$\mathcal{O}_{\mathsf{Proj} \: A}(\mathsf{Proj} \: A)=A$?

Question

It's very well known that if one takes an affine scheme $X=\mathsf{Spec} \: A$, then $\mathcal{O}_X(X)=A$. Now my question is if the same works for a scheme $X=\mathsf{Proj} \: A$.

Answer 1

This is not true. For instance, projective space over a field $k$ is defined as $\Bbb P^n:=\operatorname{Proj}k[x_0,\dots,x_n]$, but it's a basic fact that $\Gamma(\Bbb P^n,\mathcal O_{\Bbb P^n})=k$.