Galois cohomology

[TheMatrix]

https://www.zhihu.com/question/61000215 ... 4351678535

[FoxMe]

Galois cohomology就是Galois群上的cohomology。起因之一是class field tower problem：

Is there a number field with an infinite class field tower?

具体是啥？还没搞懂

[FoxMe]

Number Theory is the study of G = Gal(Q ̄ /Q), the group of automorphisms of the algebraic closure Q ̄, and the sets G naturally acts on. The following question, for example, is an open problem: is every finite group a quotient of G?

Galois cohomology involves studying the group G by applying homological algebra. This provides a natural way to classify objects, e.g. twists of a curve, and linearizes problems by defining new invariants, revealing previously hidden structure.

[FoxMe]

首先要理解Galois module的概念：

Galois module is a G-module, with G being the Galois group。

G-module就是group ring Z[G]上的模。比如Then the ring O_L of algebraic integers of L can be considered as an O_K[G]-module, and one can ask what its structure is.

然后研究G作用在Galois module上。这是Galois representation的基本问题。

[TheMatrix]

Galois cohomology就是Galois群上的cohomology。起因之一是class field tower problem：

Is there a number field with an infinite class field tower?

具体是啥？还没搞懂

我知道cohomology研究的是xx的obstruction，但是还没有建立具体的图像。

[TheMatrix]

首先要理解Galois module的概念：

Galois module is a G-module, with G being the Galois group。

G-module就是group ring Z[G]上的模。比如Then the ring O_L of algebraic integers of L can be considered as an O_K[G]-module, and one can ask what its structure is.

然后研究G作用在Galois module上。这是Galois representation的基本问题。

对。群表示，群作用，group algebra 模，这三个差不多是同一个意思。哦。在加上G-module，这四个差不多是同一个意思。

[FoxMe]

来看几个简单的例子。令G=Gal(L/K), A=L*.

H⁰(G, A) = A^G = {a ∈ A : s.a = a for all s ∈ G} = K*. 这个trivial情况没啥意思，只是定义。有意思的是：

H¹(G, A) = 0. 搞懂这个结果都不容易，首先要知道

We define H¹(G, A) as 1-cocycles modulo an equivalence relation. Here, a 1-cocycle is a set-theoretic map G → A, s → a_s such that a_st = a_s.s(a_t), and we say a_s ∼ b_s if there exists an a ∈ A with b_s = a⁻¹.a_s.s(a) for all s ∈ G.

https://wstein.org/edu/2010/582e/lectur ... mology.pdf

首先要理解啥是cocycle, coboundary... 先得理解chain complex/cohomology:

https://en.wikipedia.org/wiki/Chain_complex#Definitions

Elements of ker(dn) are called n-cocycles, while elements of im(dn−1) are called n-coboundaries.

[FoxMe]

Brauer group of a field K

Br(K) = H² (K, (K^sep)^*) 理解为

Br(K) = H² ( Gal(K^sep / K), (K^sep)^*)

这是infinite extension，留待以后再看

[TheMatrix]

来看几个简单的例子。令G=Gal(L/K), A=L*.

H⁰(G, A) = A^G = {a ∈ A : s.a = a for all s ∈ G} = K*. 这个trivial情况没啥意思，只是定义。有意思的是：

H¹(G, A) = 0. 搞懂这个结果都不容易，首先要知道

We define H¹(G, A) as 1-cocycles modulo an equivalence relation. Here, a 1-cocycle is a set-theoretic map G → A, s → a_s such that a_st = a_s.s(a_t), and we say a_s ∼ b_s if there exists an a ∈ A with b_s = a⁻¹.a_s.s(a) for all s ∈ G.

https://wstein.org/edu/2010/582e/lectur ... mology.pdf

首先要理解啥是cocycle, coboundary... 先得理解chain complex/cohomology:

https://en.wikipedia.org/wiki/Chain_complex#Definitions

Elements of ker(dn) are called n-cocycles, while elements of im(dn−1) are called n-coboundaries.

一般的group cohomology是很抽象的。和Galois group联系在一起的稍微具体一点。但也是很难的。有点抡不动的感觉。