Čech cohomology of twists on the projective line ($h^0(\mathcal{O}(n))=\max (n+1,0),h^1(\mathcal{O}(n))=\max (-n-1,0)$)

On the standard two-affine cover of the projective line, with coordinate $t$, a frame for the twisting sheaf on projective space $\mathcal{O}(n)$ on the second chart equals $t^n$ times the frame on the first. Its Čech cochain complex is $k[t]\oplus k[t^{-1}]\to k[t,t^{-1}]$, with map $(a,b)\mapsto t^{n}b-a$. The kernel has basis $1,t,\dots ,t^n$ for $n\ge 0$. The cokernel has basis $t^{n+1},\dots ,t^{-1}$ for $n\le -2$. The two-chart complex also proves vanishing in degrees at least two.