decomposition group

1 Decomposition Group

Let $A$ be a Noetherian integrally closed integral domain with fieldof fractions $K$ . Let $L$ be a Galois extension of $K$ and denote by $B$ the integral closure of $A$ in $L$ . Then, for any prime ideal ${\\mathfrak{p}}\\subset A$ , the Galois group $G:=\\operatorname{Gal}(L/K)$ acts transitively on theset of all prime ideals ${\\mathfrak{P}}\\subset B$ containing ${\\mathfrak{p}}$ . If we fix aparticular prime ideal ${\\mathfrak{P}}\\subset B$ lying over ${\\mathfrak{p}}$ , then thestabilizer of ${\\mathfrak{P}}$ under this group action is a subgroup of $G$ , called the decomposition group at ${\\mathfrak{P}}$ and denoted $D({\\mathfrak{P}}/{\\mathfrak{p}})$ . In other words,

D({\\mathfrak{P}}/{\\mathfrak{p}}):=\\{\\sigma\\in G\\mid\\sigma({\\mathfrak{P}})=({%\\mathfrak{P}})\\}.

If ${\\mathfrak{P}}^{\\prime}\\subset B$ is another prime ideal of $B$ lying over ${\\mathfrak{p}}$ , thenthe decomposition groups $D({\\mathfrak{P}}/{\\mathfrak{p}})$ and $D({\\mathfrak{P}}^{\\prime}/{\\mathfrak{p}})$ are conjugate in $G$ via any Galois automorphism mapping ${\\mathfrak{P}}$ to ${\\mathfrak{P}}^{\\prime}$ .

2 Inertia Group

Write $l$ for the residue field $B/{\\mathfrak{P}}$ and $k$ for the residue field $A/{\\mathfrak{p}}$ . Assume that the extension $l/k$ is separable (if it is not,then this development is still possible, but considerably morecomplicated; see [serre, p. 20]). Any element $\\sigma\\in D({\\mathfrak{P}}/{\\mathfrak{p}})$ , by definition, fixes ${\\mathfrak{P}}$ and hence descends to a welldefined automorphism of the field $l$ . Since $\\sigma$ 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({\\mathfrak{P}}/{\\mathfrak{p}})\\longrightarrow\\operatorname{Gal}(l/k),

and the kernel (http://planetmath.org/KernelOfAGroupHomomorphism) of this homomorphism is called the inertia group of ${\\mathfrak{P}}$ , and written $T({\\mathfrak{P}}/{\\mathfrak{p}})$ . It turns out that this homomorphism isactually surjective, so there is an exact sequence

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