Lemma 29.30.10. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s = f(x)$. Let $V \subset S$ be an affine open neighbourhood of $s$. The following are equivalent

1. The morphism $f$ is syntomic at $x$.

2. There exist an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ such that $f|_ U : U \to V$ is standard syntomic.

3. The morphism $f$ is of finite presentation at $x$, the local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and $\mathcal{O}_{X, x}/\mathfrak m_ s \mathcal{O}_{X, x}$ is a complete intersection over $\kappa (s)$ (see Algebra, Definition 10.135.5).

Proof. Follows from the definitions and Algebra, Lemma 10.136.15. $\square$

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