## 27.13 Projective space

Projective space is one of the fundamental objects studied in algebraic geometry. In this section we just give its construction as $\text{Proj}$ of a polynomial ring. Later we will discover many of its beautiful properties.

Lemma 27.13.1. Let $S = \mathbf{Z}[T_0, \ldots , T_ n]$ with $\deg (T_ i) = 1$. The scheme

$\mathbf{P}^ n_{\mathbf{Z}} = \text{Proj}(S)$

represents the functor which associates to a scheme $Y$ the pairs $(\mathcal{L}, (s_0, \ldots , s_ n))$ where

1. $\mathcal{L}$ is an invertible $\mathcal{O}_ Y$-module, and

2. $s_0, \ldots , s_ n$ are global sections of $\mathcal{L}$ which generate $\mathcal{L}$

up to the following equivalence: $(\mathcal{L}, (s_0, \ldots , s_ n)) \sim (\mathcal{N}, (t_0, \ldots , t_ n))$ $\Leftrightarrow$ there exists an isomorphism $\beta : \mathcal{L} \to \mathcal{N}$ with $\beta (s_ i) = t_ i$ for $i = 0, \ldots , n$.

Proof. This is a special case of Lemma 27.12.3 above. Namely, for any graded ring $A$ we have

\begin{eqnarray*} \mathop{\mathrm{Mor}}\nolimits _{graded rings}(\mathbf{Z}[T_0, \ldots , T_ n], A) & = & A_1 \times \ldots \times A_1 \\ \psi & \mapsto & (\psi (T_0), \ldots , \psi (T_ n)) \end{eqnarray*}

and the degree $1$ part of $\Gamma _*(Y, \mathcal{L})$ is just $\Gamma (Y, \mathcal{L})$. $\square$

Definition 27.13.2. The scheme $\mathbf{P}^ n_{\mathbf{Z}} = \text{Proj}(\mathbf{Z}[T_0, \ldots , T_ n])$ is called projective $n$-space over $\mathbf{Z}$. Its base change $\mathbf{P}^ n_ S$ to a scheme $S$ is called projective $n$-space over $S$. If $R$ is a ring the base change to $\mathop{\mathrm{Spec}}(R)$ is denoted $\mathbf{P}^ n_ R$ and called projective $n$-space over $R$.

Given a scheme $Y$ over $S$ and a pair $(\mathcal{L}, (s_0, \ldots , s_ n))$ as in Lemma 27.13.1 the induced morphism to $\mathbf{P}^ n_ S$ is denoted

$\varphi _{(\mathcal{L}, (s_0, \ldots , s_ n))} : Y \longrightarrow \mathbf{P}^ n_ S$

This makes sense since the pair defines a morphism into $\mathbf{P}^ n_{\mathbf{Z}}$ and we already have the structure morphism into $S$ so combined we get a morphism into $\mathbf{P}^ n_ S = \mathbf{P}^ n_{\mathbf{Z}} \times S$. Note that this is the $S$-morphism characterized by

$\mathcal{L} = \varphi _{(\mathcal{L}, (s_0, \ldots , s_ n))}^*\mathcal{O}_{\mathbf{P}^ n_ R}(1) \quad \text{and} \quad s_ i = \varphi _{(\mathcal{L}, (s_0, \ldots , s_ n))}^*T_ i$

where we think of $T_ i$ as a global section of $\mathcal{O}_{\mathbf{P}^ n_ S}(1)$ via (27.10.1.3).

Lemma 27.13.3. Projective $n$-space over $\mathbf{Z}$ is covered by $n + 1$ standard opens

$\mathbf{P}^ n_{\mathbf{Z}} = \bigcup \nolimits _{i = 0, \ldots , n} D_{+}(T_ i)$

where each $D_{+}(T_ i)$ is isomorphic to $\mathbf{A}^ n_{\mathbf{Z}}$ affine $n$-space over $\mathbf{Z}$.

Proof. This is true because $\mathbf{Z}[T_0, \ldots , T_ n]_{+} = (T_0, \ldots , T_ n)$ and since

$\mathop{\mathrm{Spec}}\left( \mathbf{Z} \left[\frac{T_0}{T_ i}, \ldots , \frac{T_ n}{T_ i} \right] \right) \cong \mathbf{A}^ n_{\mathbf{Z}}$

in an obvious way. $\square$

Lemma 27.13.4. Let $S$ be a scheme. The structure morphism $\mathbf{P}^ n_ S \to S$ is

1. separated,

2. quasi-compact,

3. satisfies the existence and uniqueness parts of the valuative criterion, and

4. universally closed.

