Let $A\to B$ be a morphism of commutative rings. Let $\mathcal C$ be the category of commutative $A$-algebras augmented over $B$. Let $\mathcal M\_B$ denote the category of $B$-modules. The cotangent complex can be defined like this (I will brush model-categorical considerations under the carpet). Consider the derived functor of the functor


$$s\mathcal C \to s\mathcal M\_B$$


obtained levelwise from the functor $\mathcal C\to \mathcal{M}\_B$ given by $R\mapsto B\otimes\_R \Omega\_{R|A}$. Here $\Omega$ denotes Kähler differentials.


The image of $B\in \mathcal C$ under this derived functor is the cotangent complex $\mathbb L \Omega\_{A|B}$.


If $M$ is a $B$-module, one defines the André-Quillen cohomology modules of $A\to B$ with coefficients in $M$ as
$$D^i(B|A,M)=H^q(\mathcal M\_B(\mathbb L\Omega\_{B|A},M)) \cong H^q \mathrm{Der}\_A(P,M)$$


where $P\to A$ is a projective resolution of $A$ in $s\mathcal C$.


So I guess this allows one to safely say that André-Quillen cohomology are some kind of "derived functor of the derivations".


I am wondering if it is (sometimes?) possible to **resolve the module variable**, i.e. to derive the functor $\mathrm{Der}\_A(B,-):s\mathcal M\_B \to s\mathcal M\_B$, and get the same result. This feels somehow like asking whether $\mathrm Der\_A(-,-): s\mathcal C \times \mathcal sM\_B\to s\mathcal M\_B$ is balanced.



Is it possible to compute André-Quillen cohomology by resolving the module variable?
The cotangent complex $\Bbb L\_{B/A}$ is a (homologically) bounded-below complex of projective $B$-modules, and the bottom homology group is $H\_0 (\Bbb L\_{B/A}) = \Omega\_{B|A}$.


If we apply $RHom\_{B-mod}(-,M)$ to this, we get a Grothendieck spectral sequence for computing André-Quillen cohomology:
$$
Ext^p\_B(H\_q \Bbb L\_{B/A}, M) \Rightarrow D^{p+q} (B|A, M).
$$
In particular, as Jason Starr points out, the derived functors of $Der\_A(B,-) = Hom\_B(\Omega\_{B|A}, -)$ are the terms
$$
Ext^p\_B(H\_0 \Bbb L\_{B/A}, M)
$$
on the *edge* of this spectral sequence, and so we get an equality precisely when the rest of the spectral sequence collapses to zero. The most common way that this could happen is if $H\_q \Bbb L\_{B/A} = 0$ for all $q > 0$ (smoothness, or certain local complete intersections), or maybe if the higher André-Quillen homology groups and $M$ are supported in different places, but is unlikely otherwise.


