## 27.21 Projective bundles

Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent sheaf of $\mathcal{O}_ S$-modules. By Modules, Lemma 17.21.6 the symmetric algebra $\text{Sym}(\mathcal{E})$ of $\mathcal{E}$ over $\mathcal{O}_ S$ is a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras. Note that it is generated in degree $1$ over $\mathcal{O}_ S$. Hence it makes sense to apply the construction of the previous section to it, specifically Lemmas 27.16.5 and 27.16.11.

Definition 27.21.1. Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ S$-module1. We denote

$\pi : \mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_ S(\text{Sym}(\mathcal{E})) \longrightarrow S$

and we call it the projective bundle associated to $\mathcal{E}$. The symbol $\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n)$ indicates the invertible $\mathcal{O}_{\mathbf{P}(\mathcal{E})}$-module of Lemma 27.16.11 and is called the $n$th twist of the structure sheaf.

According to Lemma 27.15.5 there are canonical $\mathcal{O}_ S$-module homomorphisms

$\text{Sym}^ n(\mathcal{E}) \longrightarrow \pi _*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n) \quad \text{equivalently}\quad \pi ^*\text{Sym}^ n(\mathcal{E}) \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(n)$

for all $n \geq 0$. In particular, for $n = 1$ we have

$\mathcal{E} \longrightarrow \pi _*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1) \quad \text{equivalently}\quad \pi ^*\mathcal{E} \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$

and the map $\pi ^*\mathcal{E} \to \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$ is a surjection by Lemma 27.16.11. This is a good way to remember how we have normalized our construction of $\mathbf{P}(\mathcal{E})$.

Warning: In some references the scheme $\mathbf{P}(\mathcal{E})$ is only defined for $\mathcal{E}$ finite locally free on $S$. Moreover sometimes $\mathbf{P}(\mathcal{E})$ is actually defined as our $\mathbf{P}(\mathcal{E}^\vee )$ where $\mathcal{E}^\vee$ is the dual of $\mathcal{E}$ (and this is done only when $\mathcal{E}$ is finite locally free).

Let $S$, $\mathcal{E}$, $\mathbf{P}(\mathcal{E}) \to S$ be as in Definition 27.21.1. Let $f : T \to S$ be a scheme over $S$. Let $\psi : f^*\mathcal{E} \to \mathcal{L}$ be a surjection where $\mathcal{L}$ is an invertible $\mathcal{O}_ T$-module. The induced graded $\mathcal{O}_ T$-algebra map

$f^*\text{Sym}(\mathcal{E}) = \text{Sym}(f^*\mathcal{E}) \to \text{Sym}(\mathcal{L}) = \bigoplus \nolimits _{n \geq 0} \mathcal{L}^{\otimes n}$

corresponds to a morphism

$\varphi _{\mathcal{L}, \psi } : T \longrightarrow \mathbf{P}(\mathcal{E})$

over $S$ by our construction of the relative Proj as the scheme representing the functor $F$ in Section 27.16. On the other hand, given a morphism $\varphi : T \to \mathbf{P}(\mathcal{E})$ over $S$ we can set $\mathcal{L} = \varphi ^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$ and $\psi : f^*\mathcal{E} \to \mathcal{L}$ equal to the pullback by $\varphi$ of the canonical surjection $\pi ^*\mathcal{E} \to \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. By Lemma 27.16.11 these constructions are inverse bijections between the set of isomorphism classes of pairs $(\mathcal{L}, \psi )$ and the set of morphisms $\varphi : T \to \mathbf{P}(\mathcal{E})$ over $S$. Thus we see that $\mathbf{P}(\mathcal{E})$ represents the functor which associates to $f : T \to S$ the set of $\mathcal{O}_ T$-module quotients of $f^*\mathcal{E}$ which are locally free of rank $1$.

