Graded Ring and Projective Spaces.

Question

Suppose that $R_0 = K$ is a field and $R_n$ is a finite-dimensional vector space over $K$ such that there is a product $R_m \times R_n \to R_{m+n}$ for $m, n \geq 0$. Then $R \colon= \underset{n \geq 0}{\bigoplus} R_n$ is a graded ring over $R_0 = K$. Suppose that $R$ is finitely generated by some elements $r_i \in R_i$.

On the other hand, there is a graded ring $P \colon= K[X_0,\cdots,X_N] = \underset{n \geq 0}{\bigoplus} P_n$, where $P_n$ is the degree $n$-part of $P$. The $N$-dimensional projective space ${\Bbb P}_K^N \colon= \mathrm{Proj} \,P$ is associated to $P$. I have difficulty to make a dictionary between geometries and graded rings unlike affine scheme between rings (by taking ${\mathrm{Spec}}$.)

Q. How can I get some geometric scheme from $R$ like I got ${\Bbb P}_{K}^N = {\mathrm{Proj}}\,P$ from $P$?

Answer 1

Comment: "Great thanks. I have been wondering what will happen when R is not necessarily generated by R1. In this case, is it also possible to get some embedding or surjection with projective spaces?"

Answer: You find a definition of $Proj(R)$ in Chapter II in Hartshorne - the definition in HH is valid for any graded ring $R:=⊕_{n≥0} R_n$. In many of the results presented in HH you must assume that $R$ is generated by $R_1$ as an $R_0$-algebra. The exercises in chapter II.5 in HH proves some basic properties of the $Proj(R)$-construction. In particular if $R_0:=A$ it follows there is a canonical map $π:X:=Proj(R)→Spec(A)$. Prop. HH.II.7.2 gives a criterium for the existence of a closed immersion $ϕ:X→P^n_A$ for some integer $n≥1$.

The exercises in HH are time consuming but you find many of the details in the following book (and the other books published in Publ Math IHES-series):

Éléments de géométrie algébrique. I. (English) Zbl 0203.23301 Die Grundlehren der mathematischen Wissenschaften. 166. Berlin-Heidelberg-New York: Springer-Verlag. IX, 466 p. (1971).

Note: It is in fact possible to read the EGA book series without knowing french language.

Answer 2

It is possible to get a map from $\operatorname{Proj} R$ to projective space when $R$ finitely generated but not necessarily by $R_1$. The key is that $\operatorname{Proj} R \cong \operatorname{Proj} R^{(d)}$, where $R^{(d)}=\bigoplus_{n=0}^\infty R_{nd}$: if $R$ is finitely generated over $R_0$, we can find some $d>0$ so that $R^{(d)}$ is generated by $R^{(d)}_1$ (take $d$ to be the least common multiple of the degrees of the generators, for instance). Then you may take a surjective graded morphism from $R_0[x_0,\cdots,x_N]\to R^{(d)}$ and get a closed embedding of $\operatorname{Proj} R\cong\operatorname{Proj} R^{(d)}$ in $\Bbb P_{R_0}^N$.