= Laurent-monomial description of top cohomology on projective space
{title2=$H^n(\mathbf P^n,\mathcal O(m))$}
= Top Čech cohomology from missing-denominator monomials
{synonym}
For $n\ge1$, 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 $\binom{-m-1}{n}$ for $m\le-n-1$.
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