The following is an exercise that I have attempted to work through but whose statement I find difficult to parse through:
For an open set $U$ in $\mathbb{R}^n$, let $R$ be the ring of $C^\infty$ real-valued funcions on $U$. Let $\Omega^1(U)$ be the vector space of differential ($C^\infty)$ forms in $U$, which is the free $R$-module with basis $dx_1$, $\ldots, dx_n$. Let $\Omega^k(U)$ be the space of $k$-forms on $U$, which is the $k$-th exterior power of $\Omega^1(U)$ over $R$; it has basis over $R$ of expressions $dx_{i_1} \wedge \cdots \wedge dx_{i_k}$ for $1 \le i_1 < \cdots < i_k \le n$.
Define linear mappings $d^k: \Omega^k(U) \rightarrow \Omega^{k+1}(U)$, which, for $k=0$, takes a function $f$ to its differential $df = \sum_{i=1}^n (\partial f/ \partial x_i) d x_i$. Show that:
(a) $d^{k+1} \circ d^k = 0$;
(b) For $U$ the complement of the origin in $\mathbb{R}^2$, $Ker(d^1)/Im(d^0)$ is not zero. In this case, can you compute it.
This exercise looks interesting but I am not confident on how to begin working through it, so I post it here to see if it draws anyone's interest and if anyone visiting could suggest a strategy for one or both parts.