# How to compute the stalks of $\mathbb{G}_{m}/\mathbb{G}_{m}^{n}$ over infinite points?

I am currently studying Mazur's paper.

In the paper, we assume that a number field $K$ is totally imaginary. I think that this is because he did not want to deal with the infinite points.

My question is

How to compute the stalks of $\mathbb{G}_{m}/\mathbb{G}_{m}^{n}$ over infinite points for totally non-imaginary field?

Note that $\mathbb{G}_{m}/\mathbb{G}_{m}^{n}$ is the multiplicative group scheme of mod $n$.

For a totally imaginary case, we know that the sheaf is 0 outside of the points whose residue characteristic divides $n$ by using the pull-back to $spec(\mathbb{F}_{\#k(x)})$.

I guess that it could depend on the characteristic of the number field, but not explicitly.

Furthermore, after the computation, can we show that the $q$th-cohomology group of the sheaf for totally non-imaginary group vanishes for $q\ge1$ with the same way in the paper?

Thank you for your time and effort.

Reference

Mazur, Barry, Notes on etale cohomology of number fields, Ann. Sci. Éc. Norm. Supér. (4) 6, 521-552 (1973). ZBL0282.14004.

• i have cross-posted this question to MSE. – Junhyeong Kim Jan 3 '18 at 18:52
• In the "completed" etale site of ${\rm{Spec}}(O_K)$ for a number field $K$ one has extra "points" just for real places. For the sheaf $F$ on this site that plays the role of $\mathbf{G}_m$, the "stalk" of $F$ at a real place $v$ is the multiplicative group of positive (nonzero) elements of the algebraic closure of $K$ in $K_v \simeq \mathbf{R}$. More specifically, from the definition of $F$ one sees $F/[n](F)$ vanishes over the enhanced etale site of ${\rm{Spec}}(O_K[1/n])$. See sections 2.2-2.4 and 3.3 (esp. Definition 3.3.1) in math.stanford.edu/~conrad/BSDseminar/Notes/L4.pdf – nfdc23 Jan 3 '18 at 20:06