Let $\displaystyle\varphi:\bigoplus_{d=0}^\infty S_d\rightarrow \bigoplus_{d=0}^\infty T_d$ be a morphism of commutative graded rings (with identity) and suppose that there exists $d_0$ such that $\varphi_d:S_d\rightarrow T_d$ is a bijection for all $d\geq d_0$. Can we say something about the ring homomorphism $\varphi_0:S_0\rightarrow T_0$?, for example, does it take prime ideals into prime ideals?.
Actually, my aim is to prove that there exists a bijection between the set $\mbox{Proj}\ S$ of homogeneous prime ideals of $S$ which do not contain $S_+$ and the set $\mbox{Proj}\ T$ homogeneous prime ideals of $T$ which do not contain $T_+$. One side is easy: take $f:\mbox{Proj}\ T\rightarrow \mbox{Proj}\ S$ the map $f(\mathfrak{q})=\varphi^{-1}(\mathfrak{q})$. I have problem to define an inverse map of this map.
Any help will be strongly appreciated.
Diego
Best Answer
Your map $f$ is not well-defined (when $\phi$ is not surjective), since it may happen that $\phi^{-1}(\mathfrak{q})$ contains the irrelevant ideal. Instead of defining $f$ globally, you can use the affine covering of the Proj scheme. It suffices to prove that a) the ring homomorphism $S_{(f)} \to T_{(\phi(f))}$ induced by $\phi$ is an isomorphism for every homogeneous $f \in S$ of positive degree, b) the basic open subsets $D(\phi(f))$ cover $\mathrm{Proj}(T)$.
Hint for b): If $g \in T$ is homogeneous of positive degree, choose some $n \geq 1$ such that $g^n$ has degree $\geq d_0$.
Hint for a): This is easy when you have already digested the idea of b).