Let $k$ be a field and $S=k[x]$ a polynomial ring with $\deg x = 1$. Then we know that $H_S(t) = \frac{1}{1-t}.$ Let $S'=k[y]$ with $\deg y = -1$ or $y=x^{-1}.$ Then I think that $H_{S'}(t)= \frac{-t}{1-t}.$
1) If $S$ and $S'$ are isomorphic shouldn't they have the same Hilbert series?
2) What is the Hilbert series for $k[x,y]$ and the Hilbert series for $k[x,x^{-1}]?$