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Gaps in Hochschild cohomology imply smoothness for commutative algebras

Luchezar Avramov·Srikanth Iyengar
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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 HH^n(S|K,M) in sufficiently long intervals imply the smoothness of S_q over K for all prime ideals q in the support of M. In particular, S is smooth if HH^n(S|K,S)=0 for (dim S+2) consecutive non-negative integers n.

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Luchezar Avramov, Srikanth Iyengar. 2005-08-06. Gaps in Hochschild cohomology imply smoothness for commutative algebras. https://arxiv.org/abs/math/0412259

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