I am trying to rewrite the theory of decomposition/inertia/ramification groups independently of the theory of Dedekind or valuation rings (I believe this has been done elsewhere, but I found only few results here or there, so any reference will be greatly appreciated). The result is a more general theory, that contains a large part of the same theory in the context of valuations or Dedekind domains.
The setup is quite simple : Let $A$ be an integrally closed domain, $K$ its quotient field, $L$ a Galois extension of $K$, $B$ the integral closure of $A$ in $L$, $p$ a prime ideal of $A$, and $P$ a prime ideal of $B$ lying above $A$. The residual field of $p$ is the quotient field of $A/p$, which embeds inside the residual field $P$, defined also as the quotient field of $B/P$.
The decomposition group $D$ of $P$ over $K$ is the set of automorphisms $\sigma$ of Gal($L/K$) that stabilize $P$, and the group of inertia $I$ of $P$ in $L$ is the subgroup of elements $\sigma$ of $D$ such that $\sigma x - x$ belong to $P$ for every $x$ in $B$. Everthings works well in the sense that many results of the theory of valuations hold in this setting.
Now, suppose that $L/K$ is finite. The residual degre is defined by $f=[{\rm quot}(B/P) : {\rm quot}(A/p)]$ (assuming it is finite). I would like to define a priori the index of ramification of $P$ over $K$ by $|D| = ef$. Here, there are some problems that are my questions :
1) It is not clear if $f$ may be infinite : I can show that $f$ must be finite if [quot($B/P$) : quot($A/p$)] is separable or $B$ is Noetherian. But in the general case, is there an exemple of infinite residual degree in this setting?
2) Somewhat related to this question: if it can be shown that there is a valuation that extends the canonical surjection $A\to A/p$ to $K$, whose residual field is is a finite extension of quot($A/p$), then (1) could be shown to be true. But in fact, I would be happy to have an example of such an extension by valuation whose residual field is an infinite algebraic extension over $K$, just to know if this can hold.
2) Assuming $f$ finite, it is not clear if or if not $e$ must be an integer (I can show that it must be if the residual field extension is separable, because it is then equal to the cardinal of the inertia group). This does not prevent the multiplicativity of this index, but it is important for further developments to know if or if not it must be an integer (in other words, if or if not the residual degree divides |D|). Any idea will be welcome.
N.B : 1) If this can help, I have proved that $|I| = e p^s$, where $p={\rm char}({\rm quot}(A/p))$ and $p^s=[{\rm quot}(B/P):{\rm quot}(A/p)]_{\rm insep}$.
2) Even if the question is weakened by assuming $p$ maximal in $A$ and $P$ maximal in $B$, (so $A/p$ and $B/P$ are fields), it remains of interest for me.
