The group-cohomology tag has no wiki summary.

**1**

vote

**2**answers

413 views

### Interesting representations/cohomology of surface groups?

For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not ...

**2**

votes

**1**answer

671 views

### Relation between Sheaf and Group Cohomology

Let $E=\mathbb{C}/L$ be an elliptic curve. Then $\mathbb{C}$ is contractible, and $L$ is the fundamental group of $E$. What's interesting is that we can find the cohomology of $E$, which is the same ...

**12**

votes

**2**answers

741 views

### Geometric model for classifying spaces of alternating groups

The classifying space of the nth symmetric group $S_n$ is well-known to be modeled by the space of subsets of $R^\infty$ of cardinality $n$. Various subgroups of $S_n$ have related models. For ...

**4**

votes

**1**answer

149 views

### How large a subset do you need to uniquely determine a 2-cocycle?

Suppose A and B are abelian groups. I want to find subsets D of $A \times A$ such that any 2-cocycle $c:A \times A \to B$ for the trivial action is uniquely determined by what it does on D. ...

**14**

votes

**1**answer

1k views

### Are all Galois cohomology groups also étale cohomology groups?

Let $K$ be a field and $K^s$ a separable closure of $K$, and let $\mathcal{F}$ be a sheaf on $\mathrm{Spec}(K)$ (in the étale topology).
By Grothendieck's Galois Theory, we have the isomorphism
...

**10**

votes

**2**answers

667 views

### Hilbert 90 for algebras

Let $L\diagup K$ be a Galois extension of fields satisfying $\left[L:K\right] < \infty$. Let $B$ be a finite-dimensional (as a $K$-vector space) $K$-algebra. Then, the Galois group $G$ of $L\diagup ...

**6**

votes

**4**answers

522 views

### Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields

My motivation is to understand the following situation: Given absolutely and almost simple algebraic group $G$ defined over a number field $k$ and a finite valuation $v$ on $k$, when $G(k_v)$ can be ...

**9**

votes

**3**answers

955 views

### Zariski tangent spaces to representation varieties

In Bill Goldman's paper "The Symplectic Nature of the Fundamental Groups of Surfaces" (Advances, 54, 200-225, '84) it is stated that the "Zariski tangent space" to a representation space Hom$(\pi, ...

**0**

votes

**1**answer

402 views

### Must finite groups with isomorphic commutators and quotients be isomorphic?

Let G and H be finite groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. I am looking for an example where G' is isomorphic to H' and G/G' is ...

**8**

votes

**1**answer

398 views

### Cohomology of orthogonal and symplectic groups

Hello,
in their book Cohomology of Finite Groups Adem and Milgram investigate the cohomology of the finite orthogonal and symplectic groups only in case $\mathbb{F}_2$.
Let $p$ be a prime dividing ...

**2**

votes

**2**answers

242 views

### How to fit res map into a long exact sequence?

Let G be a finite group, H a subgroup and V a G-module. Then the embedding H in G induces a restriction map on $H^{n}(G,A)$ to $H^{n}(H,A)$. My question is that is there any long exact sequence which ...

**5**

votes

**2**answers

876 views

### Question about computing group cohomology using cochains

In Milne's notes on Class Field Theory (http://www.jmilne.org/math/CourseNotes/CFT.pdf), he initially defines group cohomology in terms of injective resolutions, then he talks about computing ...

**2**

votes

**0**answers

909 views

### group cohomology and cohomology of classifying space [closed]

Let $G$ be a discrete group, and $BG$ is the classifying space.
It is well-known that the group cohomology of $G$-module M, is the same as the cohomology on $BG$ with coefficient in $\tilde{M}$, which ...

**8**

votes

**2**answers

1k views

### In what sense (if any) does the cohomology of profinite groups commute with projective limits?

Background:
Let $G$ be a profinite group. If $M$ is a discrete $G$-module, then $M=\varinjlim_U M^U$, where the direct limit is taken with respect to inclusions over all open normal subgroups of $G$, ...

**3**

votes

**1**answer

519 views

### Injectivity of Transfer (Verlagerung) map

Let $ K $ be a finite extension of a $p$-adic field or a number field, L a finite extension of $K$. The following fact holds: $
\text{Gal}(K^{\text{ab}} / K) \rightarrow \text{Gal}(L^{\text{ab}} / L) ...

**6**

votes

**2**answers

434 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 primes. Meaning that ...

**6**

votes

**3**answers

769 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 topological cohomology if A ...

**7**

votes

**2**answers

297 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 $\chi\in H^1(G,\mathbb{F}_p)$, we have ...

**10**

votes

**0**answers

363 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 group $G^\delta$. ...

**11**

votes

**5**answers

1k 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 ...

**1**

vote

**2**answers

276 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) which split when ...

**14**

votes

**7**answers

2k 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 the lines of
...

**5**

votes

**4**answers

1k 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 group defined over ...

**34**

votes

**9**answers

6k 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 ideal-theoretic arguments). I've seen the proof of the long exact sequence ...

**8**

votes

**1**answer

466 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's twisting, but in ...

**18**

votes

**5**answers

1k 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 with no references ...

**24**

votes

**6**answers

3k 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 in ordinary group ...