Can anyone explain Galois Cohomology Groups and their significance in Fermat Last Theorem?

[full shell]

Confused on the definition of Cohomology

[scenic turtle]

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[flat mountain]

I'm not an expert, but your notes on the right seem quite clear, which part of the definition do you struggle with ?

[full shell]

I’m confused on why we use Cocyles, n-coderiviations, and n-boundaries to define Galois Cohomology group H^n(G,A)

[simple canyon]

What do you mean exactly? That’s just the way they are defined

you have a cochain complex you compute its cohomology that’s basically it

here you can even see it as the right derived functor of the fixed point functor as they said

[flat mountain]

actually, that's how it is defined (H^n is the n-th derived functor of the fixed point functor under the galois group), and using a particular injective resolution you can compute it as the cohomology of this particular cochain complex

this is standard when computing derived functors

and you want to see it as a derived functor using the fundamental property that if 0 -> A -> B -> C -> 0 is exact, you get a long exact sequence 0 -> A^G -> B^G -> C^G -> H¹(A,G) -> H¹(B,G) -> H¹(C,G) -> H²(A,G) -> … with H^i(·,G) the i-th derived functors

(i.e. to measure the obstruction of A->A^G to be right exact)

[full shell]

Do you specialize in any field?

[flat mountain]

Wdym ? You work over k[G] modules

(k is the base field)

[scenic turtle]

@full shell

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[full shell]

+close

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