Andre-Quillen homology of algebra retracts

Details Andre-Quillen homology of algebra retracts Andre-Quillen homology of algebra retracts

2002-10-02

arxiv.org/abs/math/0210037v1

Mathematics

Commutative Algebra

30 pages. To be published in Ann. Sci. Ecole Norm. Sup. (4)

Scientific paper

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.

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