}$ is flat.

Proof. Choose a commutative diagram

\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]

where $U$ and $V$ are schemes, $a, b$ are étale, and $u \in U$ mapping to $x$. We can find a geometric point $\overline{u} : \mathop{\mathrm{Spec}}(k) \to U$ lying over $u$ with $\overline{x} = a \circ \overline{u}$, see Properties of Spaces, Lemma 66.19.4. Set $\overline{v} = h \circ \overline{u}$ with image $v \in V$. We know that

\[ \mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh} \quad \text{and}\quad \mathcal{O}_{Y, \overline{y}} = \mathcal{O}_{V, v}^{sh} \]

see Properties of Spaces, Lemma 66.22.1. We obtain a commutative diagram

\[ \xymatrix{ \mathcal{O}_{U, u} \ar[r] & \mathcal{O}_{X, \overline{x}} \\ \mathcal{O}_{V, v} \ar[u] \ar[r] & \mathcal{O}_{Y, \overline{y}} \ar[u] } \]

of local rings with flat horizontal arrows. We have to show that the left vertical arrow is flat if and only if the right vertical arrow is. Algebra, Lemma 10.39.9 tells us $\mathcal{O}_{U, u}$ is flat over $\mathcal{O}_{V, v}$ if and only if $\mathcal{O}_{X, \overline{x}}$ is flat over $\mathcal{O}_{V, v}$. Hence the result follows from More on Flatness, Lemma 38.2.5. $\square$

Lemma 67.30.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is flat if and only if the morphism of sites $ (f_{small}, f^\sharp ) : (X_{\acute{e}tale}, \mathcal{O}_ X) \to (Y_{\acute{e}tale}, \mathcal{O}_ Y) $ associated to $f$ is flat.

Proof. Flatness of $(f_{small}, f^\sharp )$ is defined in terms of flatness of $\mathcal{O}_ X$ as a $f_{small}^{-1}\mathcal{O}_ Y$-module. This can be checked at stalks, see Modules on Sites, Lemma 18.39.3 and Properties of Spaces, Theorem 66.19.12. But we've already seen that flatness of $f$ can be checked on stalks, see Lemma 67.30.8. $\square$

Lemma 67.30.10. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module with scheme theoretic support $Z \subset X$. If $f$ is flat, then $f^{-1}(Z)$ is the scheme theoretic support of $f^*\mathcal{F}$.

Proof. Using the characterization of the scheme theoretic support as given in Lemma 67.15.3 and using the characterization of flat morphisms in terms of étale coverings in Lemma 67.30.5 we reduce to the case of schemes which is Morphisms, Lemma 29.25.14. $\square$

Lemma 67.30.11. Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$. Let $V \to Y$ be a quasi-compact open immersion. If $V$ is scheme theoretically dense in $Y$, then $f^{-1}V$ is scheme theoretically dense in $X$.

Proof. Using the characterization of scheme theoretically dense opens in Lemma 67.17.2 and using the characterization of flat morphisms in terms of étale coverings in Lemma 67.30.5 we reduce to the case of schemes which is Morphisms, Lemma 29.25.15. $\square$

Lemma 67.30.12. Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$. Let $g : V \to Y$ be a quasi-compact morphism of algebraic spaces. Let $Z \subset Y$ be the scheme theoretic image of $g$ and let $Z' \subset X$ be the scheme theoretic image of the base change $V \times _ Y X \to X$. Then $Z' = f^{-1}Z$.

Proof. Let $Y' \to Y$ be a surjective étale morphism such that $Y'$ is a disjoint union of affine schemes (Properties of Spaces, Lemma 66.6.1). Let $X' \to X \times _ Y Y'$ be a surjective étale morphism such that $X'$ is a disjoint union of affine schemes. By Lemma 67.30.5 the morphism $X' \to Y'$ is flat. Set $V' = V \times _ Y Y'$. By Lemma 67.16.3 the inverse image of $Z$ in $Y'$ is the scheme theoretic image of $V' \to Y'$ and the inverse image of $Z'$ in $X'$ is the scheme theoretic image of $V' \times _{Y'} X' \to X'$. Since $X' \to X$ is surjective étale, it suffices to prove the result in the case of the morphisms $X' \to Y'$ and $V' \to Y'$. Thus we may assume $X$ and $Y$ are affine schemes. In this case $V$ is a quasi-compact algebraic space. Choose an affine scheme $W$ and a surjective étale morphism $W \to V$ (Properties of Spaces, Lemma 66.6.3). It is clear that the scheme theoretic image of $V \to Y$ agrees with the scheme theoretic image of $W \to Y$ and similarly for $V \times _ Y X \to Y$ and $W \times _ Y X \to X$. Thus we reduce to the case of schemes which is Morphisms, Lemma 29.25.16. $\square$

Comments (2)

Comment #5960 by Lukas on

Def. 65.30.1 , (1) and (2): 'if the equivalent conditions ... HOLD'

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