Assume

- Let $R$ be a Noetherian normal excellent domain, $F$ its field of fractions.
- Let $S$ be a finite $R$-algebra, $L$ its field of fractions.
- $L/F$ a (finite) Galois extension
- $S$ normal in $L$
- Suppose $P$ is a prime in $S$ (but not necessarily height 1), $p$ the prime below in $R$.

The notion of a decomposition group $D(P/p)$ makes sense.

Now take $\widehat{R_p}$ and $\widehat{S_P}$ ($p$-adic and $P$-adic completion of the local rings at $P$, $p$ respectively). By normality (and excellence) both are domains. Let $\hat{L}$ and $\hat{F}$ denote the corresponding fields of fractions. This is a finite field extension (hopefully Galois...)

- it seems to me the standard proof for Dedekind domains actually carries over and shows that the natural map $D(P/p) \rightarrow Aut(\hat{L}/\hat{F})$ is an isomorphism. I cannot find any reference, so I am feeling a bit suspicious...
- does every pair $P/p$ of primes admit a pair of
*height 1*primes $V/v$ such that they have the same decomposition groups $D(P/p)=D(V/v)$?

normal domain(by excellence!). The 2nd question seems dubious. – BCnrd Nov 29 '10 at 6:42