= Cohomology of the projective plane minus a section
{title2=$H^1(X,\mathcal O_X)=\bigoplus_{m,n\geq1}R[x_0^{m+n}/(x_1^mx_2^n)]$}
For a <commutative ring> $R$, remove the section $[1:0:0]$ from $\mathbb P_R^2$. The two remaining standard charts have <coordinate rings> $R[a,b]$ and $R[a/b,b^{-1}]$ inside the <Laurent polynomial ring> $R[a,b,b^{-1}]$. Their intersection consists of constants: their <monomial> exponent regions are $i\geq0,j\geq0$ and $i\geq0,j\leq-i$, respectively. The <Čech cohomology> in degree one is free over $R$ on the missing <monomials> $a^ib^{-k}$ with $1\leq k<i$. Equivalently its <basis> consists of the classes of $x_0^{m+n}/(x_1^mx_2^n)$ for $m,n\geq1$. Degree zero is $R$ and every degree at least two is zero. The charts and their intersection are <affine schemes>, so this computes <sheaf cohomology>. If $R$ is nonzero, the first <sheaf cohomology> is nonzero and this <scheme> is not an <affine scheme>.
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