Koszul Duality and Andre-Quillen homology relationship

Question

This may be a very open ended question but what is the relationship between koszul duality and Andre-Quillen homology? For example if I consider the André-Quillen homology $AQ_k(B)$ of an associative differential (positively) graded $R$-algebra then there is a grading on $AQ_k(B)$ which we may denote as $AQ_{k,n}(B)$. Now if I define $B$ to be koszul when $AQ_{k,n}(B) = 0$ for all $k\neq n$ is this the same as the usual definition of koszulness using $Ext$/$Tor$? Has this been studied?

Answer 1

You should not mix up André--Quillen homology and Quillen homology. One is the derived functor of derivations, the other of indecomposables.

You want the latter, and then Koszulness is defined by the vanishing of $(d-1,w)$ terms ($d$ is the homological degree, $w$ the weight degree) with $d-1\neq w$, since there is a shift with respect to the usual grading for $\operatorname{Tor}$: $s^{-1}\operatorname{Tor}$ is Quillen homology, with the same weight grading, where $s^{-1}$ is the desuspension.

The extra grading for André--Quillen homology is useful, but for different reasons and related to deducing statements about deformations, for example see this paper.