Articles of group cohomology

Prerequisites to study cohomology?

Work related I have to deal with cohomology theory fairly soon. Unfortunately, I never had any classes on this, so I’d like to study it on my own. Before I dive into a book or two, I’d like to make sure that I have all required previous knowledge to actually understand it. It would be […]

Show that image of $res$ lies in $H^n(H,A)^{G/H}$

Let $G$ and $G^{\prime}$ be groups, $A$ and $A^{\prime}$ be $G$-module and $G^{\prime}$-module respectively, $C^n(G,A)$ be set of all maps from $G \times \cdots \times G$ ($n$ times) to $A$, $d_n :C^n(G,A) \rightarrow C^{n+1}(G,A)$ be coboundary operator, $H^n(G,A)$ be $n$ dimensional cohomology group. The group homomorphisms $\phi:G^{\prime} \rightarrow G$ and $\psi : A \rightarrow A^{\prime}$ […]

Showing that $\operatorname {Br}(\Bbb F_q)=0$

I want to prove that $\operatorname {Br}(\Bbb F_q)=0$ using the cohomological description of the Brauer group. We have: $\operatorname {Br}(\Bbb F_q)=H^2(\operatorname {Gal}(\overline {\Bbb F_q}/\Bbb F_q), \overline {\Bbb F_q}^*)$. Since $\operatorname {Gal}(\overline {\Bbb F_q}/\Bbb F_q)$ is just $\widehat {\Bbb Z}$, this is just $H^2(\widehat{\Bbb Z}, \overline {\Bbb F_q}^*) = \operatorname{Ext}_{\Bbb Z[\widehat{\Bbb Z}]}^2(\Bbb Z, \overline {\Bbb F_q}^*)$. […]

Lang's Algebra: Herbrand quotient

I’ve looked around a lot and couldn’t find much help (at least that I could understand) on this question – it is 1.45 in Lang’s Algebra book: Let $G$ be a cyclic group of order $n$, generated by $\sigma$. Assume $G$ acts on an abelian group $A$ as groups s.t. $\sigma(x+y) = \sigma(x)+\sigma(y)$ for $x,y […]

What is the motivation for defining both homogeneous and inhomogeneous cochains?

In my few months of studying group cohomology, I’ve seen two “standard” complexes that are introduced: We let $X_r$ be the free $\mathbb{Z}[G]$-module on $G^r$ (so, it has as a $\mathbb{Z}[G]$-basis the $r$-tuples $(g_1,\ldots,g_r)$ of elements of $G$). The $G$-module structure of $X_r$ comes by virtue of being a $\mathbb{Z}[G]$-module. The boundary maps $\partial_r:X_r\to X_{r-1}$ […]

Interpretations of the first cohomology group

I’ve been revisiting group cohomology, and I realized that there is something I never quite understood. Let $G$ be a finite group, and let $A$ be a $G$-module (i.e. $\mathbb{Z}[G]$-module). Then the second cohomology $H^2(G,A)$ classifies group extensions $1\rightarrow A\rightarrow E\rightarrow G\rightarrow 1$ such that the action of $G$ on $A$ by left conjugation jibes […]

What is the intuition between 1-cocycles (group cohomology)?

This is, I’m sure, an incredibly naive question, but: is there a simple explanation for why one should be interested in 1-cocycles? Let me explain a bit. Given an action of a group $G$ on another group $A$ (the group structure of $A$ is respected by the action, in the sense that $\tau(ab)=(\tau a)(\tau b)$ […]

Completion and algebraic closure commutable

The following corollary of Krasner´s Lemma says: Let k be a global field and p a prime of k. Then $(\overline{k})_p=\overline{k_p}$. Im wondering if $(\overline{k})_p$ means the completion of $\overline{k}$ because i know that $\overline{\mathbb{Q}_p}$ is not complete. So i think it means $\bigcup L_{ip}$ with the $L_{ip}$ ranging over all finite extensions of k. […]

$H^1$ of $\Bbb Z$ as a trivial $G$-module is the abelianization of $G$

This question already has an answer here: Isomorphism between $I_G/I_G^2$ and $G/G'$ 1 answer

What exactly is a trivial module?

Yes, this is a quite basic answer, but I have to admit to be absolutely confused about this notion. Searching on the web, I managed to found two possible definition of trivial modules, referring actually to two different mathematical objects. The first one is just the singleton set with the only possible module structure, also […]