Definition 29.30.1. Let $f : X \to S$ be a morphism of schemes.

1. We say that $f$ is syntomic at $x \in X$ if there exists an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset X$ of $x$ and affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is syntomic.

2. We say that $f$ is syntomic if it is syntomic at every point of $X$.

3. If $S = \mathop{\mathrm{Spec}}(k)$ and $f$ is syntomic, then we say that $X$ is a local complete intersection over $k$.

4. A morphism of affine schemes $f : X \to S$ is called standard syntomic if there exists a global relative complete intersection $R \to R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ (see Algebra, Definition 10.136.5) such that $X \to S$ is isomorphic to

$\mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)) \to \mathop{\mathrm{Spec}}(R).$

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