The etale site of a closed subscheme and its etale Grothendieck subtopology

Question

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.

Answer 1

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 $Y$.

Answer 2

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.