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I'm sorry if my question is rather trivial, but I can't figure it out.. Given $A$ a ring and $P=Proj(A[X_0,\cdots,X_n])$, I know that $\oplus_n H^0(P,\mathcal{O}(n))=A[X_0,\cdots,X_n]$. This equality is not true in general for every graded algebra $B$ for which $P=Proj(B)$. Under which hypothesis is still verified $\oplus_n H^0(P,\mathcal{O}(n))=B$?

Thank you

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    $\begingroup$ I think the keyword you are looking for is projective normality $\endgroup$ Jan 30, 2015 at 15:28
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    $\begingroup$ Or more specifically than projective normality, you need a saturation hypothesis of $B$ w/ respect to $B_{\geq 1}$: $\Gamma(\text{Spec} B \setminus V(B_{\geq 1}), \mathcal{O}_{\text{Spec} B}) = B$ should be enough (though I didn't check details). See Exercises 5.9, 5.10, 5.13 and 5.14 of Chapter II of Hartshorne for special cases and some examples. Of course, one also has to be careful how you define $\mathcal{O}(n)$, this comes from the $\text{Proj}$ and different $B$s gives can give very different $\mathcal{O}(n)$ (think about Veronese subrings, or see the aforementioned Exercise 5.13). $\endgroup$ Jan 30, 2015 at 16:37

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