The Stacks project

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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