Andre-Quillen homology of algebra retracts

Abstract

Given a homomorphism of commutative noetherian rings $ϕ: R \to S$, 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 $ψ: S \to R$ such that $ϕ\circψ=\id_S$. More precisely, in this case we show that $ψ$ is complete intersection at $ϕ^{-1}(\fn)$ 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.