User lorenzo - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:26:33Z http://mathoverflow.net/feeds/user/973 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53036/books-you-would-like-to-read-if-somebody-would-just-write-them/53040#53040 Answer by Lorenzo for Books you would like to read (if somebody would just write them...) Lorenzo 2011-01-24T10:41:15Z 2011-01-25T08:43:44Z <p>I would like to read an SGA-like book on <strong>Étale cohomology</strong> to replace as a reference SGA 4½. I also have an idea about who could write such a text: Luc Illusie. I'd really love that.</p> http://mathoverflow.net/questions/52404/locally-constant-sheaves-for-the-etale-topology-lack-of-intuition-about-etale-l Locally constant sheaves for the étale topology, lack of intuition about "étale-localness" Lorenzo 2011-01-18T13:25:01Z 2011-01-18T18:04:11Z <p>I have started studying some étale cohomology and I am trying to build up some intuition about the concept of <em>local for the étale topology</em>. I can understand some nice examples (like Kummer exact sequence) but I am still quite confused by some "easy" notions such as locally constant sheaves.</p> <p>I believe that an étale sheaf which is étale locally isomorphic to <em>the same</em> constant sheaf should be also globally isomorphic to that constant sheaf if the isomorphisms verify some cocycle condition, but here is a toy <strong>example</strong> which seems to contradict this:</p> <p>Let $k$ be a field, $n$ an integer invertible in $k$ and assume that $k$ does not contain all $n$-th roots of unity. Now consider the two following étale sheaves on $X=Spec\; k$:</p> <ul> <li>The sheaf of n-th roots of unity $\mu_n$;</li> <li>The constant sheaf $\mathbb Z/n \mathbb Z$.</li> </ul> <p>They are not isomorphic since their sections on $Spec\; k$ are different, but they become isomorphic after some finite separable extension of scalars so they are isomorphic étale locally. To be precise, $U=Spec(k[T]/(T^n-1))$ is an étale cover of $X$ such that the pullbacks of the two sheaves are isomorphic.</p> <p>Why are this two sheaves locally isomorphic but not isomorphic?<br> Is it normal that this isomorphism doesn't "patch"? (which would imply that the sheaves over the small étale site on $Spec\; k$ don't form a prestack)</p> <hr> <p>If I try to think to all this "stalkwise", changing to the point of view of topoi, (I'm not very familiar with the theory of topoi so please correct me if I am writing nonsense) I believe that:<br> the topos of sheaves over $Spec\;k$ with the small étale site has enough points, a family of conservative points consisting of just one element (the étale local ring is some separable closure $k^{sep}$ of $k$); and on this local ring the two sheaves above coincide.<br> It should follow that as soon as we have a morphism of sheaves inducing this isomorphism on the stalk the two sheaves should be isomorphic, which is not the case. </p> <p>Is it just because we don't have such a morphism or am I missing something more fundamental here?</p> http://mathoverflow.net/questions/56591/what-are-the-monomorphisms-in-the-category-of-schemes/56608#56608 Comment by Lorenzo Lorenzo 2011-02-25T08:16:49Z 2011-02-25T08:16:49Z Nice and complete answer! I particularly like the criterion (2). I add an example parallel to the last one: In EGA IV, 17.9.1 it's proved that the &#233;tale monomorphisms are exactly the open immersions. http://mathoverflow.net/questions/52164/where-can-you-find-grothendiecks-recoltes-et-semailles/52166#52166 Comment by Lorenzo Lorenzo 2011-01-19T08:28:38Z 2011-01-19T08:28:38Z They removed the links from the webpage but apparently most files are still online, look for instance <a href="http://www.math.jussieu.fr/~leila/grothendieckcircle/pubtexts.php" rel="nofollow">math.jussieu.fr/~leila/grothendieckcircle/&hellip;</a> and <a href="http://www.math.jussieu.fr/~leila/grothendieckcircle/unpubtexts.php" rel="nofollow">math.jussieu.fr/~leila/grothendieckcircle/&hellip;</a>. http://mathoverflow.net/questions/52404/locally-constant-sheaves-for-the-etale-topology-lack-of-intuition-about-etale-l/52420#52420 Comment by Lorenzo Lorenzo 2011-01-18T19:07:41Z 2011-01-18T19:07:41Z Thanks, I like this point of view. http://mathoverflow.net/questions/52404/locally-constant-sheaves-for-the-etale-topology-lack-of-intuition-about-etale-l/52416#52416 Comment by Lorenzo Lorenzo 2011-01-18T19:04:10Z 2011-01-18T19:04:10Z That's indeed what was confusing me, now everything is much more clear, thanks! http://mathoverflow.net/questions/52404/locally-constant-sheaves-for-the-etale-topology-lack-of-intuition-about-etale-l Comment by Lorenzo Lorenzo 2011-01-18T14:30:58Z 2011-01-18T14:30:58Z Sorry my previous comment was an answer for Daniel, but you were faster than me. Tom, I'll think about your suggestion, I did'n think about that. http://mathoverflow.net/questions/52404/locally-constant-sheaves-for-the-etale-topology-lack-of-intuition-about-etale-l Comment by Lorenzo Lorenzo 2011-01-18T14:28:10Z 2011-01-18T14:28:10Z thanks for the remark, you're right the question wasn't very clear, I just edited it. In the Zariski topology if you have two sheaves F and G on the scheme X, the presheaf $Isom_{F,G}$ associating to a Zariski open U the isomorphisms between the restrictions of F and G to U is indeed a sheaf, i.e. you can patch local isomorphisms as soon as they verify a cocycle condition. I thought this was the case for the &#233;tale site as well but the above example leaves me quite confused, I'd like to understand what's going on.