Algebraic Geometry 2, Summer term 2023.
Lecture course notes.

Content

5 Projective schemes *

(5.1) The Proj construction

Reference: [ GW1 ] Ch. 13.

Definition 5.1

  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 \(d\).

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

  3. 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 \(d\).

  4. 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 \(d\).

  5. 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 \(d\).

  6. 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
Let \(R\) be a ring. Then the polynomial ring \(R[T_0, \dots , T_n]\) is a graded \(R\)-algebra if we set \(R[T_0, \dots , T_n]_d\) to be the \(R\)-submodule of homogeneous polynomials of degree \(d\).

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

\[ M_{f, d} = \{ \frac{m}{f^i};\ m\in M_{d+ei} \} . \]

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

\[ M_{(f)} := M_{f, 0}, \]

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

Example 5.3
Let \(R\) be a ring. Then
\[ R[T_0, \dots , T_n]_{(T_i)} = R[\frac{T_0}{T_i}, \dots , \frac{T_n}{T_i}]. \]

Definition 5.4
We write \(A_+ := \bigoplus _{d{\gt}0} A_d\), an ideal of \(A\). A homogeneous prime ideal \({\mathfrak p}\subset A\) is called relevant if \(A_+ \not \subseteq {\mathfrak p}\).

Definition 5.5
We denote by \(\operatorname{Proj}(A)\) the set of all relevant homogeneous prime ideals of \(A\). We equip \(\operatorname{Proj}(A)\) with the Zariski topology, by saying that the closed subsets are the subsets of the form
\[ V_+(I) := \{ {\mathfrak p}\in \operatorname{Proj}(A);\ I \subseteq {\mathfrak p}\} . \]
for homogeneous ideals \(I\subseteq A\).

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

Lemma 5.6
Let \(f\in A\) be a homogeneous element. Then the map
\[ D_+(f) \to \operatorname{Spec}A_{(f)}, \quad {\mathfrak p}\mapsto ({\mathfrak p}A_f)\cap A_{(f)} \]
is a homeomorphism.

Proposition 5.7
There is a unique sheaf \({\mathscr O}\) of rings on \(\operatorname{Proj}(A)\) such that
\[ \Gamma (D_+(f), {\mathscr O}) = A_{(f)} \]
for every homogeneous element \(f\in A\) and with restriction maps given by the canonical maps between the localizations. The ringed space \((\operatorname{Proj}(A), {\mathscr O})\) is a separated scheme which we again denote by \(\operatorname{Proj}(A)\).

Definition 5.8
Let \(R\) be a ring, and let \(X\) be an \(R\)-scheme. We say that \(X\) is 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
Let \(R\) be a ring, and let \(X\) be a projective \(R\)-scheme. Then \(X\) is proper over \(R\).

(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

\[ \Gamma (D_+(f), \widetilde{M}) = M_{(f)} \]

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

\[ {\mathscr O}_{X|D_+(f)} \overset {\sim }{\to }{\mathscr O}_X(n)_{|D_+(f)}. \]

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

\[ \Gamma _*({\mathscr F}) = \bigoplus _{n\in \mathbb {Z}} \Gamma (X, {\mathscr F}(n)). \]

Lemma 5.11
For a graded \(A\)-module \(M\), there is a natural map \(M\to \Gamma _*(\widetilde{M})\). For an \({\mathscr O}_X\)-module \({\mathscr F}\), there is a natural map \(\widetilde{\Gamma _*({\mathscr F})}\to {\mathscr F}\). If \({\mathscr F}\) is quasi-coherent, then the latter map is an isomorphism.

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

Proposition 5.12
The functors \(M\to \widetilde{M}\) and \({\mathscr F}\to \Gamma _*({\mathscr F})\) define an equivalence of categories between the category of saturated graded \(A\)-modules and the category of quasi-coherent \({\mathscr O}_X\)-modules.