Gaps in Hochschild cohomology imply smoothness for commutative algebras

Details Gaps in Hochschild cohomology imply smoothness for commutative algebras Gaps in Hochschild cohomology imply smoothness for commutative algebras

2004-12-13

arxiv.org/abs/math/0412259v2

Mathematics

Commutative Algebra

Minor revisions. To appear in Math. Res. Letters. 16 pages; available also at http://www.math.unl.edu/~siyengar

Scientific paper

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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