Lemma 15.110.10 (0BRK)—The Stacks project

Lemma 15.110.10. Let $A$ be a normal domain with fraction field $K$ . Let $L/K$ be a (possibly infinite) Galois extension. Let $G = \text{Gal}(L/K)$ and let $B$ be the integral closure of $A$ in $L$ .

For any two primes $\mathfrak q, \mathfrak q' \subset B$ lying over the same prime in $A$ there exists a $\sigma \in G$ with $\sigma (\mathfrak q) = \mathfrak q'$ .
Let $\mathfrak q \subset B$ be a prime lying over $\mathfrak p \subset A$ . Then $\kappa (\mathfrak q)/\kappa (\mathfrak p)$ is an algebraic normal extension and the map

$D = \{ \sigma \in G \mid \sigma (\mathfrak q) = \mathfrak q\} \longrightarrow \text{Aut}(\kappa (\mathfrak q)/\kappa (\mathfrak p))$

is surjective.

Proof. Proof of (1). Consider pairs $(M, \sigma )$ where $K \subset M \subset L$ is a subfield such that $M/K$ is Galois, $\sigma \in \text{Gal}(M/K)$ with $\sigma (\mathfrak q \cap M) = \mathfrak q' \cap M$ . We say $(M', \sigma ') \geq (M, \sigma )$ if and only if $M \subset M'$ and $\sigma '|_ M = \sigma$ . Observe that $(K, \text{id}_ K)$ is such a pair as $A = K \cap B$ since $A$ is a normal domain. The collection of these pairs satisfies the hypotheses of Zorn's lemma, hence there exists a maximal pair $(M, \sigma )$ . If $M \not= L$ , then we can find $M \subset M' \subset L$ with $M'/M$ nontrivial and finite and $M'/K$ Galois (Fields, Lemma 9.16.5). Choose $\sigma ' \in \text{Gal}(M'/K)$ whose restriction to $M$ is $\sigma$ (Fields, Lemma 9.22.2). Then the primes $\sigma '(\mathfrak q \cap M')$ and $\mathfrak q' \cap M'$ restrict to the same prime of $B \cap M$ . Since $B \cap M = (B \cap M')^{\text{Gal}(M'/M)}$ we can use Lemma 15.110.8 to find $\tau \in \text{Gal}(M'/M)$ with $\tau (\sigma '(\mathfrak q \cap M')) = \mathfrak q' \cap M'$ . Hence $(M', \tau \circ \sigma ') > (M, \sigma )$ contradicting the maximality of $(M, \sigma )$ .

Part (2) is proved in exactly the same manner as part (1). We write out the details. Pick $\overline{\sigma } \in \text{Aut}(\kappa (\mathfrak q)/\kappa (\mathfrak p))$ . Consider pairs $(M, \sigma )$ where $K \subset M \subset L$ is a subfield such that $M/K$ is Galois, $\sigma \in \text{Gal}(M/K)$ with $\sigma (\mathfrak q \cap M) = \mathfrak q \cap M$ and

$\xymatrix{ \kappa (\mathfrak q \cap M) \ar[r] \ar[d]_\sigma & \kappa (\mathfrak q) \ar[d]_{\overline{\sigma }} \\ \kappa (\mathfrak q \cap M) \ar[r] & \kappa (\mathfrak q) }$

commutes. We say $(M', \sigma ') \geq (M, \sigma )$ if and only if $M \subset M'$ and $\sigma '|_ M = \sigma$ . As above $(K, \text{id}_ K)$ is such a pair. The collection of these pairs satisfies the hypotheses of Zorn's lemma, hence there exists a maximal pair $(M, \sigma )$ . If $M \not= L$ , then we can find $M \subset M' \subset L$ with $M'/M$ finite and $M'/K$ Galois (Fields, Lemma 9.16.5). Choose $\sigma ' \in \text{Gal}(M'/K)$ whose restriction to $M$ is $\sigma$ (Fields, Lemma 9.22.2). Then the primes $\sigma '(\mathfrak q \cap M')$ and $\mathfrak q \cap M'$ restrict to the same prime of $B \cap M$ . Adjusting the choice of $\sigma '$ as in the first paragraph, we may assume that $\sigma '(\mathfrak q \cap M') = \mathfrak q \cap M'$ . Then $\sigma '$ and $\overline{\sigma }$ define maps $\kappa (\mathfrak q \cap M') \to \kappa (\mathfrak q)$ which agree on $\kappa (\mathfrak q \cap M)$ . Since $B \cap M = (B \cap M')^{\text{Gal}(M'/M)}$ we can use Lemma 15.110.9 to find $\tau \in \text{Gal}(M'/M)$ with $\tau (\mathfrak q \cap M') = \mathfrak q \cap M'$ such that $\tau \circ \sigma$ and $\overline{\sigma }$ induce the same map on $\kappa (\mathfrak q \cap M')$ . There is a small detail here in that the lemma first guarantees that $\kappa (\mathfrak q \cap M')/\kappa (\mathfrak q \cap M)$ is normal, which then tells us that the difference between the maps is an automorphism of this extension (Fields, Lemma 9.15.10), to which we can apply the lemma to get $\tau$ . Hence $(M', \tau \circ \sigma ') > (M, \sigma )$ contradicting the maximality of $(M, \sigma )$ . $\square$

The Stacks project

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