My question is what is the relation (if any) between the following two notions.


1. Starting from an augmented algebra $A$ over a field $k$, one way to compute the homology of $A$ is to find a projective resolution of $k$ as an $A$-module.
2. On the other hand, one could also see $A$ as a (let's say positively graded) dg-algebra (concentrated in degree 0), and take a cofibrant replacement of $A$ in this category, using the projective model structure.


Both notions give some notions homotopical information about $A$, and I would be interested in knowing if it is possible to deduce one from the other (presumably, to recover a resolution of $k$ from a cofibrant replacement of $A$), at least in "good" cases.


The reason I ask is because I have two very similar constructions building these objects for suitable algebras, and I am wondering whether one of these constructions can be deduced from the other one.



Algebras: Homology vs. Resolution as a dg-algebra
From your first description, I guess that what you call "the homology of $A$" is in fact the Hochschild homology of $A$ with constant coefficients, i.e. $HH\_\*(A;k)$. Please tell me if I got this wrong.


Then yes, it's possible to compute it from a projective resolution of $A$. Suppose that $R \xrightarrow{\sim} A$ is a projective resolution of $A$. Let $\Omega\_R$ be the $R$-module of Kähler differentials. Then you can compute the Hochschild homology of $A$ as the André–Quillen homology, i.e. the homology of the complex:
$$k \otimes\_{A \otimes A^{op}} (A \otimes\_R \Omega\_R).$$
The object $\mathbb{L}\_{R/A} = A \otimes\_R \Omega\_R$ is called the cotangent complex of $A$.


