Projective spaces as schemes

The primary example of a non-affine scheme is a projective space. We will describe the scheme and in the process obtain an understanding of its category of quasi-coherent sheaves.

We start in general with a ($\mathbb{Z}$-) graded ring.

Definition. A ring $R$ is $\mathbb{Z}$-graded if there is a decomposition of $R$ \(R = \bigoplus_{i \in \mathbb{Z}}^{\infty} R_i\) as an abelian group such that \(R_i R_j \subseteq R_{i+j}\) $R_i$ is called the i-th graded piece of $R$. An element $r \in R_i$ is homogeneous of degree i.

We will often be concerned with case that $R_i = 0$ for $i > 0$. Here we will say that $R$ is $\mathbb{N}$-graded. In the $\mathbb{N}$-graded case, we have a canonical maximal ideal \(R_+ := \bigoplus_{i > 0} R_i\) Often $R_+$ is called the irrelevant ideal.

The primary example of a graded ring is the polynomial ring $k[x_0,\ldots,x_n]$ where we grade each $x_i$ as $a_i$ for some choice and extend multiplicatively.

The standard grading is $a_i = 1$ for each $i$.

A graded $R$-module $M$ is an $R$-module with \(M = \bigoplus_{i \in \mathbb{Z}} M_i\) such that \(R_i M_j \subseteq M_{i+j}\)

An ideal $I$ is homogeneous if \(I = \bigoplus_{i \in \mathbb{Z}} I \cap R_i\) is a graded module.

Definition. Let $R$ be a $\mathbb{N}$-graded ring. Then \(\operatorname{Proj} R := \lbrace \mathfrak p \not \supseteq R_+ \mid \mathfrak{p} \text{ is homogenous and prime} \rbrace\) with the topology whose closed subsets are $V(J)$ for homogeneous $J$.

Example. For $R = k[x_0,\ldots,x_n]$ with the standard grading we write \(\mathbb{P}_k^n := \operatorname{Proj} R\) and call it n-dimensional projective space.

For homogeneous elements $a \in R$, we have the corresponding principal opens $U_a$.

Pick element $a \in R_i$ with $i > 0$. Then, we get a function \(\begin{aligned} U_a & \to \operatorname{Spec} (R_a)_0 \\ \mathfrak p & \mapsto (\mathfrak p R_a)_0 \end{aligned}\)

Lemma. This is well-defined and a homeomorphism.

We equip $\operatorname{Proj} R$ with the sheaf of rings determined by \(\mathcal O(U_a) := (R_a)_0\) for positive degree $a$, plus restriction agreeing with localization as before.

Given this, we see $\operatorname{Proj} R$ is a scheme.

Example. Let’s look at $\mathbb{P}_k^n$. Then, we have the principal opens corresponding to the monomials $x_i$. There \(k[x_0,\ldots,x_n]_{x_i} = k[x_0,\ldots,x_i^{\pm 1}, \ldots, x_n]\) To get degree zero elements, we need a rational function $f/x_i^d$ where $f$ has degree $d$. Dividing through by $x_i$, we get \(f/x_i^d \in k[x_0/x_i,\ldots,x_n/x_i]\) Thus, \(U_{x_i} \cong \mathbb{A}^n_k\)

Looking at the overlap \(U_{x_i} \cap U_{x_j} = U_{x_ix_j}\) we have the inclusions of graded rings \(k[x_0,\ldots,x_i^{\pm 1},\ldots,x_n] \subset k[x_0,\ldots,x_i^{\pm 1},\ldots,x_j^{\pm 1},\ldots,x_n] \supset k[x_0,\ldots,x_j^{\pm 1},\ldots,x_n]\)

Let’s write $y_l = x_l/x_i$ and $z_l = x_l/x_j$. Then, we have \(y_j z_i = 1\) We have \(U_{x_i} \cong \operatorname{Spec} k[y_0,\ldots,\hat{y_i},\ldots,y_n]\) and \(U_{x_j} \cong \operatorname{Spec} k[z_0,\ldots,\hat{z_j},\ldots,z_n]\) In the first set of coordinates, \(U_{x_i} \cap U_{x_j} \cong \operatorname{Spec} k[y_0,\ldots,\hat{y_i},\ldots,y_n]_{y_j}\) while in the second \(U_{x_i} \cap U_{x_j} \cong \operatorname{Spec} k[z_0,\ldots,\hat{z_j},\ldots,z_n]_{z_i}\) Changing from one set of coordinates to another is an isomorphism on $U_{x_i} \cap U_{x_j}$ which can be described explicitly by \(y_l z_i = z_l\)

One can extract these charts and their overlap maps as gluing data for the scheme $\mathbb{P}_k^n$. We won’t talk about this in general but it is another useful way to think of schemes - as glued from affine schemes.

Let’s note that $\mathbb{P}_k^n$ cannot be an affine scheme. Suppose we have $s \in \Gamma(\mathbb{P}^k,\mathcal O)$. Then, we get \(s|_{U_{x_i}} \in k[x_0/x_i,\ldots,x_n/x_i]\) for each $i$. Thus, \(s \in \bigcap_{i=0}^n k[x_0/x_i,\ldots,x_n/x_i] = k\) We know that if $X$ is affine then \(X \cong \operatorname{Spec}(\Gamma(X,\mathcal O))\) It is clear that \(\mathbb{P}_k^n \neq \operatorname{Spec} k\) for $n > 0$. (We have a lot more points.)

Quasi-coherent sheaves and graded modules

Next, for a graded $R$-module $M$, we get a quasi-coherent sheaf $\widetilde{M}$ by \(\widetilde{M}(U_a) := (M_a)_0\)

This produces an exact functor \(\widetilde{(-)} : \operatorname{GrMod} R \to \operatorname{Qcoh} (\operatorname{Proj} R)\)

The category of graded modules admits more operations than ungraded modules. In particular, we have grading twists \(M(i)_j = M_{i+j}\) analogous to shifts of complexes.

We have defined \(\mathcal O := \widetilde{R}\) and we set \(\mathcal O(i) := \widetilde{R(i)}\) for each $i \in \mathbb{Z}$.

Lemma. The functor $\widetilde{(-)}$ commutates with forming tensor products: \(\widetilde{M \otimes_R N} \cong \widetilde{M} \otimes_{\mathcal O} \widetilde{N}\)

Definition. Given a quasi-coherent sheaf $\mathcal F$ on $\operatorname{Proj} R$, its i-th twist is \(\mathcal F(i) := \mathcal F \otimes_{\mathcal O} \mathcal O(i)\)

Combining twists with global sections gives us a way to produce a graded module from a quasi-coherent sheaf.

Let \(\underline{\Gamma}(\operatorname{Proj} R, \mathcal F) := \bigoplus_{i \in \mathbb{Z}} \Gamma(\operatorname{Proj} R, \mathcal F(i))\) Lemma. Evaluation at $1$ is an isomorphism \(\operatorname{Hom}_{\mathcal O}(\mathcal O, \mathcal F) \to \Gamma(X,\mathcal F)\)

Thus we can view \(\underline{\Gamma}(\operatorname{Proj} R, \mathcal F) := \bigoplus_{i \in \mathbb{Z}} \operatorname{Hom}(\mathcal O, \mathcal F (i))\) Using the isomorphism \(\mathcal O(i) \otimes_{\mathcal O} \mathcal O(j) \cong \mathcal O(i+j)\) $\underline{\Gamma}(\operatorname{Proj} R, \mathcal O)$ acquires a natural ring structure.

Moreover, given a $r \in R_i$, we get \(r : R \to R(i)\) in $\operatorname{GrMod} R$. Applying $\widetilde{(-)}$ and then $\Gamma$ gives a map \(R \to \underline{\Gamma}(\operatorname{Proj} R, \mathcal O)\) which is homomorphism of graded rings.

For each $\mathcal F$, $\underline{\Gamma}(\operatorname{Proj} R, \mathcal F)$ naturally is a $\underline{\Gamma}(\operatorname{Proj} R, \mathcal O)$-module. Thus, is a graded $R$-module.

Proposition. We have an adjunction \(\widetilde{(-)} \vdash \underline{\Gamma}(\operatorname{Proj} R,-)\)

We will revisit this at the start of the next semester. But for now, let’s note that this adjunction in general is not coming from an equivalence.

Example. Let $R = k[x_0,\ldots,x_n]$ with its standard grading and take \(k := k[x_0,\ldots,x_n]/(x_0,\ldots,x_n)\) as a graded module. Then \(\widetilde{k}(U_{x_i}) = (k_{x_i})_0\) As $x_i$ acts by zero on $k$ and must be an isomorphism, we have \((k_{x_i})_0 = 0\) for each $i$. Thus, \(\widetilde{k} = 0\) giving an nonzero object which is mapped to $0$.

More generally, we say that $M$ is $R_+$-torsion if for each $m \in M$ there is some $l > 0$ with \((R_+)^l m = 0\) The full subcategory of $R_+$-torsion modules is denoted by $\operatorname{Tors}_{R_+}(R)$ or simply $\operatorname{Tors}(R)$ if the context is clear.

The same argument as in the example of $\mathbb{P}_k^n$ shows that \(\widetilde{M} = 0\) for any $M \in \operatorname{Tors}(R)$.

In fact, we will see that $\operatorname{Qcoh}(X)$ is the quotient of $\operatorname{GrMod} R$ by the subcategory $\operatorname{Tors}(R)$.