The Stacks project

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).

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 $A$.

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0ECA. Beware of the difference between the letter 'O' and the digit '0'.