proj

DEFINITION 07.4   An $\mathbb{N}$ -graded ring $S$ is a commutative ring with a direct sum decomposition

$$S=\bigoplus_{i\in \mathbb{N}} S_i$$   (as a module)

such that $S_i S_j \subset S_{i+j} \quad (\forall i,j\in \mathbb{N})$ holds. We define its irrelevant ideal $S_+$ as

$$S_+=\bigoplus_{i>0} S_i.$$

An element $f$ of $S$ is said to be homogenous if it is an element of $\cup S_i$ . An ideal of $S$ is said to be homogeneous if it is generated by homogeneous elements. Homogeneous subalgebras are defined in a same way.

DEFINITION 07.5  

$$\operatorname{Proj}(S)=\{\mathfrak{p};$$$ $ is a homogeneous prime ideal of $S$$$, \mathfrak{p}\not \supset S_+ \}$$

For any homogeneous element $f$ of $S$ , we define a subset $D_f$ of $\operatorname{Proj}(S)$ as

$$D_f=\{ \mathfrak{p}\in \operatorname{Proj}S; \mathfrak{p}\notin f\} .$$

$\operatorname{Proj}S$ has a topology (Zariski topology) which is defined by employing $\{D(f)\}$ as an open base.

PROPOSITION 07.6   For any graded ring $S$ and its homogeneous element $f$ , $S[\frac{1}{f}]$ also carries a structure of graded ring. There is a homeomorphism

$$D_f \sim \operatorname{Spec}(S[\frac{1}{f}])_0).$$

We may define, via these homeo altogether, a locally ringed space structure on $\operatorname{Proj}(S)$ .