## 66.36 Syntomic morphisms

The property “syntomic” 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 29.30.4 and Descent, Lemma 35.23.26. Hence, by Lemma 66.22.1 above, we may define the notion of a syntomic morphism of algebraic spaces as follows and it agrees with the already existing notion defined in Section 66.3 when the morphism is representable.

Definition 66.36.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

1. We say $f$ is syntomic if the equivalent conditions of Lemma 66.22.1 hold with $\mathcal{P} =$“syntomic”.

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

Lemma 66.36.4. 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 syntomic,

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

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

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

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

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

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 syntomic,

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 syntomic, 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 syntomic.

Proof. Omitted. $\square$

Lemma 66.36.5. A syntomic morphism is locally of finite presentation.

Proof. Follows immediately from the case of schemes (Morphisms, Lemma 29.30.6). $\square$

Proof. Follows immediately from the case of schemes (Morphisms, Lemma 29.30.7). $\square$

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