Proof. All these properties are stable under base change (this is clear for the last two and for the other two see Schemes, Lemmas 26.21.12 and 26.19.3). Hence it suffices to prove them for the morphism $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$. Separatedness is Lemma 27.8.8. Quasi-compactness follows from Lemma 27.13.3. Existence and uniqueness of the valuative criterion follow from Lemma 27.8.11. Universally closed follows from the above and Schemes, Proposition 26.20.6. $\square$

Remark 27.13.5. What's missing in the list of properties above? Well to be sure the property of being of finite type. The reason we do not list this here is that we have not yet defined the notion of finite type at this point. (Another property which is missing is “smoothness”. And I'm sure there are many more you can think of.)

Lemma 27.13.6 (Segre embedding). Let $S$ be a scheme. There exists a closed immersion

$\mathbf{P}^ n_ S \times _ S \mathbf{P}^ m_ S \longrightarrow \mathbf{P}^{nm + n + m}_ S$

called the Segre embedding.

Proof. It suffices to prove this when $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$. Hence we will drop the index $S$ and work in the absolute setting. Write $\mathbf{P}^ n = \text{Proj}(\mathbf{Z}[X_0, \ldots , X_ n])$, $\mathbf{P}^ m = \text{Proj}(\mathbf{Z}[Y_0, \ldots , Y_ m])$, and $\mathbf{P}^{nm + n + m} = \text{Proj}(\mathbf{Z}[Z_0, \ldots , Z_{nm + n + m}])$. In order to map into $\mathbf{P}^{nm + n + m}$ we have to write down an invertible sheaf $\mathcal{L}$ on the left hand side and $(n + 1)(m + 1)$ sections $s_ i$ which generate it. See Lemma 27.13.1. The invertible sheaf we take is

$\mathcal{L} = \text{pr}_1^*\mathcal{O}_{\mathbf{P}^ n}(1) \otimes \text{pr}_2^*\mathcal{O}_{\mathbf{P}^ m}(1)$

The sections we take are

$s_0 = X_0Y_0, \ s_1 = X_1Y_0, \ldots , \ s_ n = X_ nY_0, \ s_{n + 1} = X_0Y_1, \ldots , \ s_{nm + n + m} = X_ nY_ m.$

These generate $\mathcal{L}$ since the sections $X_ i$ generate $\mathcal{O}_{\mathbf{P}^ n}(1)$ and the sections $Y_ j$ generate $\mathcal{O}_{\mathbf{P}^ m}(1)$. The induced morphism $\varphi$ has the property that

$\varphi ^{-1}(D_{+}(Z_{i + (n + 1)j})) = D_{+}(X_ i) \times D_{+}(Y_ j).$

Hence it is an affine morphism. The corresponding ring map in case $(i, j) = (0, 0)$ is the map

$\mathbf{Z}[Z_1/Z_0, \ldots , Z_{nm + n + m}/Z_0] \longrightarrow \mathbf{Z}[X_1/X_0, \ldots , X_ n/X_0, Y_1/Y_0, \ldots , Y_ n/Y_0]$

which maps $Z_ i/Z_0$ to the element $X_ i/X_0$ for $i \leq n$ and the element $Z_{(n + 1)j}/Z_0$ to the element $Y_ j/Y_0$. Hence it is surjective. A similar argument works for the other affine open subsets. Hence the morphism $\varphi$ is a closed immersion (see Schemes, Lemma 26.4.2 and Example 26.8.1.) $\square$

The following two lemmas are special cases of more general results later, but perhaps it makes sense to prove these directly here now.

Lemma 27.13.7. Let $R$ be a ring. Let $Z \subset \mathbf{P}^ n_ R$ be a closed subscheme. Let

$I_ d = \mathop{\mathrm{Ker}}\left( R[T_0, \ldots , T_ n]_ d \longrightarrow \Gamma (Z, \mathcal{O}_{\mathbf{P}^ n_ R}(d)|_ Z)\right)$

Then $I = \bigoplus I_ d \subset R[T_0, \ldots , T_ n]$ is a graded ideal and $Z = \text{Proj}(R[T_0, \ldots , T_ n]/I)$.

