## Tag `01NF`

Chapter 26: Constructions of Schemes > Section 26.13: Projective space

Definition 26.13.2. The scheme $\mathbf{P}^n_{\mathbf{Z}} = \text{Proj}(\mathbf{Z}[T_0, \ldots, T_n])$ is called

projective $n$-space over $\mathbf{Z}$. Its base change $\mathbf{P}^n_S$ to a scheme $S$ is calledprojective $n$-space over $S$. If $R$ is a ring the base change to $\mathop{\rm Spec}(R)$ is denoted $\mathbf{P}^n_R$ and calledprojective $n$-space over $R$.

The code snippet corresponding to this tag is a part of the file `constructions.tex` and is located in lines 2771–2780 (see updates for more information).

```
\begin{definition}
\label{definition-projective-space}
The scheme
$\mathbf{P}^n_{\mathbf{Z}} = \text{Proj}(\mathbf{Z}[T_0, \ldots, T_n])$
is called {\it projective $n$-space over $\mathbf{Z}$}.
Its base change $\mathbf{P}^n_S$ to a scheme $S$ is called
{\it projective $n$-space over $S$}. If $R$ is a ring the base change
to $\Spec(R)$ is denoted $\mathbf{P}^n_R$ and called
{\it projective $n$-space over $R$}.
\end{definition}
```

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