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?

homological algebra – $Omega^1_{B_bullet/A_bullet}$ is acyclic if $A_bullet to B_bullet$ is quasi-isomorphism

Let $A_bullet to B_bullet$ be a quasi-isomorphism of simplicial rings in the sense of (P.62, I.3.1.7, Complexe Cotangent et Déformations I, Luc Illusie).

Then, we define the simplicial $B_bullet$-module of Kähler differentials $Omega^1_{B_bullet/A_bullet}$ by $left(Omega^1_{B_bullet/A_bullet}right)_n := Omega^1_{B_n/A_n}$ (P.119, ibid.).

My question is whether it follows that $Omega^1_{B_bullet/A_bullet}$ is acyclic (i.e. quasi-isomorphic to the zero module) from these assumptions.


The broader context is when I am trying to show that the cotangent complex is independent of the free resolution taken.

Suppose we are given an algebra $R to S$ which has two free simplicial resolutions $P_bullet to Q_bullet$ (i.e. they are quasi-isomorphic to $S$, and $P_n$ and $Q_n$ are free $R$-algebras).

Then, using these two simplicial resolutions, we define the cotangent complex $L_{S/R}$ to be $Omega_{P_bullet/R} otimes_{P_bullet} S$ and $Omega_{Q_bullet/R} otimes_{Q_bullet} S$, the former of which is isomorphic to $Omega_{P_bullet/R} otimes_{P_bullet} Q_bullet otimes_{Q_bullet} S$. My approach was to break down the question of whether $Omega_{P_bullet/R} otimes_{P_bullet} S$ and $Omega_{Q_bullet/R} otimes_{Q_bullet} S$ are quasi-isomorphic into two sub-questions:

  1. Whether the map $Omega_{P_bullet/R} otimes_{P_bullet} Q_bullet to Omega_{Q_bullet/R}$ is a quasi-isomorphism.
  2. Whether $- otimes_{Q_bullet} S$ sends quasi-isomorphisms of free $Q_bullet$-modules to quasi-isomorphisms.

For 1, I have fitted them in a short exact sequence (in general $0 to$ is not present, but for my case it is ok):
$$0 to Omega_{P_bullet/R} otimes_{P_bullet} Q_bullet to Omega_{Q_bullet/R} to Omega_{Q_bullet/P_bullet} to 0$$

So it suffices to show that $Omega_{Q_bullet/P_bullet}$ is acyclic, which is what I asked above.

For 2, I guess the usual proof with cones would go through ($f$ quasi-isomorphism $implies$ $operatorname{cone}(f)$ acyclic $implies$ $operatorname{cone}(f) otimes_{Q_bullet} S = operatorname{cone}(f otimes_{Q_bullet} S)$ acyclic $implies$ $otimes_{Q_bullet} S$ quasi-isomorphism), but I have not checked the details yet. I would appreciate if someone would give me a pointer on this, but I could also ask this in a later question.