《Galois cohomology》

少一点思考就多一点效率 ~

* * * 9:50

Let K'1/K1 and K'2/K2 be two Galois extensions, with Galois groups G1 and G2.

---- 取两个 Galois 扩展，及对应的 Galois 群.

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Assume we are given an injection K1 -i-> K2.

---- 考虑 K1 和 K2 之间的单射.

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Let us suppose that there exists an injection K'1 -j-> K'2 which extents the inclusion i. 

---- 假定 K'1 和 K'2 之间存在单射，它扩展 i.

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Using j, we get a homomorphism G2 --> G1 and a morphism A(K'1) --> A(K'2);...

---- 利用 j, 可得到 同态 G2 --> G1 (?) 和 态射 A(K'1) --> A(K'2);...

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... these two maps are compatible, and define maps H^q(G1, A(K'1)) --> H^q(G2, A(K'2));...

---- 此二映射相容，并定义映射 H^q(G1, A(K'1)) --> H^q(G2, A(K'2)).

注：后一个 “maps” 也是复数 (q 值不只一个).

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... these maps do not depend on the choice of j (cf. [145], p. 164).

---- 这些映射不依赖于 j 的选择.

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Thus we have maps H^q(K'1/K1, A) --> H^q(K'2/K2, A) which depend only on i (and on the existence of j).

---- 总之有映射 H^q(K'1/K1, A) --> H^q(K'2/K2, A) 并且只依赖于 i (及 j 的存在).

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评论：有了 Galois cohomology，就该考虑它们之间的映射.(正如有了集合，就该考虑他们之间的映射那样)

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小结：引入了映射 H^q(K'1/K1, A) --> H^q(K'2/K2, A).

* * * 10:40