The Stacks project

Exercise 111.51.5. 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”?

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View Exercise 111.51.5 as pdf