How does the first group cohomology classify torsors?

Asked Jun 16 '14 at 00:27

Active Jun 16 '14 at 00:42

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I know this is true, but I'm having some trouble finding any references on this.

I'm in particular interested in the nonabelian case.

Specifically, let $G$ be a group acting on a group $A$ (both possibly nonabelian). Then $H^1(G,A)$ is defined to be the quotient of the pointed set of functions $f : G\rightarrow A$ satisfying $f(st) = f(s)t(f(t))$ for all $s,t\in G$, by the equivalence relation that $f\sim g$ if there is an $a\in A$ with $g(s) = a^{-1}f(s)s(a)$.

My first question is, can someone describe how this classifies $A$-torsors?

Also, given an action of $G$ on $A$, is there a relationship between $H^1(G,A)$ and $H^1_{\text{triv}}(G,A)$, where the latter is the cohomology with respect to the trivial action of $G$ on $A$? For example, is one of them always contained in the other?

edited Jun 16 '14 at 00:42

asked Jun 16 '14 at 00:27

oxeimon

12,279

1

Do you understand the classical case of $H^1 (X, A)$ classifying principal $A$-bundles on a (nice) space $X$? – Zhen Lin Jun 16 '14 at 00:58
No, though I had always seen group cohomology done from a purely algebraic point of view... – oxeimon Jun 16 '14 at 01:12
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Pity. Both are special cases of the result for topos cohomology. See here. – Zhen Lin Jun 16 '14 at 02:45
@ZhenLin thanks! thats really helpful. – oxeimon Jun 16 '14 at 18:17

How does the first group cohomology classify torsors?

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