Galois cohomology of $\mathbf{Z}_\ell(m)$ over finite fields

Question

My question is surely a classical one in the algebraic number theory, but I'm not working in it and I do not know the results and references. Please excuse me.

Let $k = \mathbf{F}_q$ be a finite field and let $\ell$ be a prime number invertible in $k$. Let $\mathbf Z_\ell(1) = \varprojlim_{r\to\infty} \mu_{\ell^r}$ be the Tate module, which is a Galois module of $k$, and let $\mathbf{Z}_\ell(m) = \mathbf{Z}_\ell(1)^{\otimes m}$ be the $m$-th tensor power over $\mathbf{Z}_\ell$ for $m\in \mathbf{Z}$.

Question: What is $H^n(k, \mathbf{Z}_\ell(m))$? When is it an $\ell$-torsion abelian group?

Answer 1

The Galois cohomology of finite fields is pretty straightforward: a $G_k$-module $M$ is entirely determined by the action of Frobenius $\varphi$, and we have $H^0(k, M) = M^{\varphi = 1}$, $H^1(k, M) = M/(1 - \varphi)M$ (and it's zero in all other degrees).

On $\mathbf{Z}_\ell(m)$, the Frobenius acts as multiplication by $q^m$. So $H^0(k, \mathbf{Z}_\ell(m)) = \mathbf{Z}_\ell$ for $m = 0$ and is zero otherwise; and $H^1(k, \mathbf{Z}_\ell(m))$ is $\mathbf{Z}_\ell / (q^m - 1) \mathbf{Z}_\ell$. In particular, it's $\ell$-torsion for all $m \ne 0$.