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Let S be a scheme. Let (Sch/S) be a small category of schemes over S (including essentially all finitely presented schemes affine over S). Let E = (Sch/S)zar denote the gros Zariski topos with its local ring object A1.

Is there a nice way to construct the petit Zariski topos X = Szar out of the locally ringed topos E? (By nice I mean, for example, that there is a universal property that the locally ringed topos X possesses with respect to E.)

There are variations of this question in which I am also interested: For example, one can replace E by the gros étale (or fppf or fpqc) topos (Sch/S)ét and ask for the construction of Szar out of (Sch/S)ét. Or one can replace X by the petit étale (or fppf or fpqc) topos Sét and ask for the construction of it out of E = (Sch/S)zar.

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There was quite a long discussion on this topic over at the n-Category Cafe, here: golem.ph.utexas.edu/category/2009/01/… Have you seen it? –  Manny Reyes Apr 16 '10 at 22:54
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@Manny: Thanks for pointing out that discussion to me. Denis-Charles Cisinski's comments there were quite helpful to me. Furthermore, the discussion contains a link to a paper by Mathieu Anel, "Grothendieck topologies from unique factorisation systems" where my question is completely answered in the case that my scheme S is affine, i.e. Spec A: E classifies local rings in topoi while X classifies local rings that are localisations of A, so X is the subtopos of those objects Y of E such that A^1 restricted to Y is a localisation of A. This should generalise easily to general schemes S. –  Marc Nieper-Wißkirchen Nov 19 '10 at 11:20
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1 Answer

I will deal with étale toposes because they behave much better in every possible way. They are also easier to define, althought they require substantially more commutative algebra to work with in practice:

The gros étale topos for $S$ is just $((Shv(Aff_{\acute Et})\downarrow S).$ We can construct from it the petit étale topos by considering $Shv(\acute Et \downarrow S)$, where $(\acute Et \downarrow S)$ is the subcategory of the gros étale topos consisting of étale morphisms $A\to S$ where $A$ is affine. This site is equipped with the induced topology.

Now for the ring object. For the petit topos, we let $\mathcal{O}_S$ be defined simply the sheaf sending any affine scheme to its corresponding ring (exercise: Show that this is a sheaf). This defines a ring object in the category of sheaves on the small site (exercise: Prove this. (Hint: Think of the definition of a group object and recall that the Yoneda embedding is full.)). For the large topos, we just let it be the base change of the affine line. It's not hard to show that they agree on étale morphisms $A\to S$ for A affine.

It turns out that the gros and petit toposes have a geometric morphism induced by the inclusion of the small site into the large site. I don't know if there is a specific universal property, per se, but it turns out that they are "homotopy equivalent" in a suitable sense.

For an explanation of the homotopy condition, see

Mac Lane and Moerdijk - Sheaves in Geometry and Logic Chapter 7.

Edit: If I remember correctly, the statement about "homotopy equivalence" does not work in the fppf or fpqc topologies. The small flat sites are too small, in some sense.

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Harry, thanks for your answer. However, it is not the answer I was looking for. It is clear that one can construct the petit (étale) topos out of the gros (étale) topos and that the ring object can be constructed the way you described it. But the question was whether this construction can be viewed as a natural one or whether the geometric morphisms between the gros and the petit toposes fulfill some universal properties. (For example, if I am not mistaken, constructing the petit étale topos out of the petit Zariski topos has a nice description; see M. Hakim's thesis.) Marc –  Marc Nieper-Wißkirchen Apr 13 '10 at 9:11
    
Where in M. Hakim's thesis? –  Harry Gindi Apr 13 '10 at 10:13
    
See for example in section IV.5 of her thesis. A good account on what it says about the étale spectrum can be found here: springerlink.com/content/10x9002103602132/fulltext.pdf –  Marc Nieper-Wißkirchen Apr 16 '10 at 15:24
    
I was able to find her thesis in one of the usual places =). –  Harry Gindi Apr 16 '10 at 15:37
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