= Holomorphic first cohomology of punctured complex two-space
{title2=$H^1(\mathbb C^2\setminus\{0\},\mathcal O)\cong w_1w_2\mathcal O(\mathbb C_w^2)$}
For $X=\mathbb C^2\setminus\{0\}$, the cover $\mathbb C\times\mathbb C^*$ and $\mathbb C^*\times\mathbb C$ is acyclic for $\mathcal O_X$. The first <Čech cohomology> quotient is represented uniquely by normally convergent doubly negative <Laurent series>. Under $w_i=1/z_i$ it is $w_1w_2\mathcal O(\mathbb C_w^2)$, the entire functions divisible by both coordinates. This is an infinite-dimensional analytic vector space; restricting representatives to finite Laurent polynomials loses classes. The class of $1/(z_1z_2)$ is nonzero, so $X$ is not a <Stein manifold>.
Back to article page
