Rigidity properties of the cotangent complex

TL;DR

This paper proves the rigidity of Andre9-Quillen homology for certain ring maps, showing that vanishing at one degree implies complete intersection properties, extending previous theorems and confirming conjectures.

Contribution

It establishes the rigidity of Andre9-Quillen functors for ring maps, generalizing prior results and confirming conjectures about cotangent modules and homology.

Findings

Proves rigidity of Andre9-Quillen homology for ring maps.

Shows vanishing at one degree implies vanishing at all degrees.

Answers a question on higher cotangent modules and confirms Vasconcelos's conjecture.

Abstract

This work concerns maps $\varphi \colon R\to S$ of commutative noetherian rings, locally of finite flat dimension. It is proved that the Andr\'e-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 $\varphi$ 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 Andr\'e-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 Andr\'e-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of $\varphi$.