  Given a homomorphism of commutative noetherian rings $\phi: R \to S$, Daniel
Quillen conjectured in 1970 that if the Andre-Quillen homology functors
$D_n(S|R,-)$ 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 $\phi\circ\psi=\id_S$. More
precisely, in this case we show that $\psi$ is complete intersection at
$\phi^{-1}(\fn)$ for every prime ideal $\fn$ of $S$. Using these results, we
describe all algebra retracts $S\to R\to S$ for which the $S$-algebra
$Tor^R(S,S)$ is finitely generated.
