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GAPS IN HOCHSCHILD COHOMOLOGY IMPLY SMOOTHNESS FOR COMMUTATIVE ALGEBRAS
 

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 K-module. For a finite S-module 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 K-algebra, and
M an S-module. We let HH(S|K; M) and HH
(S|K; M) denote, respectively,
the Hochschild homology and the Hochschild cohomology of the K-algebra S with
coefficients in M. For each n Z there are canonical homomorphisms
M
n : (n
SS |K) S M - HHn(S|K; M)

  

Source: Avramov, Luchezar L.- Department of Mathematics, University of Nebraska-Lincoln

 

Collections: Mathematics