The Stacks project

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

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

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

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

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

where $k$ is a field and

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

where

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

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

and

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

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

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

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

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

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