 
Summary: GAPS IN HOCHSCHILD COHOMOLOGY IMPLY SMOOTHNESS
FOR COMMUTATIVE ALGEBRAS
LUCHEZAR L. AVRAMOV AND SRIKANTH IYENGAR
Abstract. The paper concerns Hochschild cohomology of a commutative al
gebra S, which is essentially of finite type over a commutative noetherian ring
K and projective as a Kmodule. For a finite Smodule M it is proved that
vanishing of HHn
(S K; M) in sufficiently long intervals imply the smoothness
of Sq over K for all prime ideals q in the support of M. In particular, S is
smooth if HHn
(S K; S) = 0 for (dim S + 2) consecutive n 0.
Introduction
Let K be a commutative noetherian ring, S a commutative Kalgebra, and
M an Smodule. We let HH(SK; M) and HH
(SK; M) denote, respectively,
the Hochschild homology and the Hochschild cohomology of the Kalgebra S with
coefficients in M. For each n Z there are canonical homomorphisms
M
n : (n
SS K) S M  HHn(SK; M)
