(*I asked this question a couple of days back on Stackexchange but with no success, it seems elementary, but I am struggling to go about attempting it.*)


Let $X$ be a smooth geometrically integral variety over a number field $k$. We denote by $\bar{k}[X]^\*$ the group of invertible functions on $\bar{X}$, and let $$G = \varinjlim\_{n \in \mathbb{Z}\_{>0}}I\_n$$
where $I\_n=\bar{k}[X]^\*$ for all $n$ and for $m$ such that $m|n$, we have $$I\_m \rightarrow I\_n:\bar{k}[X]^\* \rightarrow \bar{k}[X]^\*:x \mapsto x^{n/m}.$$
**Question 1.** How do we show that this direct limit is $\bar{k}[X]^\* \otimes\_\mathbb{Z} \mathbb{Q}$? Most examples of computing direct limits are related to multiplicative maps between $\mathbb{Z}$, so I'm struggling to find any similarities in methods.


Furthermore, we have
$$H^i(k,\bar{k}[X]^\*\otimes \mathbb{Q}) \cong H^i(k,\bar{k}[X]^\*) \otimes \mathbb{Q},$$ where $H^i(k,-)$ denotes the Galois cohomology functor.


**Question 2.** How do we compute $H^0(k,\bar{k}[X]^\*)$, i.e., the invertible functions fixed by Galois action?


**Question 3.** I believe that $H^i(k,\bar{k}[X]^\*)$ is torsion for $i \geq 1$, is there any direct way to show this?



The direct limit of invertible functions on a variety
As for the first question:


It is an elementary exercise that $\mathbb{Q}$ is the colimit of the diagram of copies of $\mathbb{Z}$ indexed by the positive integers with the divisibility relation. The transition maps are given by multiplying with the corresponding fraction, and a number $z$ in the $n$th copy represents $z/n$.


Since the tensor product is cocontinuous in both variables, it follows that for every abelian group $A$ the tensor product $A \otimes \mathbb{Q}$ is the colimit of the diagram of copies of $A \otimes \mathbb{Z} \cong A$ with the evident transition maps.


