Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems'

2012-05-23T20:25:40Z

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.

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

$$ H^0(\mathbb{F}_p, \mathbb{G}_m)\to H^1_c(U, \mathbb{G}_m)\to H^1_c(X, \mathbb{G}_m) $$

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.

If we use (a) of Prop 2.3, we have

$$ H^0(U, \mathbb{G}_m)\to H^0(K_p, \mathbb{G}_m)\to H^1_c(U,\mathbb{G}_m)\to H^1(U,\mathbb{G}_m)$$

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.

So I wonder, did I miss anything? Thanks very much in advance!

Answer by SGP for Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems'

2012-05-23T22:14:05Z

Apologies: (as pointed out by Minhyong Kim), the answer below does not address the OP's question

$K_p$ 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.

Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems' - MathOverflow

Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems'

Answer by SGP for Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems'