Rigidity properties of the cotangent complex

This work concerns a map φ : R →<!-- → <mml:mi>S \varphi \colon R\to S of commutative noetherian rings, locally finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if alttext="normal D Subscript n Baseline left-parenthesis S slash semicolon minus right-parenthesis equals 0"> D n stretchy="false">( / ; −<!-- − stretchy="false">) = 0 encoding="application/x-tex">\mathrm {D}_n(S/R;-)=0 for some alttext="n greater-than-or-equal-to 1"> ≥<!-- ≥ <mml:mn>1 encoding="application/x-tex">n\ge 1 , then i i {D}_i(S/R;-)=0 all alttext="i 2"> 2 encoding="application/x-tex">i\ge 2 and alttext="phi"> encoding="application/x-tex">{\varphi } complete intersection. extends Avramov’s theorem draws same conclusion assuming right-parenthesis"> {D}_n(S/R;-) vanishes much-greater-than ≫<!-- ≫ encoding="application/x-tex">n\gg 0 confirming conjecture Quillen. The rigidity deduced from more general result about higher cotangent modules which answers question raised by Avramov Herzog, subsumes Vasconcelos was recently first author. new insight leading to these results equivariance cohomology Hochschild defined using universal Atiyah class encoding="application/x-tex">\varphi .