TITLE: Why is Proj of any graded ring isomorphic to Proj of a graded ring generated in degree one?
QUESTION [16 upvotes]: 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.

REPLY [17 votes]: 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.