Gaps in Hochschild cohomology imply smoothness for commutative algebras

Luchezar L. Avramov and Srikanth Iyengar

Abstract. The paper concerns Hochschild cohomology of a commutative
algebra $S$, which is essentially of finite type over a commutative
noetherian ring $K$ and projective as a $K$-module, with coefficients
in an $S$-module $M$. It is proved that vanishing of $\textrm{HH}^n(S|K;M)$
in sufficiently long intervals imply the smoothness of $S_\mathfrak q$ over
$K$ for all prime ideals $\mathfrak q$ in the support of $M$. In 
particular, $S$ is smooth if $\textrm{HH}^n(S|K;S)=0$ for ($\dim S + 2$)
consecutive $n \ge 0$.