= Cohomology of twists on projective space
{title2=$H^i(\mathbf P^r,\mathcal O(n))$}
For $r\ge1$, a <twisting sheaf on projective space> has $H^0(\mathbf P_k^r,\mathcal O(n))=k[X_0,\ldots,X_r]_n$ when $n\ge0$, 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 $\binom{n+r}{r}$ and $\binom{-n-1}{r}$ 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 $\operatorname{Spec}k$ and has $H^0=k$.
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