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This question is closely related to Projective space is not affine. I want to show that the projective space is not affine and to this end I want to prove that $\Gamma(\mathbb P^n_R, \mathcal O_{P_R^n})=R$. Intuitively, I get that the only "admitted" functions on the projective space must be constant since they have to be inside all $R[\frac{X_0}{X_i}, \dots,\widehat{\frac{X_i}{X_i}}, \dots, \frac{X_n}{X_i} ]$ at once. I just can't seem to formulate the proof in terms of scheme theory.

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    This is proved in Hartshorne, Prop. II.5.13, for example. Let me know if you want a more thorough explanation. – Takumi Murayama Feb 10 '16 at 17:46
  • Hartshorne uses quite a different construction than Görtz. I would prefer a more "hands-on" proof that shows that the global sections are constant. – user306194 Apr 28 '16 at 10:55
  • If you're using Görtz/Wedhorn, then you could look at Ex. 13.16, which refers to the calculation in Ex. 11.43. That example is written for $R$ equal to a field, but the calculation works for any commutative ring $R$. On the other hand, this proof is pretty similar to Hartshorne's, so I'm not sure you'd be satisfied… – Takumi Murayama Apr 28 '16 at 17:37

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