TITLE: Does eventually polynomial imply finitely generated?
QUESTION [7 upvotes]: I was considering converses to the theorem of Hilbert that the Hilbert function of a finitely generated graded module over $k[x_0,\dots,x_n]$ is eventually polynomial.  I asked the following question and I'm curious if it's true or to have a counterexample.

Let $k$ be a field and $R\subseteq k[x_0,\dots,x_n]$ be generated by homogeneous elements.  Let $\varphi(l)=\dim_k R_l$.  Suppose that $\varphi(l)$ is eventually polynomial.  Does this imply that $R$ is finitely generated?  Feel free to add extra conditions, if necessary.

REPLY [5 votes]: No!
Take $R=k[x,xy,xy^2,\cdots,xy^n,\cdots]\subset k[x,y]$, which is not finitely generated as a $k$-algebra. 
Nevertheless its Hilbert function $\varphi(l)=\dim_k(\operatorname {vect} (x^l,x^{l-1}y,\cdots  ,xy^{l-1}))=l$ is a polynomial in $l$.