The Stacks project

Lemma 107.8.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{CM} \subset \mathcal{C}\! \mathit{urves}$ 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 $\mathcal{C}\! \mathit{urves}^{CM}$,

    2. the morphism $X \to S$ is Cohen-Macaulay,

  2. given a scheme $X$ proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent

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

    2. $X$ is Cohen-Macaulay.

Proof. Let $f : X \to S$ be a family of curves. By More on Morphisms of Spaces, Lemma 74.26.7 the set

\[ W = \{ x \in |X| : f \text{ is Cohen-Macaulay at }x\} \]

is open in $|X|$ and formation of this open commutes with arbitrary base change. Since $f$ is proper the subset

\[ S' = S \setminus f(|X| \setminus W) \]

of $S$ is open and $X \times _ S S' \to S'$ is Cohen-Macaulay. Moreover, formation of $S'$ commutes with arbitrary base change because this is true for $W$ Thus we get the open substack with the desired properties by the method discussed in Section 107.6. $\square$


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