Lemma 31.31.6. Let $X$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ X$-module. There is a bijection
\[ \left\{ \begin{matrix} \text{sections }\sigma \text{ of the }
\\ \text{morphism } \mathbf{P}(\mathcal{E}) \to X
\end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{surjections }\mathcal{E} \to \mathcal{L}\text{ where}
\\ \mathcal{L}\text{ is an invertible }\mathcal{O}_ X\text{-module}
\end{matrix} \right\} \]
In this case $\sigma $ is a closed immersion and there is a canonical isomorphism
\[ \mathop{\mathrm{Ker}}(\mathcal{E} \to \mathcal{L}) \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes -1} \longrightarrow \mathcal{C}_{\sigma (X)/\mathbf{P}(\mathcal{E})} \]
Both the bijection and isomorphism are compatible with base change.
Proof.
Recall that $\pi : \mathbf{P}(\mathcal{E}) \to X$ is the relative proj of the symmetric algebra on $\mathcal{E}$, see Constructions, Definition 27.21.1. Hence the descriptions of sections $\sigma $ follows immediately from the description of the functor of points of $\mathbf{P}(\mathcal{E})$ in Constructions, Lemma 27.16.11. Since $\pi $ is separated, any section is a closed immersion (Constructions, Lemma 27.16.9 and Schemes, Lemma 26.21.11). Let $U \subset X$ be an affine open and $k \in \mathcal{E}(U)$ and $s \in \mathcal{E}(U)$ be local sections such that $k$ maps to zero in $\mathcal{L}$ and $s$ maps to a generator $\overline{s}$ of $\mathcal{L}$. Then $f = k/s$ is a section of $\mathcal{O}_{\mathbf{P}(\mathcal{E})}$ defined in an open neighbourhood $D_+(s)$ of $s(U)$ in $\pi ^{-1}(U)$. Moreover, since $k$ maps to zero in $\mathcal{L}$ we see that $f$ is a section of the ideal sheaf of $s(U)$ in $\pi ^{-1}(U)$. Thus we can take the image $\overline{f}$ of $f$ in $\mathcal{C}_{\sigma (X)/\mathbf{P}(\mathcal{E})}(U)$. We claim (1) that the image $\overline{f}$ depends only on the sections $k$ and $\overline{s}$ and not on the choice of $s$ and (2) that we get an isomorphism over $U$ in this manner (see below). However, once (1) and (2) are established, we see that the construction is compatible with base change by $U' \to U$ where $U'$ is affine, which proves that these local maps glue and are compatible with arbitrary base change.
To prove (1) and (2) we make explicit what is going on. Namely, say $U = \mathop{\mathrm{Spec}}(A)$ and say $\mathcal{E} \to \mathcal{L}$ corresponds to the map of $A$-modules $M \to N$. Then $k \in K = \mathop{\mathrm{Ker}}(M \to N)$ and $s \in M$ maps to a generator $\overline{s}$ of $N$. Hence $M = K \oplus A s$. Thus
\[ \text{Sym}(M) = \text{Sym}(K)[s] \]
Consider the identification $\text{Sym}(K) \to \text{Sym}(M)_{(s)}$ via the rule $g \mapsto g/s^ n$ for $g \in \text{Sym}^ n(K)$. This gives an isomorphism $D_+(s) = \mathop{\mathrm{Spec}}(\text{Sym}(K))$ such that $\sigma $ corresponds to the ring map $\text{Sym}(K) \to A$ mapping $K$ to zero. Via this isomorphism we see that the quasi-coherent module corresponding to $K$ is identified with $\mathcal{C}_{\sigma (U)/D_+(s)}$ proving (2). Finally, suppose that $s' = k' + s$ for some $k' \in K$. Then
\[ k/s' = (k/s) (s/s') = (k/s) (s'/s)^{-1} = (k/s) (1 + k'/s)^{-1} \]
in an open neighbourhood of $\sigma (U)$ in $D_+(s)$. Thus we see that $s'/s$ restricts to $1$ on $\sigma (U)$ and we see that $k/s'$ maps to the same element of the conormal sheaf as does $k/s$ thereby proving (1).
$\square$
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