# proving closure of a set of automorphisms in the Krull topology in the context of integral ring extensions

Let $A$ be an integrally closed integral domain with field of fractions $K$ and let $p \in Spec(A)$. Let $L/K$ be a Galois extension with group $G$ and $L'/K$ be a finite Galois subextension. Let $B$ be the integral closure of $A$ in $L$ and $P_1,P_2 \in Spec(B)$ lying over $p$. Define the set $F(L')=\left\{\sigma \in G: \sigma(P_1 \cap L')=P_2 \cap L'\right\}$. Then Matsumura, in his Commutative Ring Theory, p. 67, says that $F(L')$ is closed in the Krull topology for infinite Galois extensions. How can we see that?

The basic neighborhoods of an element $\sigma \in G$ in the Krull topology are given by fixing its behavior on some finite subextension and putting no restriction on what's going on outside this subextension. If $M$ is some finite subextension of $K$, then the collection of all those $\sigma'$ which agree with $\sigma$ on $M$ is a basic neighborhood of $\sigma$.
If $\sigma$ is not in $F(L')$ then any other element which agrees with $\sigma$ on $L'$ will also be outside $F(L')$, so the complement of $F(L')$ is open. In fact, the same reasoning shows that $F(L')$ is a clopen set (every set $C$ with the property, that whether or not $\sigma$ is in $C$ is determined entirely by the behavior of $\sigma$ on some fixed finite subextension will be clopen)