algebraic local charts - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T12:10:41Z http://mathoverflow.net/feeds/question/53488 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53488/algebraic-local-charts algebraic local charts unknown 2011-01-27T13:23:54Z 2011-01-27T19:31:44Z <p>Hi, in differential geometry or in complex geometry one of basic stuff to prove something is to do it on the local charts and then to check that the construction glues with the others charts. Which is the anologous of local charts (ball in \$C^n\$) in algebraic geometry (not only complex algebraic geometry) which are "more small" or more local then the affine schemes?</p> <p>The question arise from the fact that many time I see that something is proved assuming that these local charts are Spec of some complete ring.</p> <p>Thank you in advice</p> http://mathoverflow.net/questions/53488/algebraic-local-charts/53528#53528 Answer by Mattia Talpo for algebraic local charts Mattia Talpo 2011-01-27T19:31:44Z 2011-01-27T19:31:44Z <p>What you're looking for is the étale topology (I think).</p> <p>The fact is that Zariski opens are way too big to grasp stuff which should be more local than "Zariski local", for example any two smooth points on two \$n\$-dimensional varieties over \$\mathbb{C}\$ have isomorphic completed local rings (power series ring in \$n\$ indeterminates), but you will never find two isomorphic zariski open neighborhoods, unless the varieties are birational!</p> <p>The way to go more locally is to consider étale morphisms to a scheme (which should morally be locally invertible, but they aren't in the algebraic context) to be "open subsets" of that scheme. Of course this doesn't make sense unless you modify the notion of topology, and in fact this is what led to the notion of Grothendieck topologies.</p> <p>A good introduction to this stuff is Milne's book "étale cohomology", first chapters (I hope this is what you were looking for).</p>