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I have seen a lot of places where the group cohomology when a group acts on a module, is extensively used. But beyond seeing the definition and some claims of partial results, I havent seen any uses of the case when we replace action on a module by action on noncommutative group. It seems hard to believe that much can be made out of it as H_1 is just a set, not even a group. Can somebody explain some uses of bringing up and studying this notion?

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up vote 10 down vote accepted

Often elements of $H^1$ are used to classify objects of a certain kind. E.g. if $k$ is a field and $k^s$ its separable closure, and $G_k = Gal(k^s/k)$, then if $X$ is a variety over $k$, and $Aut(X)$ is the group of automorphisms of $X_{/k^s}$, then the set $H^1(G_k,Aut(X))$ classifies varieties over $k$ that become isomorphic to $X$ when both are regarded as varieties over $k^s$. The base point of $H^1$ corresponds to the original variety $X$ itself. (Such varieties are called twists of $X$.)

This is an important construction, which (along with variants) is used all the time in arithmetic geometry, including (perhaps especially) the theory of algebraic groups.

As one concrete example, let me mention a famous example, namely the result that $H^1(G_k, GL_n(k^s)) = 1$. (A generalized form of Hilbert's Thm. 90.) If we now consider the short exact sequence $1 \to (k^s)^{\times} \to GL_n(k^s) \to PGL_n(k^s) \to 1,$ we then obtain an injection $H^1(G_k, PGL_n(k^s))\hookrightarrow H^2(G_k,(k^s)^{\times}).$

Now $PGL_n(k^s)$ acts as automorphisms of $\mathbb P^{n-1}$, so we see that the twists of projective spaces (such twists are called Brauer--Severi varieties) are classified by certain elements of the Brauer group. For example, smooth conics in the plane (which are twists of $\mathbb P^1$, because over $k^s$ they acquire rational points, and so are isomorphic to $\mathbb P^1$) correspond to quaternion algebras. (To see this concretely --- or rather, to go the other way --- if $D$ is a quaternion algebra, then $D^{tr = 0}$, the subspace of elements of reduced trace zero, is 3-dimensional, and the reduced norm gives a quadratic form on this space. Passing to the associated two-dimensional projective space, we get a conic in a projective plane. This is more concrete than the cohomological story, but the cohomological story has the advantage of being very general, and relies just on a general fact, namely Hilbert's Thm. 90, rather than specific knowledge of the geometric situation. This is one typical advantage of cohomological arguments and constructions, when you can find them/make them.)

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Matt E's response is the canonical first answer. As someone (else) who works with nonabelian group (in particular, Galois) cohomology frequently, let me give a second answer.

Certain interesting maps between commutative cohomology groups are defined via a non-commutative intermediary. The justification for this is that, though one does not in general have anything like a "long exact sequence" in non-commutative cohomology, if one has an extension of $\mathfrak{g}$-modules

$$1 \rightarrow Z \rightarrow G \rightarrow A \rightarrow 1$$

where $A$ is commutative and $Z$ is central, then one gets a connecting map in cohomology

$$\Delta: H^1(\mathfrak{g},A) \rightarrow H^2(\mathfrak{g},Z).$$

Often elements in an $H^2$ may be viewed as "obstructions" to something desirable happening at the $H^1$-level. In particular, this is the case for the period-index obstruction map in the Galois cohomology of abelian varieties. See for instance

http://math.uga.edu/~pete/wc1.pdf

and also publications [12], [14], [16] on

http://www.math.uga.edu/~pete/papers.html

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