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For writing a local class field theory using Galois cohomology,
maybe first step is to determine a Braue group of a local field.
it is known that Brauer group of a local field is isomorphic to $\mathbb {Q/Z}$ .
is it a coincidence? How can it be exactly $\mathbb {Q/Z}$? .
It seems that God of math says "you have to take a Pontryagin dual!"
When Tate was finding local duality,
How could he know cup product make dual relation exactly?
For me, all of these are quite mysterious.
Does anybody know natural reason that local Tate duality is logical consequence?

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Dear Wan Lee, it is not at all clear what you are actually asking. Please try to ask a precise question that actually admits a definitive answer. See the "how to ask" link at the top, and browse around a bit to get a feel for what sorts of questions this site is intended for. – Alex Bartel Dec 31 at 14:00
"When Tate was finding local duality, How could he know cup product make dual relation exactly?" Actually, he didn't. He originally proved a duality theorem for abelian varieties over local fields, observed that it implied a local duality for modules occurring in the abelian varieties, and only later realized that the local duality held for all modules. Concerning your general question of why all this holds. Well, we can prove it. I'm sure there a vague philosophical reasons why it must hold, but they are probably not very helpful. – anon Jan 1 at 4:45

closed as spam by Chandan Singh Dalawat, Franz Lemmermeyer, François Brunault, Alex Bartel, Fernando Muro Dec 31 at 12:30

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