The Stacks project

Theorem 53.29.3. Let $Y$ be an excellent regular scheme over a field. Let $f : X \to Y$ be a finite type morphism of schemes with $X$ normal. Let $V \subset X$ be the maximal open subscheme where $f$ is ├ętale. Then the inclusion morphism $V \to X$ is affine.

Proof. Let $x \in X$ with image $y \in Y$. It suffices to prove that $V \cap W$ is affine for some affine open neighbourhood $W$ of $x$. Since $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ is the limit of the schemes $W$, this holds if and only if

\[ V_ x = V \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \]

is affine (Limits, Lemma 31.4.13). Thus, if the theorem holds for the morphism $X \times _ Y \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y})$, then the theorem holds. In particular, we may assume $Y$ is regular of finite dimension, which allows us to do induction on the dimension $d = \dim (Y)$. Combining this with the same argument again, we may assume that $Y$ is local with closed point $y$ and that $V \cap (X \setminus f^{-1}(\{ y\} ) \to X \setminus f^{-1}(\{ y\} )$ is affine.

Let $x \in X$ be a point lying over $y$. If $x \in V$, then there is nothing to prove. Observe that $f^{-1}(\{ y\} ) \cap V$ is a finite set of closed points (the fibres of an ├ętale morphism are discrete). Thus after replacing $X$ by an affine open neighbourhood of $x$ we may assume $y \not\in f(V)$. We have to prove that $V$ is affine.

Let $e(V)$ be the maximum $i$ with $H^ i(V, \mathcal{O}_ V) \not= 0$. As $X$ is affine the integer $e(V)$ is the maximum of the numbers $e(V_ x)$ where $x \in X \setminus V$, see Local Cohomology, Lemma 48.3.6 and the characterization of cohomological dimension in Local Cohomology, Lemma 48.3.1. We have $e(V_ x) \leq \dim (\mathcal{O}_{X, x}) - 1$ by Local Cohomology, Lemma 48.3.7. If $\dim (\mathcal{O}_{X, x}) \geq 2$ then purity of ramification locus (Lemma 53.28.3) shows that $V_ x$ is strictly smaller than the punctured spectrum of $\mathcal{O}_{X, x}$. Since $\mathcal{O}_{X, x}$ is normal and excellent, this implies $e(V_ x) \leq \dim (\mathcal{O}_{X, x}) - 2$ by Hartshorne-Lichtenbaum vanishing (Local Cohomology, Lemma 48.15.7). On the other hand, since $X \to Y$ is of finite type and $V \subset X$ is dense (after possibly replacing $X$ by the closure of $V$), we see that $\dim (\mathcal{O}_{X, x}) \leq d$ by the dimension formula (Morphisms, Lemma 28.50.1). Whence $e(V) \leq \max (0, d - 2)$. Thus $V$ is affine by Lemma 53.29.2 if $d \geq 2$. If $d = 1$ or $d = 0$, then the punctured spectrum of $\mathcal{O}_{Y, y}$ is affine and hence $V$ is affine. $\square$


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