## Tag `02FK`

Chapter 102: Exercises > Section 102.48: Schemes, Final Exam, Fall 2007

Exercise 102.48.5. Let $X \to \mathop{\mathrm{Spec}}({\mathbf Z})$ be a morphism of finite type. Suppose that there is an infinite number of primes $p$ such that $X \times_{\mathop{\mathrm{Spec}}({\mathbf Z})} \mathop{\mathrm{Spec}}({\mathbf F}_p)$ is not empty.

- Show that $X \times_{\mathop{\mathrm{Spec}}({\mathbf Z})}\mathop{\mathrm{Spec}}(\mathbf{Q})$ is not empty.
- Do you think the same is true if we replace the condition ''finite type'' by the condition ''locally of finite type''?

The code snippet corresponding to this tag is a part of the file `exercises.tex` and is located in lines 4902–4913 (see updates for more information).

```
\begin{exercise}
\label{exercise-finite-type-over-Z}
Let $X \to \Spec({\mathbf Z})$ be a morphism of finite type.
Suppose that there is an infinite number of primes $p$ such that
$X \times_{\Spec({\mathbf Z})} \Spec({\mathbf F}_p)$ is not empty.
\begin{enumerate}
\item Show that $X \times_{\Spec({\mathbf Z})}\Spec(\mathbf{Q})$
is not empty.
\item Do you think the same is true if we replace the condition
``finite type'' by the condition ``locally of finite type''?
\end{enumerate}
\end{exercise}
```

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