Graded Rings

Here we define the projectivization of a graded ring and the projectivization of a sheaf of graded rings, and prove some important lemmas about them. We also record some important facts about graded rings. This basically follows the discussion in Hartshorne’s Algebraic Geometry, Chapter II, Section 2.

Preliminaries

Definition (Graded Ring):

A graded ring is a ring $A$, together with a decomposition

of the underlying abelian group of $A$, subject to the condition $A_iA_j\subseteq A_{i+j}$ on the multiplicative structure of $A$. A homogeneous element of the ring is an element contained in $A_i$ for some $i$; the degree of that element is then $i$. A homogeneous ideal is an ideal which can be generated by homogeneous elements (note, in particular, this does not mean that each element of the ring is homogeneous. Additionally, the homogeneous elements may not all be of the same degree). A homogeneous ideal can be prime or maximal if it also satisfies the conditions of a prime or maximal ideal. The irrelevant ideal $A_+$ is the ideal

Lemma (Primality on Homogeneous Elements):

A homogeneous ideal $\mf a$ of $A$ is prime if and only if for any homogeneous elements $a, b$ of $A$, $ab\in A$ implies $a\in A$ or $b\in A$.

Proof. Suppose first $\mf a$ is prime. Then for any pair of elements $a, b\in A$, we have $ab\in \mf a$ implies that either $a\in\mf a$ or that $b\in\mf a$. But

A morphism of graded rings is a morphism of rings $\psi: A\to B$ where

and $\psi(A_i)\subseteq B_i$. Graded rings form a category; the standard identity morphism is a morphism of graded rings, the composition of morphisms of graded rings is a morphism of graded rings, and composition is associative. A sheaf of graded rings is a sheaf with values in the category of graded rings.

Definition (Affine Case)

Let $A$ be a graded ring. We seek to define a scheme $\proj(A)$. The underlying set of $\proj(A)$ is the set of all homogeneous prime ideals of $A$. If $\mf a$ is a homogeneous ideal of $A$, let $V(\mf a)$ be the set of all homogeneous prime ideals $\mf p$ such that $\mf a\subseteq \mf p$. We then wish to establish the

Lemma 1:

The sets $V(\mf a)$ satisfy the following conditions:

1. If $\mf a$ and $\mf b$ are homogeneous ideals, then $V(\mf a\mf b) = V(\mf a) \cup V(\mf b)$
2. If $\mf a_i$ are a family of homogeneous ideals, then

Proof. Follows from the fact that we can check primality on homogeneous elements.