All Questions
8 questions
3
votes
3
answers
714
views
Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$
I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...
7
votes
0
answers
140
views
Is there a homological interpretation for the cokernel of the kernel of a map between complexes induced by tensor product?
Let $A$ be a free abelian group of rank 2, and let $S = \mathbb{Z}[A]\cong\mathbb{Z}[a_1^{\pm1},a_2^{\pm1}]$ the group algebra for $A$.
Let $t : S\times S\rightarrow S$ be the $S$-module map given by ...
- 7,527
1
vote
0
answers
397
views
A functor on the category of commutative rings, algebras or Banach algebras
Edit: According to the comments of abx and Yemon Choi I revise the question as follows:
Let $G$ be a group and $\mathcal{A_G}$ be the category of $G$-module commutative algebras, that is the ...
- 356
7
votes
1
answer
716
views
How to compute the group cohomology of $\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$ with coefficient in a trivial module?
The group cohomology of cyclic groups can be computed easily due to the periodity. Now how can one compute the group cohomology $H^r(\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z},M)$? As least ...
- 997
2
votes
0
answers
176
views
Trivial Tate modules
Let $A$ be an abelian group, and $p$ a prime.
I'll call $$T_p(A) := \text{Hom}_{\mathbf{Z}}(\mathbf{Q}_{p}/\mathbf{Z}_{p}, A).$$
If $A$ is finite, then $T_p(A)$ is trivial, but the converse is not ...
user119470
7
votes
0
answers
344
views
Short exact sequence in nonabelian group cohomology and finitness condition
Let $1\to A\to B\to C\to 1$ be an exact sequence of (nonabelian) $G$-groups. Then there is a well-known exact sequence of pointed sets
$
1\to A^G\to B^G\to C^G\to H^1(G,A)\to H^1(G,B)\to H^1(G,C)
$
...
- 71
14
votes
2
answers
807
views
1-st cohomology of multiplicative group in a vector space
Let $\mathbb k$ be a field of characteristic $p$ and let $\mathbb k_n$ be a 1-dimensional representation of $\mathbb k^\times$, where the action is given by $t\circ v= t^n v$. Is it known what are the ...
- 790
4
votes
3
answers
518
views
Computing the cardinality of cohomology groups
I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite.
Let us assume we are given a group $G$ and a ...