I read Example 9.1. Of "Differential forms in Algebraic Topology" Bott&Tu's.
We consider the following Good cover of the circle
And we study they Čech cohomology which is the cohomology induced by the coboundary operator $(\delta\omega)_{\alpha_{0}...\alpha_{n+1}}=\sum(-1)^{i}\omega_{\alpha_{0}...\hat\alpha{i}...\alpha_{n+1}}$
And where a p-cochain is a constant function on a p-intersection of open sets from the good covering.
For instance, we can write $C^{1} = \{(\eta_{01},\eta_{02},\eta_{12}),$ $\eta_{ij}$ constant on $U_{i} \cap U_{j}\}$
It states that $\eta = (1,0,0)$ is a nontrivial 1-cocycle on the circle for the Čech cohomology, which means it is not a coboundary.
But cocycle are supposed to be element of the Kernel of $\delta$ which is not the case for $\eta$.
Why can we state that $\eta$ is a cocylce please ?
I now there is a similar question on the forum but it doesn't answer mine sadly.
Dearly
