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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 $B_0$?

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    @QiaochuYuan I don't know, I can only handle the case $A=B_0$. I also checked his errata on website, but get no luck.2012-06-09

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I think the author just forgot to add this kind of assumption. Otherwise it is even unclear that $B_d$ has a structure of $A$-module. So you can suppose $B_0$ is an $A$-algebra and $B_1$ is finitely generated over $A$.

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    @M.N.Yes, this is correct.2012-06-10