Rigidity properties of the cotangent complex

Abstract

This work concerns maps $\phi :R\to S$ of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if ${\mathrm{D}}_n(S/R;-)=0$ for some $n\ge 2$, then ${\mathrm{D}}_n(S/R;-)=0$ for all $n\ge 2$ and $\phi$ is locally complete intersection. This extends Avramov's theorem that draws the same conclusion assuming ${\mathrm{D}}_n(S/R;-)$ vanishes for all $n\gg 0$, confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a question raised by Avramov and Herzog, and subsumes a conjecture of Vasconcelos that was proved recently by the first author. The new insight leading to these results concerns the equivariance of a map from André-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of $\phi$.

arXiv comment20 pages. Major revision; to appear in the Journal of the American Mathematical Society