Rigidity properties of the cotangent complex.

This work concerns a map \varphi \colon 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 1, then \mathrm {D}_i(S/R;-)=0 for all i\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 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 \varphi. [ABSTRACT FROM AUTHOR]