Global section morphism is surjective?

Question

Let $X$ be a scheme and $f:P_X^n\to X$ denotes the projective $n-$space. $f^{\#}:O_X\to f_*O_{P_X^n}$. Is $O_X(X)\to O_{P_X^n}(P_X^n)$ surjective?

Answer 1

It's better than surjective, it's an isomorphism. This problem is local on $X$: we have a map of sheaves and we want to verify it's an isomorphism. It suffices to check on affine opens.

On any affine open $\operatorname{Spec} R\subset X$, we can base change our map along the natural open immersion $\operatorname{Spec} R \to X$ so that we're in the situation of $f: \Bbb P^n_R\to \operatorname{Spec} R$. As $f_*\mathcal{O}_{\Bbb P^n_R}$ is a sheaf on the affine scheme $\operatorname{Spec} R$, it suffices to compute the global sections. But these are just $R$ (cf here for example), and the map is the identity, so we're finished.