Sections of etale morphisms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:00:35Z http://mathoverflow.net/feeds/question/17955 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17955/sections-of-etale-morphisms Sections of etale morphisms Yuhao Huang 2010-03-12T06:32:29Z 2010-04-13T19:10:16Z <p>We all know that smooth morphisms have sections etale locally. However, the following similar statement is not obvious for me:</p> <p>If X->Y->Z, X is etale over Y, Y is finite and surjective over Z, then a section of X->Y exists etale locally on Z, i.e. there exists an etale cover U of Z such that X_U->Y_U has a section. Where _U means pullback on U.</p> <p>I think it is supposed to be easy.</p> <p>Can anyone explain this to me? Thanks.</p> http://mathoverflow.net/questions/17955/sections-of-etale-morphisms/21250#21250 Answer by Ben Webster for Sections of etale morphisms Ben Webster 2010-04-13T19:10:16Z 2010-04-13T19:10:16Z <p>Brian Conrad says: "First reduce to the case when Y is also finitely presented over Z. Then you can use limit arguments, so it becomes an easy application of basic facts about strictly henselian local rings and finite algebras over them. It will be instructive for you to think about it some more for yourself in view of these hints.</p> <p>I should have also added that you must be assuming $X\to Y$ is surjective, or else it clearly isn't true. That condition enters into the argument when doing analysis of the situation with strictly henselian local rings."</p>