Lemma 27.13.3 (01NG)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 27: Constructions of Schemes
Section 27.13: Projective space
Lemma 27.13.3 (cite)

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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. $\square$

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