etale fundamental group and etale cohomology of curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:02:39Z http://mathoverflow.net/feeds/question/16566 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16566/etale-fundamental-group-and-etale-cohomology-of-curves etale fundamental group and etale cohomology of curves John Doe 2010-02-27T00:29:56Z 2010-02-27T00:44:41Z <p>Given a curve $C$. Is there any relation between the etale fundamental group $\pi_1(C)$ and the first etale cohomology of the constant sheaf , say $Z/nZ$, on $C$ ?</p> <p>For example, if $C$ is a complex curve, then the singular cohomology $H^1(C,Z)$ is the dual of the topological fundamental group divided by the commutators ( which is the same as Hom$(\pi_1(C),Z) )$.</p> <p>So it seems that there should be some relation between Hom$(\pi_1(C),Z/nZ)$ and $H^1(C,Z/nZ)$ in the etale case, but how?</p> http://mathoverflow.net/questions/16566/etale-fundamental-group-and-etale-cohomology-of-curves/16568#16568 Answer by Jared Weinstein for etale fundamental group and etale cohomology of curves Jared Weinstein 2010-02-27T00:44:41Z 2010-02-27T00:44:41Z <p>The two groups you want to compare are canonically isomorphic, so long as C is connected. See Example 11.3 of Milne's notes:</p> <p><a href="http://www.jmilne.org/math/CourseNotes/lec.html" rel="nofollow">http://www.jmilne.org/math/CourseNotes/lec.html</a></p>