Let $f\colon X \rightarrow Y$ be a proper morphism with $Y$ Noetherian (and even affine, if you wish), and let $\mathscr{A} = \bigoplus\_{n \ge 0} \mathscr{A}\_n$ be a quasi-coherent graded $\mathscr{O}\_X$-algebra of finite type. Is the quasi-coherent graded $\mathscr{O}\_Y$-algebra $f\_\* \mathscr{A}$ of finite type? 


An analogue of this for graded modules is EGA III, 3.3.1. Perhaps the version for algebras that I am asking about can somehow be deduced, but I could't see how. I'd be happy to accept any precise reference (as well as any proof or counterexample, of course) as an answer.



Push-forward of a quasi-coherent graded algebra under a proper map
No, that is not true, and there are several similar (and standard) examples. For one, let $Y$ be $\text{Spec}(k)$, let $X$ be an elliptic curve over $k$ (there are many other varieties one could use), let $f$ be the unique $k$-morphism, let $\mathcal{L}$ be an invertible sheaf on $X$ that is algebraically equivalent to $0$ but not torsion, and let $\mathcal{M}$ be an invertible sheaf of positive degree. Let $\mathcal{E}$ be $\mathcal{L}\oplus \mathcal{M}$, and let $\mathcal{A}$ be the symmetric algebra, $\text{Sym}^{\bullet}\_{\mathcal{O}\_X}(\mathcal{E})$. Thus, we have a direct sum decomposition,
$$
\mathcal{A}\_n = \text{Sym}^n\_{\mathcal{O}\_X}(\mathcal{L}\oplus \mathcal{M}) = \bigoplus\_{l,m\geq 0, \ l+m=n} \left(\mathcal{L}^{\otimes l}\otimes\_{\mathcal{O}\_X} \mathcal{M}^{\otimes m}\right).
$$
As is evident from this decomposition, $\mathcal{A}$ is actually $\mathbb{Z}\_{\geq 0} \times \mathbb{Z}\_{\geq 0}$-graded; the decomposition is,
$$
\mathcal{A} = \bigoplus\_{l,m\geq 0} \mathcal{A}\_{l,m}, \ \ 
\mathcal{A}\_{l,m} = \mathcal{L}^{\otimes l}\otimes\_{\mathcal{O}\_X} \mathcal{M}^{\otimes m}.
$$
Since cohomology commutes with direct sum of coherent sheaves on a Noetherian scheme, we have a $\mathbb{Z}\_{\geq 0}\times \mathbb{Z}\_{\geq 0}$-grading of the corresponding algebra of global sections,
$$
H^0(X,\mathcal{A}) = \bigoplus\_{l,m\geq 0} H^0(X,\mathcal{A}\_{l,m}), \ \ 
H^0(X,\mathcal{A}\_{l,m}) = H^0(X,\mathcal{L}^{\otimes l}\otimes\_{\mathcal{O}\_X} \mathcal{M}^{\otimes m}).
$$
Using Riemann-Roch and the hypotheses on $\mathcal{L}$ and $\mathcal{M}$, the graded piece $H^0(X,\mathcal{A}\_{l,m})$ is nonzero if and only if either $(l,m) = (0,0)$ or $m>0$. In particular, this gives a subsemigroup of $\mathbb{Z}\_{\geq 0} \times \mathbb{Z}\_{\geq 0}$ that is not finitely generated as a subsemigroup. Therefore $H^0(X,\mathcal{A})$ cannot be finitely generated as a graded $k$-algebra.


**Edit**. Probably this example has appeared previously on MathOverflow -- it is a standard example. There are other examples with various other interesting features.


