Lemma 109.79.2. 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

By Descent, Lemma 35.13.5 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

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. $\square$

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## Comments (2)

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