decomposition group


1 Decomposition Group

Let A be a NoetherianPlanetmathPlanetmathPlanetmath integrally closedMathworldPlanetmath integral domainMathworldPlanetmath with field of fractionsMathworldPlanetmath K. Let L be a Galois extensionMathworldPlanetmath of K and denote by B the integral closureMathworldPlanetmath of A in L. Then, for any prime idealMathworldPlanetmathPlanetmath 𝔭A, the Galois groupMathworldPlanetmath G:=Gal(L/K) acts transitively on the set of all prime ideals 𝔓B containing 𝔭. If we fix a particular prime ideal 𝔓B lying over 𝔭, then the stabilizerMathworldPlanetmath of 𝔓 under this group actionMathworldPlanetmath is a subgroupMathworldPlanetmathPlanetmath of G, called the decomposition groupMathworldPlanetmath at 𝔓 and denoted D(𝔓/𝔭). In other words,

D(𝔓/𝔭):={σGσ(𝔓)=(𝔓)}.

If 𝔓B is another prime ideal of B lying over 𝔭, then the decomposition groups D(𝔓/𝔭) and D(𝔓/𝔭) are conjugatePlanetmathPlanetmathPlanetmath in G via any Galois automorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath mapping 𝔓 to 𝔓.

2 Inertia Group

Write l for the residue fieldMathworldPlanetmath B/𝔓 and k for the residue field A/𝔭. Assume that the extensionPlanetmathPlanetmathPlanetmathPlanetmath l/k is separablePlanetmathPlanetmath (if it is not, then this development is still possible, but considerably more complicated; see [serre, p. 20]). Any element σD(𝔓/𝔭), by definition, fixes 𝔓 and hence descends to a well defined automorphism of the field l. Since σ also fixes A by virtue of being in G, it induces an automorphism of the extension l/k fixing k. We therefore have a group homomorphism

D(𝔓/𝔭)Gal(l/k),

and the kernel (http://planetmath.org/KernelOfAGroupHomomorphism) of this homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is called the inertia group of 𝔓, and written T(𝔓/𝔭). It turns out that this homomorphism is actually surjectivePlanetmathPlanetmath, so there is an exact sequencePlanetmathPlanetmathPlanetmathPlanetmath

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