# The Stacks Project

## Tag 02FK

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.

1. Show that $X \times_{\mathop{\mathrm{Spec}}({\mathbf Z})}\mathop{\mathrm{Spec}}(\mathbf{Q})$ is not empty.
2. 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}

There are no comments yet for this tag.

## Add a comment on tag 02FK

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).