Lemma 27.22.3. Let $n \geq 1$. There is a canonical isomorphism $\mathbf{G}(n, n + 1) = \mathbf{P}^ n_\mathbf {Z}$.
Proof. According to Lemma 27.13.1 the scheme $\mathbf{P}^ n_\mathbf {Z}$ represents the functor which assigns to a scheme $S$ the set of isomorphisms classes of pairs $(\mathcal{L}, (s_0, \ldots , s_ n))$ consisting of an invertible module $\mathcal{L}$ and an $(n + 1)$-tuple of global sections generating $\mathcal{L}$. Given such a pair we obtain a quotient
\[ \mathcal{O}_ S^{\oplus n + 1} \longrightarrow \mathcal{L},\quad (h_0, \ldots , h_ n) \longmapsto \sum h_ i s_ i. \]
Conversely, given an element $q : \mathcal{O}_ S^{\oplus n + 1} \to \mathcal{Q}$ of $G(n, n + 1)(S)$ we obtain such a pair, namely $(\mathcal{Q}, (q(e_1), \ldots , q(e_{n + 1})))$. Here $e_ i$, $i = 1, \ldots , n + 1$ are the standard generating sections of the free module $\mathcal{O}_ S^{\oplus n + 1}$. We omit the verification that these constructions define mutually inverse transformations of functors. $\square$
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