decomposition group
1 Decomposition Group
Let be a Noetherian integrally closed integral domain with field of fractions . Let be a Galois extension of and denote by the integral closure of in . Then, for any prime ideal , the Galois group acts transitively on the set of all prime ideals containing . If we fix a particular prime ideal lying over , then the stabilizer of under this group action is a subgroup of , called the decomposition group at and denoted . In other words,
If is another prime ideal of lying over , then the decomposition groups and are conjugate in via any Galois automorphism mapping to .
2 Inertia Group
Write for the residue field and for the residue field . Assume that the extension is separable (if it is not, then this development is still possible, but considerably more complicated; see [serre, p. 20]). Any element , by definition, fixes and hence descends to a well defined automorphism of the field . Since also fixes by virtue of being in , it induces an automorphism of the extension fixing . We therefore have a group homomorphism
and the kernel (http://planetmath.org/KernelOfAGroupHomomorphism) of this homomorphism is called the inertia group of , and written . It turns out that this homomorphism is actually surjective, so there is an exact sequence
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