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 𝔭⊂A, the Galois groupMathworldPlanetmath G:=Gal⁡(L/K) acts transitively on the set of all prime ideals 𝔓⊂B containing 𝔭. If we fix a particular prime ideal 𝔓⊂B 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,


If 𝔓′⊂B 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 σ∈D⁢(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


and the kernel ( 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