The Stacks project

68.23 Applications of the theorem on formal functions

We will add more here as needed.

Lemma 68.23.1. (For a more general version see More on Morphisms of Spaces, Lemma 75.35.1). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $Y$ is locally Noetherian. The following are equivalent

  1. $f$ is finite, and

  2. $f$ is proper and $|X_ k|$ is a discrete space for every morphism $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is a field.

Proof. A finite morphism is proper according to Morphisms of Spaces, Lemma 66.45.9. A finite morphism is quasi-finite according to Morphisms of Spaces, Lemma 66.45.8. A quasi-finite morphism has discrete fibres $X_ k$, see Morphisms of Spaces, Lemma 66.27.5. Hence a finite morphism is proper and has discrete fibres $X_ k$.

Assume $f$ is proper with discrete fibres $X_ k$. We want to show $f$ is finite. In fact it suffices to prove $f$ is affine. Namely, if $f$ is affine, then it follows that $f$ is integral by Morphisms of Spaces, Lemma 66.45.7 whereupon it follows from Morphisms of Spaces, Lemma 66.45.6 that $f$ is finite.

To show that $f$ is affine we may assume that $Y$ is affine, and our goal is to show that $X$ is affine too. Since $f$ is proper we see that $X$ is separated and quasi-compact. We will show that for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have $H^1(X, \mathcal{F}) = 0$. This implies that $H^1(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ by Lemmas 68.15.1 and 68.5.1. Then it follows that $X$ is affine from Proposition 68.16.7. By Lemma 68.22.8 we conclude that the stalks of $R^1f_*\mathcal{F}$ are zero for all geometric points of $Y$. In other words, $R^1f_*\mathcal{F} = 0$. Hence we see from the Leray Spectral Sequence for $f$ that $H^1(X , \mathcal{F}) = H^1(Y, f_*\mathcal{F})$. Since $Y$ is affine, and $f_*\mathcal{F}$ is quasi-coherent (Morphisms of Spaces, Lemma 66.11.2) we conclude $H^1(Y, f_*\mathcal{F}) = 0$ from Cohomology of Schemes, Lemma 30.2.2. Hence $H^1(X, \mathcal{F}) = 0$ as desired. $\square$

As a consequence we have the following useful result.

Lemma 68.23.2. (For a more general version see More on Morphisms of Spaces, Lemma 75.35.2). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$. Assume

  1. $Y$ is locally Noetherian,

  2. $f$ is proper, and

  3. $|X_{\overline{y}}|$ is finite.

Then there exists an open neighbourhood $V \subset Y$ of $\overline{y}$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.

Proof. The morphism $f$ is quasi-finite at all the geometric points of $X$ lying over $\overline{y}$ by Morphisms of Spaces, Lemma 66.34.8. By Morphisms of Spaces, Lemma 66.34.7 the set of points at which $f$ is quasi-finite is an open subspace $U \subset X$. Let $Z = X \setminus U$. Then $\overline{y} \not\in f(Z)$. Since $f$ is proper the set $f(Z) \subset Y$ is closed. Choose any open neighbourhood $V \subset Y$ of $\overline{y}$ with $Z \cap V = \emptyset $. Then $f^{-1}(V) \to V$ is locally quasi-finite and proper. Hence $f^{-1}(V) \to V$ has discrete fibres $X_ k$ (Morphisms of Spaces, Lemma 66.27.5) which are quasi-compact hence finite. Thus $f^{-1}(V) \to V$ is finite by Lemma 68.23.1. $\square$

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