Definition of the $0$-coboundary in group cohomology

Question

I'm trying to learn, what is group cohomology. Since I'm not a matematician, the general definition is too abstract to me (at least for the time being), and requires too much category theory and homological algebra. First I'm trying to catch the "technical" definition presented by Wikipedia. This definition starts with the definition of the modules of cochains for $n=0,1,...$, and the coboundary maps from the $n$-th module to the $n+1$-th as follows:

"For $n\ge0$, let $C^n(G,M)$ be the group of all functions from $G^n$ to $M$. This is an abelian group; its elements are called the (inhomogeneous) $n$-cochains. The coboundary homomorphisms

$\begin{cases} d^{n+1} : C^n (G,M) \to C^{n+1}(G,M)\\ \left(d^{n+1}\varphi\right) (g_1, \ldots, g_{n+1}) = g_1 \varphi(g_2,\dots, g_{n+1}) + \sum_{i=1}^n (-1)^i \varphi \left (g_1,\ldots, g_{i-1}, g_i g_{i+1}, \ldots, g_{n+1} \right ) (-1)^{n+1}\varphi(g_1,\ldots, g_n) \end{cases}$

"

This is OK for $n>0$, but it has no sense when $n=0$. What is the definition of the $d^1$ coboundary map?

Answer 1

For $n=0$, $G^n$ has only one element, so a function $G^n\to M$ is essentially an element $m\in M$ (the image of the element of $G^n$).

Now if $m\in M$ is such an element, or correespondingly $f: *\mapsto m$ is the map $G^0\to M$, you have $d^1f $ which is a map $G^1=G\to M$, and the formula gives $d^1f (g) = g\cdot f() + \displaystyle\sum_{i=1}^0$something $+ (-1)^{-1}f()= g\cdot m - m$.

Indeed a sum going from $1$ to $0$ of anything is $0$, and for the first and last terms, you remove on element of the list $(g_1,...,g_{n+1})$, which, for $n=0$, has only one element, so you get a list with $0$ elements : great : $f$ can take those as arguments; ans it returns $m$.

Any map of the form $g\mapsto g\cdot m- m$ is called a principal derivation $G\to M$, and the kernel of $d^2 : C^1(G,M)\to C^2(G,M)$ consists of derivations, that is, maps $d: G\to M$ such that $d(gh) = d(g) + gd(h)$. So $H^1(G,M)$ is the set of derivations modulo principal derivations.