# cohomology – Verifying the quasi-isomorphism of two complexes

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?