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Lemma 108.78.2. The scheme $\mathop{\mathrm{Spec}}(\mathbf{Z})$ is not quasi-compact in the canonical topology on the category of schemes.

Proof. With notation as above consider the family of morphisms

\[ \mathcal{W} = \{ \mathop{\mathrm{Spec}}(\mathbf{Z}_ A) \to \mathop{\mathrm{Spec}}(\mathbf{Z})\} _{A \in U} \]

By Descent, Lemma 35.10.5 and the two claims proved above this is a universal effective epimorphism. In any category with fibre products, the universal effective epimorphisms give $\mathcal{C}$ the structure of a site (modulo some set theoretical issues which are easy to fix) defining the canonical topology. Thus $\mathcal{W}$ is a covering for the canonical topology. On the other hand, we have seen above that any finite subfamily

\[ \{ \mathop{\mathrm{Spec}}(\mathbf{Z}_{A_ i}) \to \mathop{\mathrm{Spec}}(\mathbf{Z})\} _{i = 1, \ldots , n},\quad n \in \mathbf{N}, A_1, \ldots , A_ n \in U \]

factors through $\mathop{\mathrm{Spec}}(\mathbf{Z}[1/p])$ for some $p$. Hence this finite family cannot be a universal effective epimorphism and more generally no universal effective epimorphism $\{ g_ j : T_ j \to \mathop{\mathrm{Spec}}(\mathbf{Z})\} $ can refine $\{ \mathop{\mathrm{Spec}}(\mathbf{Z}_{A_ i}) \to \mathop{\mathrm{Spec}}(\mathbf{Z})\} _{i = 1, \ldots , n}$. By Sites, Definition 7.17.1 this means that $\mathop{\mathrm{Spec}}(\mathbf{Z})$ is not quasi-compact in the canonical topology. To see that our notion of quasi-compactness agrees with the usual topos theoretic definition, see Sites, Lemma 7.17.3. $\square$


Comments (2)

Comment #6530 by Jonas Ehrhard on

This might be nitpicking, but for consistency reasons I think it should be "universal effective epimorphism" instead of "universally".

There are also:

  • 2 comment(s) on Section 108.78: The spectrum of the integers is not quasi-compact

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