In $\mathbb{P}^1_{\mathbb{C}}$, given affine covers $U=(v\neq0),V=(u\neq0)$, we write element $(a,b)$ of $C^0(\{U,V\},\Omega^1)=\{(a,b): a\in \Omega^1(U), b\in \Omega^1(V)\}$ as
$a=\sum^{\infty}_{n=0}a_n u^ndu$, $b=\sum^{\infty}_{n=0}b_n v^ndv=-\sum^{\infty}_{n=0}b_n u^{-n-2} du$,
since $dv=d(u^{-1})=-u^{-2}du$.
An element $c=\sum^{\infty}_{-\infty}a_n u^n du\in \Omega^1(U\cap V)$ is expressible as $\delta(a,b)=-a+b$ if and only if $a_{-1}=0$. Then, it’s said that $H^1(\mathbb{P}^1,\Omega^1)=\mathbb{C}$. I am not sure how it follows? Hope someone could help. Thanks!