The Stacks project

65.37 Smooth morphisms

The property “smooth” of morphisms of schemes is étale local on the source-and-target, see Descent, Remark 35.29.7. It is also stable under base change and fpqc local on the target, see Morphisms, Lemma 29.34.5 and Descent, Lemma 35.20.27. Hence, by Lemma 65.22.1 above, we may define the notion of a smooth morphism of algebraic spaces as follows and it agrees with the already existing notion defined in Section 65.3 when the morphism is representable.

Definition 65.37.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  1. We say $f$ is smooth if the equivalent conditions of Lemma 65.22.1 hold with $\mathcal{P} =$“smooth”.

  2. Let $x \in |X|$. We say $f$ is smooth at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is smooth.

Lemma 65.37.4. 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 smooth,

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

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

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

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

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

  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 smooth,

  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 smooth, and

  9. there exist Zariski coverings $Y = \bigcup _{i \in I} Y_ i$, and $f^{-1}(Y_ i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_ i$ is smooth.

Proof. Omitted. $\square$

Lemma 65.37.5. A smooth morphism of algebraic spaces is locally of finite presentation.

Proof. Let $X \to Y$ be a smooth morphism of algebraic spaces. By definition this means there exists a diagram as in Lemma 65.22.1 with $h$ smooth and surjective vertical arrow $a$. By Morphisms, Lemma 29.34.8 $h$ is locally of finite presentation. Hence $X \to Y$ is locally of finite presentation by definition. $\square$

Lemma 65.37.6. A smooth morphism of algebraic spaces is locally of finite type.

Proof. Let $X \to Y$ be a smooth morphism of algebraic spaces. By definition this means there exists a diagram as in Lemma 65.22.1 with $h$ smooth and surjective vertical arrow $a$. By Morphisms, Lemma 29.34.8 $h$ is flat. Hence $X \to Y$ is flat by definition. $\square$

Proof. Let $X \to Y$ be a smooth morphism of algebraic spaces. By definition this means there exists a diagram as in Lemma 65.22.1 with $h$ smooth and surjective vertical arrow $a$. By Morphisms, Lemma 29.34.7 $h$ is syntomic. Hence $X \to Y$ is syntomic by definition. $\square$

Lemma 65.37.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. There is a maximal open subspace $U \subset X$ such that $f|_ U : U \to Y$ is smooth. Moreover, formation of this open commutes with base change by

  1. morphisms which are flat and locally of finite presentation,

  2. flat morphisms provided $f$ is locally of finite presentation.

Proof. The existence of $U$ follows from the fact that the property of being smooth is Zariski (and even étale) local on the source, see Lemma 65.37.4. Moreover, this lemma allows us to translate properties (1) and (2) into the case of morphisms of schemes. The case of schemes is Morphisms, Lemma 29.34.15. Some details omitted. $\square$

Lemma 65.37.10. Let $X$ and $Y$ be locally Noetherian algebraic spaces over a scheme $S$, and let $f : X \to Y$ be a smooth morphism. For every point $x \in |X|$ with image $y \in |Y|$,

\[ \dim _ x(X) = \dim _ y(Y) + \dim _ x(X_ y) \]

where $\dim _ x(X_ y)$ is the relative dimension of $f$ at $x$ as in Definition 65.33.1.

Proof. By definition of the dimension of an algebraic space at a point (Properties of Spaces, Definition 64.9.1), this reduces to the corresponding statement for schemes (Morphisms, Lemma 29.34.21). $\square$


Comments (2)

Comment #1122 by Evan Warner on

minor typo: "syntomic" in first sentence should be "smooth"


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