Complete intersections in tensor algebras II

Abstract

In this paper we continue the study of’ the vanishing of the lower relative Poochschild cohomology groups. This paper is ;a continuation of [3]. Recall that if A is a k-algebra, where k is a commutative ring, a two-sided ideal I in A is a camplete intersection in ~4 if H*((@‘)//I,~ = 0 for all two-sided A-modules M. The use of the terminology “complete intersection” comes from the analogy between these relative cohomology groups and the cohomology groups studied in commutative deformation theory [see 2,3] . Suppose that k is a field, R and R 03+ R”P are semi-simple k-algebras and X is a two-sided R-module. Let T be the tensor k-algebra R + X + (6b~ X) + (6~ x) + l . In [3] 9 a homological criterion was found for an ideal I in T to be a complete intersection; namely, I/I* must be a left projective (T/f) Ok (T/Z)‘P-module. Section 1 of this paper gives a homological criterion for an ideal in a factor ring of T to be a complete intersection. For the rest of the paper we restrict our attention further, to special Artin tensor k-algebras, defined in section 2, and their factor rings. After defining an appropriate concept of “minimal generating set” of an ideal, we study the question of when an ideal with one generator, in a factor ring of a special tensor k-a!gebra, is a complete intersection. Section 3 deals with the question of when a subideal of a complete intersection in a special tensor k-algebra is again a complete intersection. We show that if I is a complete intersection in T and if I has generators (more precisely, J-generators, see section 2) x1, . . . . x, then, after reordering the xi in a natural way, the ideals generated by x1, . . . . xi are also complete intersections in T, for i = 1, . . . . rn. Once this is shown, we show that the generators of complete intersections have properties similar to those of R-sequences, where R is a commutative ring (see Theorem 4.1). Unfortunately, this analogy does not go too far, as shown by example in section 4.