Decomposition group vs Galois group of completed extension for height > 1 primes

Assume

1. Let $R$ be a Noetherian normal excellent domain, $F$ its field of fractions.
2. Let $S$ be a finite $R$-algebra, $L$ its field of fractions.
3. $L/F$ a (finite) Galois extension
4. $S$ normal in $L$
5. 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 $D(P/p)=D(V/v)$?
• For the first bullet point: checking Bourbaki (Commutative Algebra), Corollary 4 to Theorem 2 in Section 8.5 of Chapter 6 has the result (p 426), stated under the assumption that the valuation has height 1. This only seems to be used to prove $[L:F]=g\cdot[\hat L:\hat F]$ ($g$ is the index of the decomposition group in the Galois group), which seems to be proven a few pages earlier merely under the assumption that $L$ is separable over $F$. The point is that you need some assumption to guarantee that the canonical map $\hat F\otimes_F L$ to $\prod_{i=1}^g \hat L_i$ is an isomorphism. – B R Nov 29 '10 at 3:29
• Dear oli: May assume $R$ is local. Excellence preserved by henselization (EGA IV$_4$, 18.6.8 or so), and to define decomposition gp must use henselization. Henselization has same completion. When extend scalars to hens. of $R$, then $S$ is replaced by a product of hens. at its maximal ideals. So may assume $R$ henselian, so $S$ henselian. You want that the Galois theories of $F$ and of $\widehat{F}$ "match". This is immediate from the Galois correspondence since $S \otimes_R \widehat{R} = \widehat{S}$ (by hensel!) and a normal domain (by excellence!). The 2nd question seems dubious. – BCnrd Nov 29 '10 at 6:42
• Some clarifications: the reference "18.6.8 or so" is more precisely 18.7.6. (The behavior of henselization with respect to module-finite algebras, which was also used above, is the content of 18.6.8.) The reason I said that to make sense of decomposition group one should use henselization is that when forming the fixed field of a decomposition group in the full Galois group (of the separable closures) one gets the fraction field of a henselization. The link of henselization with decomposition groups for normal noetherian domains is nicely discussed in "Neron Models" somewhere. – BCnrd Nov 29 '10 at 7:03
• GREAT! Thank you very much BR + BCnrd, in particular for the precise references. This was very helpful. THANK YOU! – oli Dec 1 '10 at 2:49