Exercise 111.43.1. Let $k$ be a field. Let $X \subset \mathbf{P}^ n_ k$ be the “coordinate cross”. Namely, let $X$ be defined by the homogeneous equations

$T_ i T_ j = 0\text{ for }i > j > 0$

where as usual we write $\mathbf{P}^ n_ k = \text{Proj}(k[T_0, \ldots , T_ n])$. In other words, $X$ is the closed subscheme corresponding to the quotient $k[T_0, \ldots , T_ n]/(T_ iT_ j; i > j > 0)$ of the polynomial ring. Compute $H^ i(X, \mathcal{O}_ X)$ for all $i$. Hint: use Čech cohomology.

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