Let $X$ be a smooth variety over a number field $k$, with canonical morphism $pi:X rightarrow mathrm{Spec} , k$. Let $mathcal{D}(k)$ denote the derived category of bounded complexes of discrete $Gamma_k (=mathrm{Gal}(bar{k}/k))$-modules. We have the following truncated object in $mathcal{D}(k)$:

$$KD(X) = (tau_{leq 1}Rpi_*mathbb{G}_{m,X})(1).$$

This is a complex in degrees -1 and 0 and it is well-known that in our setting, it can simply be written as

$$(bar{k}(X)^* rightarrow mathrm{Div}(bar{X})).$$

By the canonical morphism $i : mathbb{G}_{m,k} rightarrow tau_{leq 1}Rpi_*mathbb{G}_{m,X}$, we define

$$KD'(X) = mathrm{Coker}(i)(1).$$

It is easy to check that $KD'(X)$ is quasi-isomorphic to the complex $$(bar{k}(X)^*/bar{k}^* rightarrow mathrm{Div}(bar{X})) in mathcal{D}(k).$$

However, I’ve seen that $KD'(X)$ can also be written as $mathrm{Cone}(mathbb{G}_{m,k}(1) rightarrow KD(X))$. How do we compute the cohomology groups of a mapping cone and verify that we do have a quasi-isomorphism?