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decomposition group (Definition)

Decomposition Group

Let $ A$ be a Noetherian integrally closed integral domain with field of 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 the set of all prime ideals $ \P\subset B$ containing $ {\mathfrak{p}}$. If we fix a particular prime ideal $ \P\subset B$ lying over $ {\mathfrak{p}}$, then the stabilizer of $ \P$ under this group action is a subgroup of $ G$, called the decomposition group at $ \P$ and denoted $ D(\P /{\mathfrak{p}})$. In other words,

$\displaystyle D(\P /{\mathfrak{p}}) := \{\sigma \in G \mid \sigma(\P ) = (\P )\}. $
If $ \P ' \subset B$ is another prime ideal of $ B$ lying over $ {\mathfrak{p}}$, then the decomposition groups $ D(\P /{\mathfrak{p}})$ and $ D(\P '/{\mathfrak{p}})$ are conjugate in $ G$ via any Galois automorphism mapping $ \P$ to $ \P '$.

Inertia Group

Write $ l$ for the residue field $ B/\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 more complicated; see [1, p. 20]). Any element $ \sigma \in D(\P /{\mathfrak{p}})$, by definition, fixes $ \P$ and hence descends to a well defined 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

$\displaystyle D(\P /{\mathfrak{p}}) \longrightarrow \operatorname{Gal}(l/k), $
and the kernel of this homomorphism is called the inertia group of $ \P$, and written $ T(\P /{\mathfrak{p}})$. It turns out that this homomorphism is actually surjective, so there is an exact sequence
$\displaystyle \xymatrix{ 1 \ar[r] & T(\P /{\mathfrak{p}}) \ar[r] & D(\P /{\mathfrak{p}}) \ar[r] & \operatorname{Gal}(l/k) \ar[r] & 1 }$ (1)

Decomposition of Extensions

The decomposition group is so named because it can be used to decompose the field extension $ L/K$ into a series of intermediate extensions each of which has very simple factorization behavior at $ {\mathfrak{p}}$. If we let $ L^D$ denote the fixed field of $ D(\P /{\mathfrak{p}})$ and $ L^T$ the fixed field of $ T(\P /{\mathfrak{p}})$, then the exact sequence (1) corresponds under Galois theory to the lattice of fields

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ L \ar@{-}[d]^e \ L^T \ar@{-}[d]^f \ L^D \ar@{-}[d]^g \ K } } \end{xy}$
If we write $ e,f,g$ for the degrees of these intermediate extensions as in the diagram, then we have the following remarkable series of equalities:
  1. The number $ e$ equals the ramification index $ e(\P /{\mathfrak{p}})$ of $ \P$ over $ {\mathfrak{p}}$, which is independent of the choice of prime ideal $ \P$ lying over $ {\mathfrak{p}}$ since $ L/K$ is Galois.
  2. The number $ f$ equals the inertial degree $ f(\P /{\mathfrak{p}})$ of $ \P$ over $ {\mathfrak{p}}$, which is also independent of the choice of prime ideal $ \P$ since $ L/K$ is Galois.
  3. The number $ g$ is equal to the number of prime ideals $ \P$ of $ B$ that lie over $ {\mathfrak{p}}\subset A$.

Furthermore, the fields $ L^D$ and $ L^T$ have the following independent characterizations:

  • $ L^T$ is the smallest intermediate field $ F$ such that $ \P$ is totally ramified over $ \P\cap F$, and it is the largest intermediate field such that $ e(\P\cap F, {\mathfrak{p}}) = 1$.
  • $ L^D$ is the smallest intermediate field $ F$ such that $ \P$ is the only prime of $ B$ lying over $ \P\cap F$, and it is the largest intermediate field such that $ e(\P\cap F, {\mathfrak{p}}) = f(\P\cap F, {\mathfrak{p}}) = 1$.

Informally, this decomposition of the extension says that the extension $ L^D/K$ encapsulates all of the factorization of $ {\mathfrak{p}}$ into distinct primes, while the extension $ L^T/L^D$ is the source of all the inertial degree in $ \P$ over $ {\mathfrak{p}}$ and the extension $ L/L^T$ is responsible for all of the ramification that occurs over $ {\mathfrak{p}}$.

Localization

The decomposition groups and inertia groups of $ \P$ behave well under localization. That is, the decomposition and inertia groups of $ \P B_\P\subset B_\P$ over the prime ideal $ {\mathfrak{p}}A_{\mathfrak{p}}$ in the localization $ A_{\mathfrak{p}}$ of $ A$ are identical to the ones obtained using $ A$ and $ B$ themselves. In fact, the same holds true even in the completions of the local rings $ A_{\mathfrak{p}}$ and $ B_\P$ at $ {\mathfrak{p}}$ and $ \P$.

Bibliography

1
J.P. Serre, Local Fields, Springer-Verlag, 1979 (GTM 67)



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See Also: ramification, inertial degree

Other names:  splitting group
Also defines:  inertia group

Attachments:
examples of prime ideal decomposition in number fields (Example) by alozano
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Cross-references: local rings, completions, even, localization, source, decomposition, prime, totally ramified, characterizations, inertial degree, independent, ramification index, equalities, degrees, lattice of fields, Galois theory, fixed field, behavior, simple, series, field extension, exact sequence, surjective, homomorphism, group homomorphism, induces, field, well defined, development, separable, extension, residue field, mapping, automorphism, conjugate, subgroup, group action, stabilizer, fix, Galois group, prime ideal, integral closure, Galois extension, field of fractions, integral domain, integrally closed, Noetherian
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This is version 7 of decomposition group, born on 2002-06-04, modified 2005-03-05.
Object id is 3022, canonical name is DecompositionGroup.
Accessed 6465 times total.

Classification:
AMS MSC: 11S15 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Ramification and extension theory)
 13B02 (Commutative rings and algebras :: Ring extensions and related topics :: Extension theory)
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