2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73574The 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.Minor revisions. To appear in Math. Res. Letters. 16 pages; available also at http://www.math.unl.edu/~siyengarCommutative AlgebraRings and Algebras13D03; 14B25. Secondary: 14M10; 16E40text