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Results tagged with group-cohomology
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user 4008
In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group.
19
votes
Accepted
Hilbert 90 for algebras
It's actually easier to go the other way around. Finite dimensional modules over
an algebra $A$ fulfils the Krull-Remak-Schmidt theorem of being isomorphic to a
direct sum of indecomposable modules wi …
- 22.6k
16
votes
Accepted
Where can I easily look up / calculate (abelian) group cohomology?
This group is best understood in terms of the universal coefficient formula,
i.e., in terms of the homology of the involved group. Hence, if $A$ is any
abelian group we have $H_1(A)=A$ and the additio …
- 22.6k
14
votes
Accepted
Injection of Ext into H^2
You get a description from the universal coefficient theorem which gives a (split) exact sequence
$$
0\to \mathrm{Ext}(H_1(G),A) \to H^2(G,A) \to \mathrm{Hom}(H_2(G),A) \to 0
$$
and the fact that $H_1 …
- 22.6k
2
votes
How to fit res map into a long exact sequence?
There is a long exact sequence but I think it is largely useless: We have that for any $H$-module $B$ the cohomology $H^n(H,B)$ is equal to the cohomology $H^n(G,B^G_H)$ of the induced module $B^G_H$. …
- 22.6k