I post this naive question on Math stackexchange, but have got no reply, so I decide to bother the mathoverflow community.

https://math.stackexchange.com/questions/2350436/rational-structures-of-cokernel-of-linear-maps

In Nekovar's paper on Beilinson Conjecture,

http://math.stanford.edu/~conrad/BSDseminar/refs/BeilinsonintroII.pdf

I got stuck with the rational structures on cokernel, see sections(2.1) and (2.2), and the question could be stated as follows,

$M_B^+$ and $M_{dR}/F^0$ are both rational vector spaces and $\alpha_M$ is an injetive linear map, \begin{equation} \alpha_M: M_B^+ \otimes_{\mathbb{Q}} \mathbb{R} \rightarrow (M_{dR}/F^0) \otimes_{\mathbb{Q}} \mathbb{R} \end{equation} From the paper the rational structures on $M_B^+$ and $M_{dR}/F^0$ define a natural rational structure on $\text{det(Coker}~\alpha_M)$ where $\text{det}(V)$ means the highest exterior power of the vector space $V$, but I don't know how?

If $\alpha_M$ is an isomorphism, then $\text{Coker}~\alpha_M$ is trivial, from the paper $\text{det(Coker}~\alpha_M)$ now is canonically isomorphic to $\mathbb{R}$, and the rational structure is $\text{det}(\alpha_M)^{-1}\mathbb{Q}$, which seems very confusing, could someone clarify this?