Gaps in Hochschild cohomology imply smoothness for commutative algebras

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. For a finite $S$-module $M$ it is proved that vanishing of $\hcoh{n}KSM$ in sufficiently long intervals imply the smoothness of $S_\fq$ over $K$ for all prime ideals $\fq$ in the support of $M$. In particular, $S$ is smooth if $\hcoh{n}KSS=0$ for $(\dim S+2)$ consecutive $n\ge0$.

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Published 1 January 2005