The Stacks project

Lemma 107.13.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{lci} \subset \mathcal{C}\! \mathit{urves}$ such that

  1. given a family of curves $X \to S$ the following are equivalent

    1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci}$,

    2. $X \to S$ is a local complete intersection morphism, and

    3. $X \to S$ is a syntomic morphism.

  2. given $X$ a proper scheme over a field $k$ of dimension $\leq 1$ the following are equivalent

    1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci}$,

    2. $X$ is a local complete intersection over $k$.

Proof. Recall that being a syntomic morphism is the same as being flat and a local complete intersection morphism, see More on Morphisms of Spaces, Lemma 74.48.6. Thus (1)(b) is equivalent to (1)(c). In Section 107.6 we have seen it suffices to show that given a family of curves $f : X \to S$, there is an open subscheme $S' \subset S$ such that $S' \times _ S X \to S'$ is a local complete intersection morphism and such that formation of $S'$ commutes with arbitrary base change. This follows from the more general More on Morphisms of Spaces, Lemma 74.49.7. $\square$


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