5
votes
3answers
249 views
Group cohomology vs. topological cohomology in the case of non-trivial action
When A is an abelian group with trivial G-action (G being a discrete group) we get that Hn(G,A)≅Hn(BG,A). Is there a similar connection between group cohomology and topologic …
5
votes
2answers
234 views
Common Computations in Group Cohomology
Let G=A⋊B, where A and B are abelian, and of coprime order. It seems, from my computations (and correct me if I'm wrong), that Z1(Cp,Cq) is trivial, for p and q different pri …
4
votes
2answers
118 views
Coboundary Representations for Trivial Cup Products
Suppose $G$ is a pro-$p$-group, $p$ odd, and $\mathbb{F}_p$ is given the trivial $G$-action. By skew-symmetry of the cup-product in degree 1, given $\chi\in H^1(G,\mathbb{F}_p)$, …
9
votes
5answers
376 views
Tate Cohomology via Stable Categories
Situation
Let $G$ be a finite group and provide $G\text{-mod} := {\mathbb Z}G\text{-mod}$ with the Frobenius structure of ${\mathbb Z}$-split short exact sequences. Denote by $\un …
4
votes
4answers
581 views
Galois cohomology of linear groups over local fields
Let $F$ be a local field of characteristic zero (for simplicity), $\overline{F}$ an algebraic closure of $F$ and $L/F$ a fixed finite Galois extension. If $G$ is a linear algebraic …
12
votes
5answers
373 views
Essential theorems in group (co)homology
I'm trying to fill in the gaps in my understanding of group (co)homology and I'm wondering what are considered the "must know" theorems and concepts. I'm thinking of things along t …
14
votes
6answers
741 views
Intuition for Group Cohomology
I'm beginning to learn cohomology for cyclic groups in preparation for use in the proofs of global class field theory (using ideals). I've seen the proof of the long exact sequence …
16
votes
6answers
843 views
Why does non-abelian group cohomology exist?
If K is a non-abelian group on which a group G acts via automorphisms, we can define 1-cocycles and 1-coboundaries by mimicking the explicit formulas coming from the bar resolution …
7
votes
0answers
130 views
To what extent does (co)homology of groups made discrete depend on set theory?
There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete …
1
vote
2answers
111 views
Relative Frobenius Structure on the Category of G-modules
Let $G$ be a group $H\leq G$ a subgroup of finite index. Further, let ${\mathcal E}^G_H$ denote the class of those short exact sequences of $G$-modules (over some fixed base ring) …
14
votes
5answers
600 views
Why is the standard definition of cocycle the one that _always_ comes up??
This question might not have a good answer. It was something that occurred to me yesterday when I found myself in a pub, needing to do an explicit calculation with 2-cocycles but w …
4
votes
0answers
141 views
substitute for Serre’s twisting when the “twisting” is outer
Does anyone know if there is something that can be said (ideally under at most very mild hypotheses) in group cohomology (let's even restrict to degree 1) that is similar to Serre' …
