Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems' - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:32:29Z http://mathoverflow.net/feeds/question/97789 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97789/lemmas-on-etale-cohomology-with-compact-support-from-the-book-arithmetic-duality Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems' Ying Zhang 2012-05-23T20:25:40Z 2012-05-24T21:05:37Z <p>I was reading Milne's book "Arithmetic Duality Theorems". On page 166 there are a lot of useful lemmas on the etale cohomology with compact support on S-integers. However, I get confused when I tried to look at Prop 2.3 (a) and (d) at once. </p> <p>Suppose $X$ is the ring of integers of a totally complex number field $K$. $U\hookrightarrow X$ is an open subscheme, $S=X\setminus U={p}$ has only one finite prime. Consider the sheaf $\mathbb{G}_m$ and its pull-back on the various schemes involved, by (d) of Prop 2.3, we have</p> <p>$$H^0(\mathbb{F}_p, \mathbb{G}_m)\to H^1_c(U, \mathbb{G}_m)\to H^1_c(X, \mathbb{G}_m)$$</p> <p>Here $H^1_c(X, \mathbb{G}_m)=H^1(X, \mathbb{G}_m)$ since $K$ is a totally complex field, but anyway it is finite. Also $H^0(\mathbb{F}_p, \mathbb{G}_m)=\mathbb{F}_p^*$ is finite, therefore $H^1_c(U, \mathbb{G}_m)$ is finite.</p> <p>If we use (a) of Prop 2.3, we have </p> <p>$$H^0(U, \mathbb{G}_m)\to H^0(K_p, \mathbb{G}_m)\to H^1_c(U,\mathbb{G}_m)\to H^1(U,\mathbb{G}_m)$$</p> <p>Here $K_p$ is the Henselization of $K$ at $p$. Since we have shown $H^1_c(U,\mathbb{G}_m)$ is finite, and since we know $H^0(U, \mathbb{G}_m)$, which is the $S$-units on $X$, is a finite rank abelian group, we should have $H^0(K_p, \mathbb{G}_m)=K_p^*$ is also a finite rank abelian group. This seems absurd to me, since the subfields of $K_p$ which are number fields has infinite rank multiplicative groups.</p> <p>So I wonder, did I miss anything? Thanks very much in advance!</p> http://mathoverflow.net/questions/97789/lemmas-on-etale-cohomology-with-compact-support-from-the-book-arithmetic-duality/97795#97795 Answer by SGP for Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems' SGP 2012-05-23T22:14:05Z 2012-05-24T21:05:37Z <p>Apologies: (as pointed out by Minhyong Kim), the answer below does not address the OP's question</p> <p>$K_p$ is not a number field. It contains a $p$-adic field $Q_p$ and, as such, '$K_p^*$' is not a finitely generated abelian group.</p>