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.