This question has two parts, the first part will be to obtain the desired exact sequence while the second will be to study it in the corresponding derived category and try to obtain it from there.

Let $X$ be a smooth geometrically integral variety over a number field $k$ with canonical morphism $pi:X rightarrow mathrm{Spec},k$, I want to obtain the following exact sequence of $mathrm{Gal}(bar{k}/k)$-modules (all tensor products are over $mathbb{Z}$):

$$mu_infty rightarrow bar{k}(X)^* rightarrow bar{k}(X)^* otimes mathbb{Q} rightarrow H^1(bar{X},mu_infty) rightarrow mathrm{Pic}(bar{X}) rightarrow mathrm{Pic}(bar{X})_{free} rightarrow 0,$$

where $bar{X} := X times _k bar{k}$, $mu_infty = mathrm{colim}_nmu_n$, and $mathrm{Pic}(bar{X})_{free}$ denotes the maximal free quotient of $mathrm{Pic}(bar{X})$.

A very natural approach will be to apply the Galois cohomology functor $H^i(bar{X},-)$ to the exact sequence

$$1 rightarrow mu_infty rightarrow mathbb{G}_m rightarrow mathbb{G}_m otimes mathbb{Q} rightarrow 1.$$

To see why this sequence is exact refer to the answer of my post A Kummer exact sequence involving $mu_infty$.

So everything is fine except for the surjective map $mathrm{Pic}(bar{X}) rightarrow mathrm{Pic}(bar{X})_{free}$, which can be rewritten as

$$mathrm{Pic}(bar{X}) rightarrow mathrm{Ker}(H^1(bar{X},mathbb{G}_m) otimes mathbb{Q} rightarrow H^2(bar{X},mu_infty)).$$

**Question 1.** How do I show that $mathrm{Pic}(bar{X})_{free}$ is precisely the kernel written above?

One thought I have is that perhaps we can try to show that $H^2(bar{X},mu_infty) = 0$, and then we would need to prove that $mathrm{Pic}(bar{X}) otimes mathbb{Q} cong mathrm{Pic}(bar{X})_{free}$, which would not be surprising since we are sort of ‘killing off’ the torsion parts of the Picard group.

Now on to the next part of the question, we have that the inclusion $mu_infty rightarrow mathbb{G}_m$ induces the map

$$varphi: tau_{leq 1}Rpi_*mu_infty rightarrow tau_{leq 1}Rpi_*mathbb{G}_m$$

in the category of bounded sheaves of complexes of discrete Galois modules. Let $D := mathrm{Cone}(varphi)$, thus we have a distinguished triangle

$$tau_{leq 1}Rpi_*mu_infty rightarrow tau_{leq 1}Rpi_*mathbb{G}_m rightarrow D rightarrow (tau_{leq 1}Rpi_*mu_infty)(1).$$

This would give rise to the exact sequence

$$0 rightarrow tau_{leq 1}Rpi_*mathbb{G}_m rightarrow D rightarrow (tau_{leq 1}Rpi_*mu_infty)(1) rightarrow 0.$$

Let $h^i(A)$ denote the $i$-th cohomology group of the complex $A$, then we have a long exact sequence of cohomology groups

$$h^{-1}(D) rightarrow h^{-1}((tau_{leq 1}Rpi_*mu_infty)(1)) rightarrow h^0(tau_{leq 1}Rpi_*mathbb{G}_m) rightarrow h^0(D) rightarrow h^0((tau_{leq 1}Rpi_*mu_infty)(1))$$

$$rightarrow h^1(tau_{leq 1}Rpi_*mathbb{G}_m) rightarrow h^1(D) rightarrow h^1((tau_{leq 1}Rpi_*mu_infty)(1)).$$

Here we only consider $h^i(D)$ for $i = -1,0,1$ because by definition, the cohomology is zero outside these degrees. It is well-known that since $X$ is smooth over $k$, $tau_{leq 1}Rpi_*mathbb{G}_m$ is quasi-isomorphic to the complex $(bar{k}(X)^* rightarrow mathrm{Div}(bar{X}))$ in degrees 0 and 1. Thus one easily computes that $h^0(tau_{leq 1}Rpi_*mathbb{G}_m) cong bar{k}(X)^*$ and $h^1(tau_{leq 1}Rpi_*mathbb{G}_m) cong mathrm{Pic}(bar{X})$. Also, we have

$$h^i((tau_{leq 1}Rpi_*mu_infty)(1)) = h^{i+1}(tau_{leq 1}Rpi_*mu_infty)$$

and so the last term of the above long exact sequence is 0.

**Question 2.** We want to show that this long exact sequence is precisely the one mentioned in the first part of the question. All we need to do is find an injective resolution for $mu_infty$, this will enable us to get an explicit presentation of $Rpi_*mu_infty$, but this is where I have no idea how to proceed.