Let $R$ be a commutative ring with a multiplicative identity. Let $A$ be a finitely generated graded $R$-algebra. Assume that $A_0$ is a finitely generated $R$-module. Is it true that $A_i$ is a finitely generated $R$-module for any positive integer $i$? We do not require $R$ to be Noetherian or $A$ to be finitely presented.
I have tried free polynomial algebras, for them this is true: the rank of $i$-th graded piece is in fact equal to $\binom{n+i-1}{i}$ if there are $n$ variables.
If we do not assume that $A_0$ is a finitely generated $R$-module, then $k[x, y]$ with $\mathrm{deg}(x)=0$, $\mathrm{deg}(y)=1$ is a counterexample I guess.
Yes (assuming $A$ is $\mathbb{N}$-graded). Note that $A$ is generated as an $R$-algebra by finitely many homogeneous elements (just take the homogeneous parts of its generators). So, then, every element of $A_i$ is an $A_0$-linear combination of monomials in the generators of positive degree with total degree $i$. There are only finitely many such monomials, and so $A_i$ is a finitely generated $A_0$-module, and hence also a finitely generated $R$-module.