Example 27.21.2 (Projective space of a vector space). Let $k$ be a field. Let $V$ be a $k$-vector space. The corresponding projective space is the $k$-scheme

$\mathbf{P}(V) = \text{Proj}(\text{Sym}(V))$

where $\text{Sym}(V)$ is the symmetric algebra on $V$ over $k$. Of course we have $\mathbf{P}(V) \cong \mathbf{P}^ n_ k$ if $\dim (V) = n + 1$ because then the symmetric algebra on $V$ is isomorphic to a polynomial ring in $n + 1$ variables. If we think of $V$ as a quasi-coherent module on $\mathop{\mathrm{Spec}}(k)$, then $\mathbf{P}(V)$ is the corresponding projective space bundle over $\mathop{\mathrm{Spec}}(k)$. By the discussion above a $k$-valued point $p$ of $\mathbf{P}(V)$ corresponds to a surjection of $k$-vector spaces $V \to L_ p$ with $\dim (L_ p) = 1$. More generally, let $X$ be a scheme over $k$, let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module, and let $\psi : V \to \Gamma (X, \mathcal{L})$ be a $k$-linear map such that $\mathcal{L}$ is generated as an $\mathcal{O}_ X$-module by the sections in the image of $\psi$. Then the discussion above gives a canonical morphism

$\varphi _{\mathcal{L}, \psi } : X \longrightarrow \mathbf{P}(V)$

of schemes over $k$ such that there is an isomorphism $\theta : \varphi _{\mathcal{L}, \psi }^*\mathcal{O}_{\mathbf{P}(V)}(1) \to \mathcal{L}$ and such that $\psi$ agrees with the composition

$V \to \Gamma (\mathbf{P}(V), \mathcal{O}_{\mathbf{P}(V)}(1)) \to \Gamma (X, \varphi _{\mathcal{L}, \psi }^*\mathcal{O}_{\mathbf{P}(V)}(1)) \to \Gamma (X, \mathcal{L})$

See Lemma 27.14.1. If $V \subset \Gamma (X, \mathcal{L})$ is a subspace, then we will denote the morphism constructed above simply as $\varphi _{\mathcal{L}, V}$. If $\dim (V) = n + 1$ and we choose a basis $v_0, \ldots , v_ n$ of $V$ then the diagram

$\xymatrix{ X \ar@{=}[d] \ar[rr]_{\varphi _{\mathcal{L}, \psi }} & & \mathbf{P}(V) \ar[d]^{\cong } \\ X \ar[rr]^{\varphi _{(\mathcal{L}, (s_0, \ldots , s_ n))}} & & \mathbf{P}^ n_ k }$

is commutative, where $s_ i = \psi (v_ i) \in \Gamma (X, \mathcal{L})$, where $\varphi _{(\mathcal{L}, (s_0, \ldots , s_ n))}$ is as in Section 27.13, and where the right vertical arrow corresponds to the isomorphism $k[T_0, \ldots , T_ n] \to \text{Sym}(V)$ sending $T_ i$ to $v_ i$.

Example 27.21.3. The map $\text{Sym}^ n(\mathcal{E}) \to \pi _*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n))$ is an isomorphism if $\mathcal{E}$ is locally free, but in general need not be an isomorphism. In fact we will give an example where this map is not injective for $n = 1$. Set $S = \mathop{\mathrm{Spec}}(A)$ with

$A = k[u, v, s_1, s_2, t_1, t_2]/I$

where $k$ is a field and

$I = (-us_1 + vt_1 + ut_2, vs_1 + us_2 - vt_2, vs_2, ut_1).$

Denote $\overline{u}$ the class of $u$ in $A$ and similarly for the other variables. Let $M = (Ax \oplus Ay)/A(\overline{u}x + \overline{v}y)$ so that

$\text{Sym}(M) = A[x, y]/(\overline{u}x + \overline{v}y) = k[x, y, u, v, s_1, s_2, t_1, t_2]/J$

where

$J = (-us_1 + vt_1 + ut_2, vs_1 + us_2 - vt_2, vs_2, ut_1, ux + vy).$

In this case the projective bundle associated to the quasi-coherent sheaf $\mathcal{E} = \widetilde{M}$ on $S = \mathop{\mathrm{Spec}}(A)$ is the scheme

$P = \text{Proj}(\text{Sym}(M)).$

Note that this scheme as an affine open covering $P = D_{+}(x) \cup D_{+}(y)$. Consider the element $m \in M$ which is the image of the element $us_1x + vt_2y$. Note that

$x(us_1x + vt_2y) = (s_1x + s_2y)(ux + vy) \bmod I$

and

$y(us_1x + vt_2y) = (t_1x + t_2y)(ux + vy) \bmod I.$

The first equation implies that $m$ maps to zero as a section of $\mathcal{O}_ P(1)$ on $D_{+}(x)$ and the second that it maps to zero as a section of $\mathcal{O}_ P(1)$ on $D_{+}(y)$. This shows that $m$ maps to zero in $\Gamma (P, \mathcal{O}_ P(1))$. On the other hand we claim that $m \not= 0$, so that $m$ gives an example of a nonzero global section of $\mathcal{E}$ mapping to zero in $\Gamma (P, \mathcal{O}_ P(1))$. Assume $m = 0$ to get a contradiction. In this case there exists an element $f \in k[u, v, s_1, s_2, t_1, t_2]$ such that

