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?



A Naive Question about Nekovar's Paper on Beilinson's Conjecture
For every short exact sequence of $k$-vector spaces
\begin{equation\*}
0 \to V\_1 \xrightarrow{\alpha} V \xrightarrow{\beta} V\_2 \to 0
\end{equation\*}
there is an isomorphism $\det(V) \cong \det(V\_1) \otimes \det(V\_2)$. Looking at the special case $V\_2=0$, it is natural to define $\det(0)=k$. Now if $k\_0$ is a subfield of $k$ and if $V\_1$ and $V$ are equipped with a $k\_0$-rational structure then $\det(V\_1)$ are $\det(V)$ are also equipped with a $k\_0$-rational structure and it is natural to equip $\det(V\_2)$ with the rational structure $\det(\alpha)^{-1} \cdot k\_0$. In this way the isomorphism above respects the rational structures.


Actually when formulating the Beilinson conjecture for a motive $M$, under the assumptions of Nekovar's paper, it is convenient to introduce the fundamental $\mathbf{Q}$-line
\begin{equation\*}
\Xi(M) = \det(H^1\_f(M))^{-1} \otimes \det(H^1\_f(M^\*(1))^\*) \otimes \det(M\_B^+)^{-1} \otimes \det(M\_{\mathrm{dR}}/\mathrm{Fil}^0)
\end{equation\*}
(see *The equivariant Tamagawa number conjecture* by Flach). The conjecture says that there is a canonical isomorphism $\Xi(M) \otimes \mathbf{R} \cong \mathbf{R}$ and that the $\mathbf{Q}$-rational structure $\Xi(M)$ corresponds to the leading term of the $L$-function $L^\*(M)$. Note that the rational structure defined by $\Xi(M)$ is equal to $\det(\alpha\_M)$ times the determinant of the height pairing, and that the problem of defining $\det(0)$ does not appear anymore.


