= Čech cohomology of twists on the projective line
{c}
{title2=$h^0(\mathcal O(n))=\max(n+1,0),\quad h^1(\mathcal O(n))=\max(-n-1,0)$}
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 $(a,b)\mapsto t^nb-a$. The kernel has basis $1,t,\ldots,t^n$ for $n\geq0$. The cokernel has basis $t^{n+1},\ldots,t^{-1}$ for $n\leq-2$. The two-chart complex also proves vanishing in degrees at least two.
Back to article page
