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There is a very basic theorem for the Zariski topology.

Let X = Spec(R) and Y=Spec(R/I) for I some reduced ideal. Y obtains a topology two ways, one is the subspace topology as a subset of X and another as the spectrum of a ring. These topologies are the same by the correspondence between ideals of R containing I and ideals in R/I.

Is there a close statement to this in the etale toplogy? There are two natural ways to understand open sets on Y, those which come from etale neighborhoods of X base changed to Y and those which are etale neighborhoods of Y.

I did a computation today in a very special case and it seems that both of these topologies seem to be 'the same'.

Does anyone know if this statement is true in a general context and where I might locate this resource?

Thanks.

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See EGA4, Proposition 18.1.1. –  Jonathan Wise Feb 28 '11 at 5:54
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EGA IV Proposition 18.1.1 says that Zariski locally, any etale (resp. smooth) neighborhood of $Y$ is induced by an etale (resp. smooth) neighbohood of $X$. –  Anton Geraschenko Feb 28 '11 at 19:14
    
There is an improvement by Arabia of the EGA proposition in question. This answers questions of Monsky et Washnitzer. The article is called "Relèvements des algèbres lisses et de leurs morphismes" ems-ph.org/journals/… –  name Jul 7 '12 at 14:39

2 Answers 2

up vote 8 down vote accepted

The statement is at least true Zariski locally. That is, given an étale map $V\to Y$, there exists a Zariski open cover $X=\bigcup X_i$ so that the pullback of $V$ to $Y\cap X_i$ is the restriction of an étale neighborhood of $X_i$.

To see this, use the structure theorem for étale morphisms: Theorem 34.11.3 in the chapter on étale morphisms in the Stacks Project.

It says that any étale morphism to $Y=Spec(R/I)$ is Zariski locally an open subscheme $V$ of $Spec((R/I)[t]_{\bar f'}/(\bar f))$, where $\bar f\in (R/I)[t]$ is monic. Let $f\in R[t]$ be an arbitrary monic lift of $\bar f$. Then since the Zariski topoplogy of $Spec((R/I)[t]_{\bar f'}/(\bar f))$ is the restriction of the Zariski topology of $Spec(R[t]_{f'}/(f))$, there is some open subscheme $U$ of $Spec(R[t]_{f'}/(f))$ so that $V = U\cap Spec((R/I)[t]_{\bar f'}/(\bar f))$. This $U$ is étale over $X$ and pulls back to $V$.

I'm implicitly replacing $X=Spec(R)$ and $Y=Spec(R/I)$ by localizations $Spec(R_g)$ and $Spec(R_g/I\cdot R_g)$ in the rest of the argument. The open cover $X=\bigcup X_i$ consists of these $Spec(R_g)$'s and the complement of $Y$.

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Thanks. This proof was very close to my original proof. I felt that my extra assumptions had more to do with personal comfort than actual mathematics. –  Anonymous Feb 28 '11 at 5:06
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Note that one can not expect a global version on $X$: take $X=\mathbb C^2$ and $Y: xy-1=0$ (which is isomorphic to $\mathbb C^*$). As $X$ is simply connected, the only connected étale cover of $Y$ you can extend to $X$ is the identity. –  Qing Liu Feb 28 '11 at 12:37

Here is one way of addressing your question. Let $X$ be a scheme, $Y$ a closed subscheme of $X$, and $U$ its open complement. Let $i: Y \rightarrow X$ and $j: U \rightarrow X$ denote the inclusion maps. Then the pushforward functor $i_{\ast}$ determines a fully faithful embedding from the category of etale sheaves on $Y$ to the category of etale sheaves on $X$. The essential image of this embedding is the full subcategory spanned by those etale sheaves $\mathcal{F}$ on $X$ such that $j^{\ast} \mathcal{F}$ is final (in other words, such that $\mathcal{F}(V)$ has a single element for every etale map $V \rightarrow X$ which factors through $U$).

In topos-theoretic language, this says that $i$ induces a closed immersion of etale topoi, which is complementary to the open immersion of etale topoi determined by $j$. (The above discussion makes sense in an arbitrary topos, and for a topos of sheaves on a topological space it recovers the usual notion of closed embedding).

I don't know a reference for the above statement offhand, but my guess would be that you can find a discussion in SGA somewhere.

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Thanks for rephrasing it in this language. I'll browse SGA for statement like this tomorrow. –  Anonymous Feb 28 '11 at 5:08
    
Can the topoi really recover this kind of information about the original etale sites? It seems like knowing that $i$ is a closed immersion of etale topoi should prove that the canonical topology on the etale topos of $Y$ is induced by the canonical topology on the etale topos of $X$ (or something like that). –  Anton Geraschenko Feb 28 '11 at 5:21
    
The original question mentions that given a commutative ring R and an ideal I, two topological spaces (defined using the Zariski topology) are homeomorphic. My answer was addressing the question ``what is an analogous statement for the etale topology?'' (The answer being that you can extract two topoi which are equivalent.) I don't think you can formally deduce anything about defining sites from this. Rather the information would go in the other direction: to write down a proof of my claim, you'll want to think about the problem of lifting etale coverings, as in your earlier reply. –  Jacob Lurie Feb 28 '11 at 5:34
    
Thanks for clarifying. –  Anton Geraschenko Feb 28 '11 at 5:42

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