I've been learning algebraic geometry, and I'm working on some problems about projective space and projective schemes. One struggle I've come across is working with morphisms between projective schemes. In affine schemes, there are some useful tools that I use often, most notably:
- For an ideal $I\subset k[x_1,\ldots, x_n]$, we have that $V(I)\cong\operatorname{Spec} k[x_1,\ldots, x_n]/(I)$ as a subset of $\mathbb A^n_k$.
- For rings $R$ and $S$, morphisms of affine schemes $\operatorname{Spec}R\to\operatorname{Spec} S$ are in direct correspondence with ring homomorphisms $S\to R$.
My question is, to what extent do these carry over to projective space? We have been using $\mathbb P^n_k=\operatorname{Proj} k\left[x_0,\ldots, x_n\right]$ as a ringed space, and I'm specifically interested in:
- For a homogenous ideal $I\subset k\left[x_0,\ldots, x_n\right]$, is it true that $V(I)\cong\operatorname{Proj} k\left[x_0,\ldots, x_n\right]/(I)$? I believe it is true, but I'm not sure if there's a "correspondence principle" for homogenous ideals in graded rings and their quotients.
- For $R$ and $S$ graded rings, do morphisms $\operatorname{Proj} R\to\operatorname{Proj}S$ lie in direct correspondence with morphisms of graded rings $S\to R$? I do not believe this is true: The irreducible conic $xy=z^2$ in $\mathbb P^2_k$ is isomorphic to $\mathbb P^1_k$, but $k\left[x, y, z\right]/(xy-z^2)$ is not isomorphic to $k[s, t]$ (I think). That said, I do know that morphisms of graded rings $S\to R$ do induce morphisms of schemes $\operatorname{Proj}R\to\operatorname{Proj}S$.
It seems to me that these last two points can't both simultaneously be true because of that example. Any help is much appreciated!