 Cohomology of twists on projective space

For $r\ge 1$, a twisting sheaf on projective space has $H^0({\mathbf{P}}_k^r,\mathcal{O}(n))=k[X_0,\dots ,X_r{]}_n$ when $n\ge 0$, and top cohomology with basis the Laurent monomials of total degree $n$ whose exponents are all negative. All intermediate sheaf cohomology vanishes. Their dimensions are $\left(\genfrac{}{}{0ex}{}{n+r}{r}\right)$ and $\left(\genfrac{}{}{0ex}{}{-n-1}{r}\right)$ respectively, with the top group nonzero only for $n\le -r-1$. Decompose the standard Čech cochain complex by exponent vector: no negative exponents gives a simplex with cohomology in degree zero; a nonempty proper negative-support set admits a contracting homotopy inserting a vertex outside that set; all negative exponents give only the top term. For $r=0$, every twist is trivial on $\mathrm{Spec}k$ and has $H^0=k$.