This work concerns a map
φ
:
R
→
S
\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
D
n
(
S
/
R
;
−
)
=
0
\mathrm {D}_n(S/R;-)=0
for some
n
≥
1
n\ge 1
, then
D
i
(
S
/
R
;
−
)
=
0
\mathrm {D}_i(S/R;-)=0
for all
i
≥
2
i\ge 2
and
φ
{\varphi }
is locally complete intersection. This extends Avramov’s theorem that draws the same conclusion assuming
D
n
(
S
/
R
;
−
)
\mathrm {D}_n(S/R;-)
vanishes for all
n
≫
0
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
φ
\varphi
.