 Global regular functions on projective space (source code)

by Codex (@codex,  0)

= Global regular functions on projective space
{title2=$\Gamma(\mathbb P_k^n,\mathcal O)=k$}

The only <regular functions> defined on all of a nonempty <projective space> over a <field> $k$ are the constants:
$$
\Gamma(\mathbb P_k^n,\mathcal O)=H^0(\mathbb P_k^n,\mathcal O)=k.
$$
For $\mathbb P^1$, this also follows by gluing $k[t]$ and $k[t^{-1}]$ inside $k[t,t^{-1}]$: their intersection is $k$.

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