Laurent-monomial description of top cohomology on projective space ($H^n({\mathbf{P}}^n,\mathcal{O}(m))$)

For $n\ge 1$, the top Čech cohomology of the standard affine cover of projective space is the degree-$m$ quotient of the Laurent ring by the sum of rings in which at least one variable has not been inverted. Its basis consists of Laurent monomials $X_0^{a_0}\cdots X_n^{a_n}$ with every $a_i<0$ and ${\sum }_i{a}_i=m$. Hence it vanishes for $m\ge -n$ and has dimension $\left(\genfrac{}{}{0ex}{}{-m-1}{n}\right)$ for $m\le -n-1$.