Čech cohomology of the punctured affine plane ($H^1({\mathbb{A}}_k^2\setminus \{0\},\mathcal{O})={⨁}_{a,b\ge 1}k{x}^{-a}{y}^{-b}$)

The two affine opens $D(x),D(y)$ cover the punctured affine plane. Their intersection has ring $k[x^{\pm 1},y^{\pm 1}]$. The Čech cohomology in degree one is the quotient by $k[x^{\pm 1},y]+k[x,y^{\pm 1}]$. 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 $k[x,y]$.