Lemma 58.29.1. Let $(A, \mathfrak m)$ be a regular local ring which contains a field. Let $f : V \to \mathop{\mathrm{Spec}}(A)$ be étale and quasi-compact. Assume that $\mathfrak m \not\in f(V)$ and assume that $g : V \to \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} $ is affine. Then $H^ i(V, \mathcal{O}_ V)$, $i > 0$ is isomorphic to a direct sum of copies of the injective hull of the residue field of $A$.
58.29 Affineness of complement of ramification locus
Let $f : X \to Y$ be a finite type morphism of Noetherian schemes with $X$ normal and $Y$ regular. Let $V \subset X$ be the maximal open subscheme where $f$ is étale. The discussion in [Chapter IV, Section 21.12, EGA] suggests that $V \to X$ might be an affine morphism. Observe that if $V \to X$ is affine, then we deduce purity of ramification locus (Lemma 58.28.3) by using Divisors, Lemma 31.16.4. Thus affineness of $V \to X$ is a “strong” form of purity for the ramification locus. In this section we prove $V \to X$ is affine when $X$ and $Y$ are equicharacteristic and excellent, see Theorem 58.29.3. It seems reasonable to guess the result remains true for $X$ and $Y$ of mixed characteristic (but still excellent).
Proof. Denote $U = \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} $ the punctured spectrum. Thus $g : V \to U$ is affine. We have $H^ i(V, \mathcal{O}_ V) = H^ i(U, g_*\mathcal{O}_ V)$ by Cohomology of Schemes, Lemma 30.2.4. The $\mathcal{O}_ U$-module $g_*\mathcal{O}_ V$ is quasi-coherent by Schemes, Lemma 26.24.1. For any quasi-coherent $\mathcal{O}_ U$-module $\mathcal{F}$ the cohomology $H^ i(U, \mathcal{F})$, $i > 0$ is $\mathfrak m$-power torsion, see for example Local Cohomology, Lemma 51.2.2. In particular, the $A$-modules $H^ i(V, \mathcal{O}_ V)$, $i > 0$ are $\mathfrak m$-power torsion. For any flat ring map $A \to A'$ we have $H^ i(V, \mathcal{O}_ V) \otimes _ A A' = H^ i(V', \mathcal{O}_{V'})$ where $V' = V \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A')$ by flat base change Cohomology of Schemes, Lemma 30.5.2. If we take $A'$ to be the completion of $A$ (flat by More on Algebra, Section 15.43), then we see that
\[ H^ i(V, \mathcal{O}_ V) = H^ i(V, \mathcal{O}_ V) \otimes _ A A' = H^ i(V', \mathcal{O}_{V'}),\quad \text{for } i > 0 \]
The first equality by the torsion property we just proved and More on Algebra, Lemma 15.89.3. Moreover, the injective hull of the residue field $k$ is the same for $A$ and $A'$, see Dualizing Complexes, Lemma 47.7.4. In this way we reduce to the case $A = k[[x_1, \ldots , x_ d]]$, see Algebra, Section 10.160.
Assume the characteristic of $k$ is $p > 0$. Since $F : A \to A$, $a \mapsto a^ p$ is flat (Local Cohomology, Lemma 51.17.6) and since $V \times _{\mathop{\mathrm{Spec}}(A), \mathop{\mathrm{Spec}}(F)} \mathop{\mathrm{Spec}}(A) \cong V$ as schemes over $\mathop{\mathrm{Spec}}(A)$ by Étale Morphisms, Lemma 41.14.3 the above gives $H^ i(V, \mathcal{O}_ V) \otimes _{A, F} A \cong H^ i(V, \mathcal{O}_ V)$. Thus we get the result by Local Cohomology, Lemma 51.18.2.
Assume the characteristic of $k$ is $0$. By Local Cohomology, Lemma 51.19.3 there are additive operators $D_ j$, $j = 1, \ldots , d$ on $H^ i(V, \mathcal{O}_ V)$ satisfying the Leibniz rule with respect to $\partial _ j = \partial /\partial x_ j$. Thus we get the result by Local Cohomology, Lemma 51.18.1. $\square$
Lemma 58.29.2. In the situation of Lemma 58.29.1 assume that $H^ i(V, \mathcal{O}_ V) = 0$ for $i \geq \dim (A) - 1$. Then $V$ is affine.
Proof. Let $k = A/\mathfrak m$. Since $V \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(k) = \emptyset $, by cohomology and base change we have
\[ R\Gamma (V, \mathcal{O}_ V) \otimes _ A^\mathbf {L} k = 0 \]
See Derived Categories of Schemes, Lemma 36.22.5. Thus there is a spectral sequence (More on Algebra, Example 15.62.4)
\[ E_2^{p, q} = \text{Tor}_{-p}(k, H^ q(V, \mathcal{O}_ V)),\quad d_2^{p, q} : E_2^{p, q} \to E_2^{p + 2, q - 1} \]
and $d_ r^{p, q} : E_ r^{p, q} \to E_ r^{p + r, q - r + 1}$ converging to zero. By Lemma 58.29.1, Dualizing Complexes, Lemma 47.21.9, and our assumption $H^ i(V, \mathcal{O}_ V) = 0$ for $i \geq \dim (A) - 1$ we conclude that there is no nonzero differential entering or leaving the $(p, q) = (0, 0)$ spot. Thus $H^0(V, \mathcal{O}_ V) \otimes _ A k = 0$. This means that if $\mathfrak m = (x_1, \ldots , x_ d)$ then we have an open covering $V = \bigcup V \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A_{x_ i})$ by affine open subschemes $V \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A_{x_ i})$ (because $V$ is affine over the punctured spectrum of $A$) such that $x_1, \ldots , x_ d$ generate the unit ideal in $\Gamma (V, \mathcal{O}_ V)$. This implies $V$ is affine by Properties, Lemma 28.27.3. $\square$
Theorem 58.29.3. 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 étale. 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 32.4.13). 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 étale 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 51.4.6 and the characterization of cohomological dimension in Local Cohomology, Lemma 51.4.1. We have $e(V_ x) \leq \dim (\mathcal{O}_{X, x}) - 1$ by Local Cohomology, Lemma 51.4.7. If $\dim (\mathcal{O}_{X, x}) \geq 2$ then purity of ramification locus (Lemma 58.28.3) 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 51.16.7). 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 29.52.1). Whence $e(V) \leq \max (0, d - 2)$. Thus $V$ is affine by Lemma 58.29.2 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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