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the fundamental group and Galois theory

Let $o$ be a Dedekind domain, let $K:=\mathrm{Quot}(o)$ denote its quotient field, let $L/K$ be a finite separable field extension of degree $n$, let $O\supset o$ be the integral closure of $o$ in $L$. Then $O$ is in particular a Dedekind domain

Let for

$$o\stackrel{i}{\to }O\to L$$o\stackrel{i}{\to} O\to L

$f:=\mathrm{Spec}(i):\mathrm{Spec}(O)\to \mathrm{Spec}(o)$ be the induced map between the ring spectra called ramified covering

Let $p\in \mathrm{Spec}(o)$ be a maximal prime ideal. Then the ideal $\mathrm{pO}$ in $O$ has a unique product decomposition

$$\mathrm{pO}=P_1^{e_1}\dots P_r^{e_r}$$pO=P_1^{e_1}\dots P_r^{e_r}

with different $P_i\in \mathrm{Spec}(O)$. It is custom to introduce some classifying vocabulary depending on the kind of this decomposition and the degree of the field extension $f_i:=[O/P_i:o/p]$: the $e_i$ are called ramification indices and the $f_i$ are called inertia degrees (see e.g. the German wikipedia for more details). These satisfy the fundamental identity ${\Sigma }_i{e}_i{f}_i=n$ and every point $p\in \mathrm{Spec}(O)$ has $\le n$ preimages. If $p$ has $<n$ preimages then $p$ is called ramified.

Let now $L/K$ be galois then the Galois automorphisms $\sigma \in \mathrm{Gal}(L/K):={\mathrm{Aut}}_K(L)$ (i.e. the automorphisms of $L$ which restrict to the identity on $K$) induce automorphisms of schemes $\mathrm{Spec}(\sigma )$ and (since $\sigma$ fixes $o$) the diagram

$$\begin{array}{ccc}\mathrm{Spec}(O)& \stackrel{Spec(\sigma )}{\to }& Spec(O)\\ {\downarrow }^f& \swarrow \\ \mathrm{Spec}(o)\end{array}$$\array{ Spec(O)&\stackrel{\Spec(\sigma)}{\to}&\Spec(O) \\ \downarrow^f &\swarrow \\ Spec(o) }

commutes. $\tau :=Spec(\sigma )$ is an example of a cover automorphism (also called cover transformation or Deck transformation). Since $\mathrm{Spec}:\mathrm{Rings}\to \mathrm{Schemes}$ is an equivalence of categories we have an isomorphism

$$\mathrm{Gal}(L/K)\simeq {\mathrm{Aut}}_{\mathrm{Spec}(o)}(\mathrm{Spec}(O))$$Gal(L/K)\simeq Aut_{Spec(o)}(Spec(O))

where the object to the right we have the group of cover automorphism.

One can show that there is a maximal unramified extension ($e_i=1$ and the extensions $O/P_i:o/p$ being separable) $\tilde{K}$ of $K$ and the scheme $\tilde{Y}:=\mathrm{Spec}(\tilde{o})$ where $\tilde{o}$ denotes the integral closure of $o$ in $\tilde{K}$. Then $f:\tilde{Y}\to Y:=\mathrm{Spec}(O)$ satisfies the axioms of a universal covering and consequently we define on the side of schemes the fundamental group

$${\pi }_1(Y):={\mathrm{Aut}}_Y(\tilde{Y})\simeq \mathrm{Gal}(\tilde{K}/K)$$\pi_1(Y):=Aut_Y(\tilde Y)\simeq Gal(\tilde K/ K)

References

• Jürgen Neukirch, algebraic number theory, I.§13.6