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:
- Whether the map $\Omega_{P_\bullet/R} \otimes_{P_\bullet} Q_\bullet \to \Omega_{Q_\bullet/R}$ is a quasi-isomorphism.
- 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 $\implies$ $f \otimes_{Q_\bullet} S$ 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.