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| Ramification of Galois representations ： ウィキペディア英語版 |
Galois module
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.
==Examples==
*Given a field ''K'', the multiplicative group (''Ks'')× of a separable closure of ''K'' is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of ''K'' (by Hilbert's theorem 90, its first cohomology group is zero).
*If ''X'' is a smooth proper scheme over a field ''K'' then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of ''K''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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