$us_1x + vt_2y = f(ux + vy) \bmod I$

Since $I$ is generated by homogeneous polynomials of degree $2$ we may decompose $f$ into its homogeneous components and take the degree 1 component. In other words we may assume that

$f = au + bv + \alpha _1s_1 + \alpha _2s_2 + \beta _1t_1 + \beta _2t_2$

for some $a, b, \alpha _1, \alpha _2, \beta _1, \beta _2 \in k$. The resulting conditions are that

$\begin{matrix} us_1 - u(au + bv + \alpha _1s_1 + \alpha _2s_2 + \beta _1t_1 + \beta _2t_2) \in I \\ vt_2 - v(au + bv + \alpha _1s_1 + \alpha _2s_2 + \beta _1t_1 + \beta _2t_2) \in I \end{matrix}$

There are no terms $u^2, uv, v^2$ in the generators of $I$ and hence we see $a = b = 0$. Thus we get the relations

$\begin{matrix} us_1 - u(\alpha _1s_1 + \alpha _2s_2 + \beta _1t_1 + \beta _2t_2) \in I \\ vt_2 - v(\alpha _1s_1 + \alpha _2s_2 + \beta _1t_1 + \beta _2t_2) \in I \end{matrix}$

We may use the first generator of $I$ to replace any occurrence of $us_1$ by $vt_1 + ut_2$, the second generator of $I$ to replace any occurrence of $vs_1$ by $-us_2 + vt_2$, the third generator to remove occurrences of $vs_2$ and the third to remove occurrences of $ut_1$. Then we get the relations

$\begin{matrix} (1 - \alpha _1)vt_1 + (1 - \alpha _1)ut_2 - \alpha _2us_2 - \beta _2ut_2 = 0 \\ (1 - \alpha _1)vt_2 + \alpha _1us_2 - \beta _1vt_1 - \beta _2vt_2 = 0 \end{matrix}$

This implies that $\alpha _1$ should be both $0$ and $1$ which is a contradiction as desired.

Lemma 27.21.4. Let $S$ be a scheme. The structure morphism $\mathbf{P}(\mathcal{E}) \to S$ of a projective bundle over $S$ is separated.

Proof. Immediate from Lemma 27.16.9. $\square$

Lemma 27.21.5. Let $S$ be a scheme. Let $n \geq 0$. Then $\mathbf{P}^ n_ S$ is a projective bundle over $S$.

Proof. Note that

$\mathbf{P}^ n_{\mathbf{Z}} = \text{Proj}(\mathbf{Z}[T_0, \ldots , T_ n]) = \underline{\text{Proj}}_{\mathop{\mathrm{Spec}}(\mathbf{Z})} \left(\widetilde{\mathbf{Z}[T_0, \ldots , T_ n]}\right)$

where the grading on the ring $\mathbf{Z}[T_0, \ldots , T_ n]$ is given by $\deg (T_ i) = 1$ and the elements of $\mathbf{Z}$ are in degree $0$. Recall that $\mathbf{P}^ n_ S$ is defined as $\mathbf{P}^ n_{\mathbf{Z}} \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} S$. Moreover, forming the relative homogeneous spectrum commutes with base change, see Lemma 27.16.10. For any scheme $g : S \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ we have $g^*\mathcal{O}_{\mathop{\mathrm{Spec}}(\mathbf{Z})}[T_0, \ldots , T_ n] = \mathcal{O}_ S[T_0, \ldots , T_ n]$. Combining the above we see that

$\mathbf{P}^ n_ S = \underline{\text{Proj}}_ S(\mathcal{O}_ S[T_0, \ldots , T_ n]).$

Finally, note that $\mathcal{O}_ S[T_0, \ldots , T_ n] = \text{Sym}(\mathcal{O}_ S^{\oplus n + 1})$. Hence we see that $\mathbf{P}^ n_ S$ is a projective bundle over $S$. $\square$

[1] The reader may expect here the condition that $\mathcal{E}$ is finite locally free. We do not do so in order to be consistent with [II, Definition 4.1.1, EGA].

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