I am beginner in the theory of Grothendieck topologies and I have the following question. Let $X, Y$ be schemes over an algebraically closed field $k$. Denote by $X_{et}$ and $Y_{et}$ the Etale sites on $X$ and $Y$, respectively. Suppose that there is a continous morphism from $X_{et}$ to $Y_{et}$. Is there any additional conditions under which we can get a morphism (as schemes) from $X$ to $Y$?
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$\begingroup$ You will have to be more specific. For instance, let $X$ be $\operatorname{Spec} k$ and let $Y$ be $\operatorname{Spec} K$ for some algebraically closed transcendental extension of $k$. Then the corresponding small étale toposes are equivalent, but there is no morphism $X \to Y$. $\endgroup$– Zhen LinCommented Dec 7, 2014 at 16:56
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$\begingroup$ @ZhenLin: There is no such morphism "over $X$"; in general there could of course be such morphisms of abstract schemes (but I agree that the question is ill-posed). $\endgroup$– user74230Commented Dec 7, 2014 at 19:01
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2$\begingroup$ @user46578: A topos is like a topological space, ignorant of a choice of structure sheaf, so your question is analogous to asking when a continuous map between smooth manifolds is infinitely differentiable. Perhaps what you mean to ask is whether morphisms of the associated ringed topoi are the "same" as scheme morphisms, subject to an appropriate "locality" condition (as for locally ringed spaces, but adjusted for the etale topology)? This is addressed affirmatively in SGA4. You have to pay attention to the structure sheaf. $\endgroup$– user74230Commented Dec 7, 2014 at 19:03
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1$\begingroup$ @user46578: I think the question is not worth dwelling about until you have acquired a better understanding of etale morphisms and descent theory (prior to such knowledge it doesn't make sense to think about the etale topology), after which the affirmative answer is an easy exercise. You don't give a reason for posing the question in the first place; if it is idle curiosity then I recommend learning more about etale morphisms and descent. (A good reference is Chapter 2 and early parts of Chapter 6 of the book "Neron Models".) $\endgroup$– user74230Commented Dec 8, 2014 at 2:52
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1$\begingroup$ @DavidCarchedi I am not familiar with this paper. I suppose you refer to Theorem 47? It appears to me that the structure sheaf is being carried around, albeit disguised as a geometric morphism to the base topos, which is the big étale topos (i.e. the classifying topos for strictly henselian rings). $\endgroup$– Zhen LinCommented Dec 9, 2014 at 8:52
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2 Answers
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There is a full and faithful embedding of the category of schemes into the $2$-category of (edit: stricly Henselian ringed) topoi, which sends each scheme $X$ to the topos $Sh\left(X_{et}\right)$ (in fact, this is fully faithful even on Delign-Mumford stacks, see: https://eudml.org/doc/90454 Theorem 50, and Definition 43). So, given a geometric morphism $Sh\left(X_{et}\right) \to Sh\left(Y_{et}\right)$ respecting structures sheaves is the same as given a morphism of schemes $X \to Y$. In particular, any functor $X_{et} \to Y_{et}$ which induces a geometric morphism (edit: respecting structure sheaves), also induces a morphism of schemes, but not every geometric morphism can be realized as a morphism of sites.
Now, the structure sheaves are realized as a geometric morphism $Sh\left(X_{et}\right) \to \mathscr{T},$ with $\mathscr{T}$ the topos of sheaves on the site $Ftyp$ of all schemes of finite type, with the etale topology. This is the classifying topos for strictly Henselian rings. In fact, we can enlarge $Ftyp$ to all schemes locally of finite type $LFtyp$ without changing the topos. If $X$ is also locally of finite type, there is a canonical functor $\pi_X:X_{et} \to LFtyp$ sending each etale map $X' \to X$ to $X'.$ This is a morphism of sites which induces the geometric morphism $Sh(X_{et}) \to \mathscr{T}$ encoding the structures sheaf. In particular, any map of sites $f:X_{et} \to Y_{et}$ commuting over their projections to $LFtyp$ encodes a morphism $X \to Y.$ However, this is too strict, since maps of ringed topoi need not respect structure sheaves up to isomorphism, but only induced a map (like in ringed spaces), i.e. it suffices to have a natural transformation $\alpha:\pi_Y \circ f \to \pi_X$.
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$\begingroup$ Apparently I did. I misread the statement of the theorem, and have updated to correct for this. $\endgroup$ Commented Dec 9, 2014 at 9:40
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$\begingroup$ It is somewhat puzzling that the author does not comment on the choice of morphisms here. As far as I know it is conventional to ask for the relevant triangles of geometric morphisms to commute up to isomorphism. But I think one really has to go with the lax version here. $\endgroup$– Zhen LinCommented Dec 9, 2014 at 10:14
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As Zhen Lin notes, the question is too vague. However, knowing the etale site of a field is equivalent to knowing its absolute Galois group, and knowing the etale site of a scheme includes knowing its etale fundamental group. Anabelian geometry asks "how much information about the isomorphism class of a variety is contained in the knowledge of its etale fundamental group" See the Wikipedia.
