let $B=\oplus_{i\geq0}B_i$ be a graded ring with $B_d=B_1^d$ for every $d\geq1$. Suppose $B_1$ is a finitely generated $A$-module for some ring $A$. Then, is $B$ an $A$-algebra in some canonical way?
In Qing Liu's Algebraic Geometry and Arithmetic Curves, Ex.2.3.11(b), he said in the above settings the scheme $Proj(B)$ was a projective $A$-scheme, I wondered if this is a printing error, should he replace $A$ by $B_0$?