Proof. It is clear that $I$ is a graded ideal. Set $Z' = \text{Proj}(R[T_0, \ldots , T_ n]/I)$. By Lemma 27.11.5 we see that $Z'$ is a closed subscheme of $\mathbf{P}^ n_ R$. To see the equality $Z = Z'$ it suffices to check on an standard affine open $D_{+}(T_ i)$. By renumbering the homogeneous coordinates we may assume $i = 0$. Say $Z \cap D_{+}(T_0)$, resp. $Z' \cap D_{+}(T_0)$ is cut out by the ideal $J$, resp. $J'$ of $R[T_1/T_0, \ldots , T_ n/T_0]$. Then $J'$ is the ideal generated by the elements $F/T_0^{\deg (F)}$ where $F \in I$ is homogeneous. Suppose the degree of $F \in I$ is $d$. Since $F$ vanishes as a section of $\mathcal{O}_{\mathbf{P}^ n_ R}(d)$ restricted to $Z$ we see that $F/T_0^ d$ is an element of $J$. Thus $J' \subset J$.

Conversely, suppose that $f \in J$. If $f$ has total degree $d$ in $T_1/T_0, \ldots , T_ n/T_0$, then we can write $f = F/T_0^ d$ for some $F \in R[T_0, \ldots , T_ n]_ d$. Pick $i \in \{ 1, \ldots , n\}$. Then $Z \cap D_{+}(T_ i)$ is cut out by some ideal $J_ i \subset R[T_0/T_ i, \ldots , T_ n/T_ i]$. Moreover,

$J \cdot R\left[ \frac{T_1}{T_0}, \ldots , \frac{T_ n}{T_0}, \frac{T_0}{T_ i}, \ldots , \frac{T_ n}{T_ i} \right] = J_ i \cdot R\left[ \frac{T_1}{T_0}, \ldots , \frac{T_ n}{T_0}, \frac{T_0}{T_ i}, \ldots , \frac{T_ n}{T_ i} \right]$

The left hand side is the localization of $J$ with respect to the element $T_ i/T_0$ and the right hand side is the localization of $J_ i$ with respect to the element $T_0/T_ i$. It follows that $T_0^{d_ i}F/T_ i^{d + d_ i}$ is an element of $J_ i$ for some $d_ i$ sufficiently large. This proves that $T_0^{\max (d_ i)}F$ is an element of $I$, because its restriction to each standard affine open $D_{+}(T_ i)$ vanishes on the closed subscheme $Z \cap D_{+}(T_ i)$. Hence $f \in J'$ and we conclude $J \subset J'$ as desired. $\square$

The following lemma is a special case of the more general Properties, Lemmas 28.28.3 or 28.28.5.

Lemma 27.13.8. Let $R$ be a ring. Let $\mathcal{F}$ be a quasi-coherent sheaf on $\mathbf{P}^ n_ R$. For $d \geq 0$ set

$M_ d = \Gamma (\mathbf{P}^ n_ R, \mathcal{F} \otimes _{\mathcal{O}_{\mathbf{P}^ n_ R}} \mathcal{O}_{\mathbf{P}^ n_ R}(d)) = \Gamma (\mathbf{P}^ n_ R, \mathcal{F}(d))$

Then $M = \bigoplus _{d \geq 0} M_ d$ is a graded $R[T_0, \ldots , R_ n]$-module and there is a canonical isomorphism $\mathcal{F} = \widetilde{M}$.

Proof. The multiplication maps

$R[T_0, \ldots , R_ n]_ e \times M_ d \longrightarrow M_{d + e}$

come from the natural isomorphisms

$\mathcal{O}_{\mathbf{P}^ n_ R}(e) \otimes _{\mathcal{O}_{\mathbf{P}^ n_ R}} \mathcal{F}(d) \longrightarrow \mathcal{F}(e + d)$

see Equation (27.10.1.4). Let us construct the map $c : \widetilde{M} \to \mathcal{F}$. On each of the standard affines $U_ i = D_{+}(T_ i)$ we see that $\Gamma (U_ i, \widetilde{M}) = (M[1/T_ i])_0$ where the subscript ${}_0$ means degree $0$ part. An element of this can be written as $m/T_ i^ d$ with $m \in M_ d$. Since $T_ i$ is a generator of $\mathcal{O}(1)$ over $U_ i$ we can always write $m|_{U_ i} = m_ i \otimes T_ i^ d$ where $m_ i \in \Gamma (U_ i, \mathcal{F})$ is a unique section. Thus a natural guess is $c(m/T_ i^ d) = m_ i$. A small argument, which is omitted here, shows that this gives a well defined map $c : \widetilde{M} \to \mathcal{F}$ if we can show that

$(T_ i/T_ j)^ d m_ i|_{U_ i \cap U_ j} = m_ j|_{U_ i \cap U_ j}$

in $M[1/T_ iT_ j]$. But this is clear since on the overlap the generators $T_ i$ and $T_ j$ of $\mathcal{O}(1)$ differ by the invertible function $T_ i/T_ j$.

