The etale site of a closed subscheme and its etale Grothendieck subtopology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T11:19:13Z http://mathoverflow.net/feeds/question/56865 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56865/the-etale-site-of-a-closed-subscheme-and-its-etale-grothendieck-subtopology The etale site of a closed subscheme and its etale Grothendieck subtopology Anonymous 2011-02-28T01:24:17Z 2011-02-28T05:08:30Z <p>There is a very basic theorem for the Zariski topology.</p> <p>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.</p> <p>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.</p> <p>I did a computation today in a very special case and it seems that both of these topologies seem to be 'the same'.</p> <p>Does anyone know if this statement is true in a general context and where I might locate this resource?</p> <p>Thanks.</p> http://mathoverflow.net/questions/56865/the-etale-site-of-a-closed-subscheme-and-its-etale-grothendieck-subtopology/56874#56874 Answer by Anton Geraschenko for The etale site of a closed subscheme and its etale Grothendieck subtopology Anton Geraschenko 2011-02-28T04:31:02Z 2011-02-28T04:31:02Z <p>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 $X_i$.</p> <p>To see this, use the structure theorem for étale morphisms: <a href="http://www.math.columbia.edu/algebraic_geometry/stacks-git/locate.php?tag=025C" rel="nofollow">Theorem 34.11.3</a> in the chapter on étale morphisms in the <a href="http://www.math.columbia.edu/algebraic_geometry/stacks-git/index.html" rel="nofollow">Stacks Project</a>.</p> <p> It says that any étale morphism to $Y=Spec(R/I)$ is Zariski locally<sup>&dagger;</sup> 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 $V$. </p> <p><sup>&dagger;</sup> 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 $Y$.</p> http://mathoverflow.net/questions/56865/the-etale-site-of-a-closed-subscheme-and-its-etale-grothendieck-subtopology/56875#56875 Answer by Jacob Lurie for The etale site of a closed subscheme and its etale Grothendieck subtopology Jacob Lurie 2011-02-28T04:54:40Z 2011-02-28T05:08:30Z <p>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 $U$). </p> <p>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).</p> <p>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.</p>