Andre-Quillen homology of algebra retracts

Given a homomorphism of commutative noetherian rings $\varphi :R\to S$, Daniel Quillen conjectured in 1970 that if the Andre-Quillen homology functors $D_n(S\mathrm{\mid }R,-)$ vanish for all $n\gg 0$, then they vanish for all $n\ge 3$. We prove the conjecture under the additional hypothesis that there exists a homomorphism of rings $\psi :S\to R$ such that $\varphi \circ \psi ={\text{id}}_S$. More precisely, in this case we show that $\psi$ is complete intersection at ${\varphi }^{-1}(\text{fn})$ for every prime ideal $\text{fn}$ of $S$. Using these results, we describe all algebra retracts $S\to R\to S$ for which the $S$-algebra $To{r}^R(S,S)$ is finitely generated.