Injectivity of $c$. We may check for injectivity over the affine opens $U_ i$. Let $i \in \{ 0, \ldots , n\}$ and let $s$ be an element $s = m/T_ i^ d \in \Gamma (U_ i, \widetilde{M})$ such that $c(m/T_ i^ d) = 0$. By the description of $c$ above this means that $m_ i = 0$, hence $m|_{U_ i} = 0$. Hence $T_ i^ em = 0$ in $M$ for some $e$. Hence $s = m/T_ i^ d = T_ i^ e/T_ i^{e + d} = 0$ as desired.

Surjectivity of $c$. We may check for surjectivity over the affine opens $U_ i$. By renumbering it suffices to check it over $U_0$. Let $s \in \mathcal{F}(U_0)$. Let us write $\mathcal{F}|_{U_ i} = \widetilde{N_ i}$ for some $R[T_0/T_ i, \ldots , T_0/T_ i]$-module $N_ i$, which is possible because $\mathcal{F}$ is quasi-coherent. So $s$ corresponds to an element $x \in N_0$. Then we have that

$(N_ i)_{T_ j/T_ i} \cong (N_ j)_{T_ i/T_ j}$

(where the subscripts mean “principal localization at”) as modules over the ring

$R\left[ \frac{T_0}{T_ i}, \ldots , \frac{T_ n}{T_ i}, \frac{T_0}{T_ j}, \ldots , \frac{T_ n}{T_ j} \right].$

This means that for some large integer $d$ there exist elements $s_ i \in N_ i$, $i = 1, \ldots , n$ such that

$s = (T_ i/T_0)^ d s_ i$

on $U_0 \cap U_ i$. Next, we look at the difference

$t_{ij} = s_ i - (T_ j/T_ i)^ d s_ j$

on $U_ i \cap U_ j$, $0 < i < j$. By our choice of $s_ i$ we know that $t_{ij}|_{U_0 \cap U_ i \cap U_ j} = 0$. Hence there exists a large integer $e$ such that $(T_0/T_ i)^ et_{ij} = 0$. Set $s_ i' = (T_0/T_ i)^ es_ i$, and $s_0' = s$. Then we will have

$s_ a' = (T_ b/T_ a)^{e + d} s_ b'$

on $U_ a \cap U_ b$ for all $a, b$. This is exactly the condition that the elements $s'_ a$ glue to a global section $m \in \Gamma (\mathbf{P}^ n_ R, \mathcal{F}(e + d))$. And moreover $c(m/T_0^{e + d}) = s$ by construction. Hence $c$ is surjective and we win. $\square$

Lemma 27.13.9. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf and let $s_0, \ldots , s_ n$ be global sections of $\mathcal{L}$ which generate it. Let $\mathcal{F}$ be the kernel of the induced map $\mathcal{O}_ X^{\oplus n + 1} \to \mathcal{L}$. Then $\mathcal{F} \otimes \mathcal{L}$ is globally generated.

Proof. In fact the result is true if $X$ is any locally ringed space. The sheaf $\mathcal{F}$ is a finite locally free $\mathcal{O}_ X$-module of rank $n$. The elements

$s_{ij} = (0, \ldots , 0, s_ j, 0, \ldots , 0, -s_ i, 0, \ldots , 0) \in \Gamma (X, \mathcal{L}^{\oplus n + 1})$

with $s_ j$ in the $i$th spot and $-s_ i$ in the $j$th spot map to zero in $\mathcal{L}^{\otimes 2}$. Hence $s_{ij} \in \Gamma (X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L})$. A local computation shows that these sections generate $\mathcal{F} \otimes \mathcal{L}$.

Alternative proof. Consider the morphism $\varphi : X \to \mathbf{P}^ n_\mathbf {Z}$ associated to the pair $(\mathcal{L}, (s_0, \ldots , s_ n))$. Since the pullback of $\mathcal{O}(1)$ is $\mathcal{L}$ and since the pullback of $T_ i$ is $s_ i$, it suffices to prove the lemma in the case of $\mathbf{P}^ n_\mathbf {Z}$. In this case the sheaf $\mathcal{F}$ corresponds to the graded $S = \mathbf{Z}[T_0, \ldots , T_ n]$ module $M$ which fits into the short exact sequence

$0 \to M \to S^{\oplus n + 1} \to S(1) \to 0$

where the second map is given by $T_0, \ldots , T_ n$. In this case the statement above translates into the statement that the elements

$T_{ij} = (0, \ldots , 0, T_ j, 0, \ldots , 0, -T_ i, 0, \ldots , 0) \in M(1)_0$

generate the graded module $M(1)$ over $S$. We omit the details. $\square$

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