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the fundamental group and Galois theory

Let oo be a Dedekind domain, let K:=Quot(o)K:=Quot(o) denote its quotient field, let L/KL/K be a finite separable field extension of degree nn, let OoO\supset o be the integral closure of oo in LL. Then OO is in particular a Dedekind domain

Let for

oiOLo\stackrel{i}{\to} O\to L

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

Let pSpec(o)p\in Spec(o) be a maximal prime ideal. Then the ideal pOpO in OO has a unique product decomposition

pO=P 1 e 1P r e rpO=P_1^{e_1}\dots P_r^{e_r}

with different P iSpec(O)P_i\in 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]f_i:=[O/P_i:o/p]: the e ie_i are called ramification indices and the f if_i are called inertia degrees (see e.g. the German wikipedia for more details). These satisfy the fundamental identity Σ ie if i=n\Sigma_i e_i f_i =n and every point pSpec(O)p\in Spec(O) has n\le n preimages. If pp has <n\lt n preimages then pp is called ramified.

Let now L/KL/K be galois then the Galois automorphisms σGal(L/K):=Aut K(L)\sigma\in Gal(L/K):=Aut_K(L) (i.e. the automorphisms of LL which restrict to the identity on KK) induce automorphisms of schemes Spec(σ)Spec(\sigma) and (since σ\sigma fixes oo) the diagram

Spec(O) Spec(σ) Spec(O) f Spec(o)\array{ Spec(O)&\stackrel{\Spec(\sigma)}{\to}&\Spec(O) \\ \downarrow^f &\swarrow \\ Spec(o) }

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

Gal(L/K)Aut Spec(o)(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=1e_i=1 and the extensions O/P i:o/pO/P_i:o/p being separable) K˜\tilde K of KK and the scheme Y˜:=Spec(o˜)\tilde Y:=Spec(\tilde o) where o˜\tilde o denotes the integral closure of oo in K˜\tilde K. Then f:Y˜Y:=Spec(O)f:\tilde Y\to Y:=Spec(O) satisfies the axioms of a universal covering and consequently we define on the side of schemes the fundamental group

π 1(Y):=Aut Y(Y˜)Gal(K˜/K)\pi_1(Y):=Aut_Y(\tilde Y)\simeq Gal(\tilde K/ K)

References

  • Jürgen Neukirch, algebraic number theory, I.§13.6
Created on August 25, 2012 23:31:17 by Stephan Alexander Spahn (79.227.155.61)