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I have seen it stated that Proj of any graded ring $A$, finitely generated as an $A_0$-algebra, is isomorphic to Proj of a graded ring $B$ such that $B_0 = A_0$ and $B$ is generated as a $B_0$-algebra by $B_1$.

Could someone either supply a reference for or a sketch a proof of this statement?

Note: An obvious approach to this question is to make $B$ a Veronese subring of $A$. However, when I try this approach, I end up getting a terrible combinatorics problem that I do not know how to approach.

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EGA II, 2.1.6(v) – BCnrd Apr 2 '10 at 6:12
Red book(old print), 403p. – Moon Apr 2 '10 at 10:15
It's also handy to note (cf EGA2, Corollary 2.1.5) that a graded ring S is Noetherian if and only if S_0 is Noetherian and S is finitely generated over S_0. Thus the fact you refer to implies that if S is Noetherian then Proj S is projective over Spec S_0. – Chuck Hague Apr 7 '10 at 19:23

1 Answer 1

up vote 12 down vote accepted

Bourbaki Commutative Algebra Chapter 3:

Let $A$ be a non-negatively graded algebra. Assume that $A$ is finitely generated over $A_0$. There exists $e \geq 1$ such that $A^{me} = A_0[A^{me}_1]$ for any $m \geq 1$.

Here $A^{e} = \oplus_{n \in\mathbb Z} A^{e}_n$, where $A^{e}_n := A_{ne}$.

The desired result follows because replacing $A$ with $A^{me}$ does not change Proj.

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More precisely, this is Chapter III, 1.3 Proposition 3. – Charles Staats Apr 2 '10 at 13:15
Now we know how to solve that combinatorics problem ! :) – Nico Bellic Jan 12 '12 at 4:03

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