Let consider the weighted polynomial ring $S := k[T_0, T_1, T_2]$ such that $T_0,T_1$ have weight $1$, but $T_2$ is weigthed by $2$.
Therefore $S:= \oplus_i S_i$ can be canonically endowed with the structure of a graduated ring, where $S_i$ is the $S_0$-algebra of $i$-weighted components of $S$. In light of this one can also consider it as the projective scheme $Proj(S)$.
My question is why in that case the canonical morphism
$$f:S_0[S_1] \to S$$
isn't surjective.
My thoughts:
If I localize $S$ in $T_2$ I have already showed that
$$S_{(T_2)} = k[T_0^2/T_2,T_1^2/T_2,T_0T_1/T_2] \cong k[a,b,c]/(c^2-ab)$$
using Krull's Principal Ideal Theorem.
But I don't see how this information about this structure of $S_{(T_2)}$ lead me to a contradiction.