On the deformation of commutative algebras

Abstract

Introduction. Given a field k and a k-algebra A, Gerstenhaber has introduced the concept of a deformation of A [8] which is a k[[t]]-algebra structure on the module A[[t]]. One of the main problems of the theory of deformations is to determine those algebras for which the only deformation (modulo an equivalence relation) is the usual k[[t]]-algebra structure on A[[t]]. Such algebras are said to be rigid. If A is a commutative k-algebra, we may restrict our attention to those deformations of A which are commutative. This leads us to consider the commutative cohomology groups of A, Hn(A, M)s, defined by Harrison [11]. The principal purpose of this paper is to investigate the following conditions on a commutative algebra A over a field k: (1) H2(A, M)s=0 for all A-modules M. (2) H2(A,A)s=0. (3) A is a rigid k-algebra in the commutative deformation theory. (4) The module of differentials of A over k is a projective A-module. (5) A is a finite product of (not necessarily algebraic) separable extension fields of k. We may show that (1) implies (4) [9, Theorem (20.4.9)] and that (2) implies (3) [8, p. 65]. Condition (1) may be expressed by saying that A is a formally smooth k-algebra [9, ?19.4.4]. When A is a field, Gerstenhaber has shown that (1), (2), and (5) are equivalent and he conjectured that these conditions were equivalent to (3). We show that (1), (2), (3), and (5) are equivalent even for A semisimple (Theorem 6.5). Moreover, if A is artinian, then (1) and (5) are equivalent (Proposition 5.9). In another -direction, if A is a geometric complete intersection, then (1), (2), and (3) are equivalent (Theorem 6.8a). If in addition A is an integral domain with a separable quotient field, then (1), (2), (3), and (4) are equivalent (Theorem 6.8c). In the final section, we investigate the above conditions for the group algebra B[G] of an abelian group G which is a direct sum of cyclic groups with B a commutative algebra over a field k of positive characteristic p. In addition, we consider the following condition on G and k: (6) G has no element of order p = char (k).