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.