# 5 Projective schemes *

**(5.1) The Proj construction**

Reference: [ GW1 ] Ch. 13.

**Definition 5.1**

A

*graded ring*is a ring \(A\) with a decomposition \(A = \bigoplus _{d\ge 0}A_d\) as abelian groups such that \(A_d\cdot A_e\subseteq A_{d+e}\) for all \(d, e\). The elements of \(A_d\) are called*homogeneous of degree \(d\)*.Let \(R\) be a ring. A

*graded \(R\)-algebra*is a graded ring \(A\) together with a ring homomorphism \(R\to A\).A

*homomorphism \(A\to B\) of graded rings (or graded \(R\)-algebras)*is a ring homomorphism (or \(R\)-algebra homomorphism, respectively) \(f\colon A\to B\) such that \(f(A_d)\subseteq B_d\) for all \(d\).Let \(A\) be a graded ring. A

*graded \(A\)-module*is an \(A\)-module \(M\) with a decomposition \(M=\bigoplus _{d\in \mathbb {Z}} M_d\) such that \(A_d\cdot M_e\subseteq M_{d+e}\) for all \(d, e\). The elements of \(M_d\) are called*homogeneous of degree \(d\)*.A

*homomorphism \(M\to N\) of graded \(A\)-modules*is an \(A\)-module homomorphism \(f\colon M\to N\) such that \(f(M_d)\subseteq N_d\) for all \(d\).Let \(A\) be a graded ring and let \(M\) be a graded \(A\)-module. A

*homogeneous submodule of \(M\)*is a submodule \(N\subseteq M\) such that \(N = \bigoplus _{d\in \mathbb {Z}} (N\cap M_d)\). In this way, \(N\) is itself a graded \(A\)-module and the inclusion \(N\hookrightarrow M\) is a homomorphism of graded \(A\)-modules. (And conversely, every injective homomorphism of graded \(A\)-modules has a homogeneous submodule as its image.) A homogeneous submodule of \(A\) is called a*homogeneous ideal*.

**Example 5.2**

We now fix a graded ring \(A\).

For a homogeneous element \(f\in A_e\), and a graded \(A\)-module \(M\), the localization \(M_f\) is a graded \(A\)-module via

Applying this to \(A\) as an \(A\)-module, we obtain a grading on \(A_f\) giving \(A_f\) the structure of a graded ring. Then \(M_f\) is a graded \(A_f\)-module.

We define

the degree \(0\) part of \(M_f\). Then \(A_{(f)}\) is a ring and \(M_{(f)}\) is an \(A_{(f)}\)-module.

**Example 5.3**

**Definition 5.4**

*relevant*if \(A_+ \not \subseteq {\mathfrak p}\).

**Definition 5.5**

For a homogeneous element \(f\), we write \(D_+(f) := \operatorname{Proj}(A)\setminus V_+(f)\).

**Lemma 5.6**

**Proposition 5.7**

**Definition 5.8**

*projective over \(R\)*(or that the morphism \(X\to \operatorname{Spec}R\) is

*projective*), if there exist \(n\ge 0\) and a closed immersion \(X\to \mathbb {P}^n_R\) of \(R\)-schemes.

**Theorem 5.9**

**(5.2) Quasi-coherent modules on \(\operatorname{Proj}(A)\)**

Let \(A\) be a graded ring, \(X=\operatorname{Proj}A\). If \(M\) is a graded \(A\)-module, there is a unique sheaf \(\widetilde{M}\) of \({\mathscr O}_X\)-modules such that

for every homogeneous element \(f\in A\), and such that the restriction maps for inclusions of the form \(D_+(g)\subseteq D_+(f)\) are given by the natural maps between the localizations. This sheaf is a quasi-coherent \({\mathscr O}_X\)-module.

**Example 5.10**

Let \(A(n)\) be the graded \(A\)-module defined by \(A(n) = \bigoplus _{d\in \mathbb {Z}} A_{n+d}\). We set \({\mathscr O}_X(n) = \widetilde{A(n)}\). If \(A=R[T_0, \dots , T_n]\) for a ring \(R\), so that \(X=\mathbb {P}^n_R\), then this notation is consistent with our previous definition.

For \(f\in A_d\), multiplication by \(f^k\) defines an isomorphism

In particular, if \(A\) is generated as an \(A_0\)-algebra by \(A_1\), then \({\mathscr O}_X(n)\) is a line bundle.

Assume, for the remainder of this section, that \(A\) is generated as an \(A_0\)-algebra by \(A_1\). So \(X = \bigcup _{f\in A_1} D_+(f)\), and \({\mathscr O}_X(n)\) is a line bundle.

For an \({\mathscr O}_X\)-module \({\mathscr F}\), write \({\mathscr F}(n) :={\mathscr F}\otimes _{{\mathscr O}_X}{\mathscr O}_X(n)\), and define a graded \(A\)-module \(\Gamma _*({\mathscr F})\) by

**Lemma 5.11**

Call a graded \(A\)-module \(M\) *saturated*, if the map \(M\to \Gamma _*(\widetilde{M})\) is an isomorphism.

**Proposition 5.12**