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.



Why is Proj of any graded ring isomorphic to Proj of a graded ring generated in degree one?
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.


