Holomorphic Laurent-series cohomology of twists on the projective line ($H^1({\mathbb{CP}}^1,\mathcal{O}(n))=\mathcal{O}({\mathbb{C}}^{\ast })/[\mathcal{O}(\mathbb{C})+z^n\mathcal{O}(\mathbb{C})(1/z)]$)

For analytic holomorphic sections, the two-chart Čech cohomology uses convergent Laurent series, rather than only Laurent polynomials. A frame on the infinity chart equals $z^n$ times a finite-chart frame. Entire finite-chart functions remove Laurent powers $k\ge 0$, and infinity-chart functions remove powers $k\le n$. The positive and negative tails converge to entire chart functions, so only the finite gap $n<k<0$ remains. Thus the first cohomology has basis $z^{n+1},\dots ,z^{-1}$ when $n\le -2$ and is zero when $n\ge -1$. The same transition relation gives global sections of dimension $\max (n+1,0)$.