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It's relatively simple to show that the geometric morphisms $ \mathbf{Set} \to \mathrm{Sh}(\mathbf{CRing}^\mathrm{op}_{\mathrm{fp}}, \mathrm{Zar})$ correspond to local rings.

More precisely, since the site has finite limits, the inverse image functors are induced by functors $\mathbf{CRing}^\mathrm{op}_{\mathrm{fp}} \to \mathbf{Set}$ that preserve finite limits and map covers to epimorphic families, and these are precisely the functors (isomorphic to) $\mathbf{CRing}(-, \mathcal{O})$ for local rings $\mathcal{O}$.

But we often consider Zariski sheaves on larger sites. For example, one might assume a small inaccessible cardinal $\kappa$ and consider the site $(\mathbf{CRing}^\mathrm{op}_{\kappa}, \mathrm{Zar})$ of all $\kappa$-small affine schemes.

The functors $\mathbf{CRing}(-, \mathcal{O})$ still define (inverse image parts of) geometric points, and every point whose inverse image part preserves all limits of affine schemes must be of this form.

But in general, the inverse image part of a geometric point is only required to preserve finite limits, so a priori there might be additional points.

Is there an argument that every geometric point of such a site is given as above? Or can there be points not of that form?

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  • $\begingroup$ I'm curious: do you know examples of situations where it is important to work with $\mathbf{CRing}_{\kappa} $ instead of $\mathbf{CRing}_{\mathrm{fp}}$ ? $\endgroup$
    – Simon Henry
    Commented Dec 12, 2018 at 13:52

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Let's simplify and consider the presheaf topos.

I asked the same question over at the nForum a while back. There Marc Hoyois reminded me of the following quite general fact: The category of topos-theoretic points of $\mathrm{PSh}(\mathcal{C})$ is the ind-completion of $\mathcal{C}^{\mathrm{op}}$.

With this fact in mind, we can …

  • … recover what you said in your first paragraph, since the ind-completion of the category of finitely presented rings is the category of all rings.
  • … see that there's no reason to expect that every point of the (pre-)Zariski topos defined using one of the larger sites is given by a ring.

This situation is even more pronounced from the point of view of classifying toposes. The usual (pre-)Zariski topos, defined using finitely presented rings, classifies the theory of rings. In contrast, I don't believe that a nontautologous answer to the question "which theory does the topos defined using one of the larger sites classify?" has ever been written down.

For the purposes of algebraic geometry, the restriction to the rather small site of finitely presentables can be a bit cumbersome. A bit of discussion is included in Section 15 of these notes of mine.

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  • $\begingroup$ And the subclass of points of $\mathrm{PSh}(\mathcal{C})$ that are points of $\mathrm{Sh}(\mathcal{C})$ is (?) closed under filtered colimits, which suggests counterexamples. For example, take $F_i$ to be the filtered system of finitely generated subfields of $\mathcal{C}$, and $\underset{\longrightarrow}{\operatorname{colim}}_i \mathrm{CRing}(-, F_i)$ gives a counterexample, since its restriction to finitely presented rings is corepresented by $\mathbb{C}$, but $\underset{\longrightarrow}{\operatorname{colim}}_i \mathrm{CRing}(\mathbb{C}, F_i) \not\cong \mathrm{CRing}(\mathbb{C}, \mathbb{C})$ $\endgroup$
    – user13113
    Commented Sep 15, 2018 at 5:00

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