  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 $\varphi$.
