= Čech cohomology of the punctured affine plane
{c}
{title2=$H^1(\mathbb A_k^2\setminus\{0\},\mathcal O)=\bigoplus_{a,b\ge1}kx^{-a}y^{-b}$}
The two affine opens $D(x),D(y)$ cover the <punctured affine plane>. Their intersection has ring $k[x^{\pm1},y^{\pm1}]$. The <Čech cohomology> in degree one is the quotient by $k[x^{\pm1},y]+k[x,y^{\pm1}]$. Its basis consists precisely of the Laurent monomials whose two exponents are negative, so this <vector space> is infinite-dimensional. Global functions, by contrast, form $k[x,y]$.
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