The etale fundamental group and etale cohomology with compact support - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:15:49Z http://mathoverflow.net/feeds/question/79011 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79011/the-etale-fundamental-group-and-etale-cohomology-with-compact-support The etale fundamental group and etale cohomology with compact support Makhalan Duff 2011-10-24T19:00:20Z 2011-10-25T00:02:36Z <p>Before me, the following was asked: <a href="http://mathoverflow.net/questions/16566/etale-fundamental-group-and-etale-cohomology-of-curves" rel="nofollow">http://mathoverflow.net/questions/16566/etale-fundamental-group-and-etale-cohomology-of-curves</a></p> <p>However, that question dealt only with projective curves.</p> <h3>Question</h3> <p>Let $X$ be any scheme (or if you prefer something more concrete, a variety over some field), and let $l$ be some prime different from the characteristics of the residue fields of $X$ (respectively, the characteristic of the field over which the variety is defined), then is there an isomorphism $Hom_{cont}(\pi_1^{et}(X),\mathbb{Q}_l)\cong H^1_c(X,\mathbb{Q}_l)$?</p> http://mathoverflow.net/questions/79011/the-etale-fundamental-group-and-etale-cohomology-with-compact-support/79012#79012 Answer by Akhil Mathew for The etale fundamental group and etale cohomology with compact support Akhil Mathew 2011-10-24T19:04:42Z 2011-10-25T00:02:36Z <p>In general, it's always true (for a connected scheme) that $H^1_{et}(X, \mathbb{Z}/l \mathbb{Z}) = \hom(\pi_1^{et}(X), \mathbb{Z}/l\mathbb{Z})$ (not compactly supported). Taking inverse limits over $l$ then gives the claim.</p> <p>The reason this is true is that $H^1_{et}(X, \mathbb{Z}/l\mathbb{Z}$) can be computed by Cech cocycles, and from this it follows that elements of this group classify torsors over $\mathbb{Z}/l\mathbb{Z}$ (i.e. sheaves with a $\mathbb{Z}/l\mathbb{Z}$-action which are locally the constant $\mathbb{Z}/l \mathbb{Z}$ (in the etale topology)). But these, by descent theory, are the same as Galois covers of $X$ with Galois group $\mathbb{Z}/l\mathbb{Z}$, and (by Galois theory) classified by maps from the etale fundamental group into $\mathbb{Z}/l\mathbb{Z}$.</p>