Vol. 111, No. 1, 1984

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ISSN: 0030-8730
Some conditions on the homology groups of the Koszul complex

Carla Massaza and Alfio Ragusa

Vol. 111 (1984), No. 1, 137–161
Abstract

In this paper we introduce the concept of a (d,i)-sequence (d,i N) in a commutative ring A, noetherian and with identity (cf. Def.  1.1). Let K(z,A) be the Koszul complex on A, with respect to the sequence z = z1,,zn: the concept of a (d,i)-sequence is expressed in terms of the structure of H1(K(z,A)); in particular, it turns out that z is an (n,i)-sequence iff H1(K(z,A)) = 0, and such a condition implies z is a (d,i)-sequence for any d n. If z1,,zh is a (d,i)-sequence in hA = A∕(zh+1,,zn), d h n, then z is seen to be a (d,i)-sequence in A; so, in particular, if H1(K(z;dA)) = 0 in dA, then z is a (d,i)-sequence. Moreover, for i = 1, the two conditions are equivalent, so that z is a (d,1)-sequence means precisely that z1,,zd is regular in dA. For i > 1, examples show that z is a (d,i)-sequence is a condition strictly weaker than z1,,zh is a (d,i)-sequence in hA, and we investigate the relationship between those two properties. In fact, their equivalence allows us to read the depth of a quotient ring A∕(zh+1,,zn) in terms of the Koszul complex K(z;A) and implies, for (d,i)-sequences, properties which are a natural generalization of good properties satisfied by regular sequences, such as the depth-sensitivity of the Koszul complex. A characteristic condition for their equivalence is a kind of weak surjectivity of a natural map acting between syzi+1(K(z;A)) and syzi+1(K(z;hA)).

From an algebraic form of that weak surjectivity we get some sufficient conditions, in terms of weak regularity of the sequence zh+1,,zn. For instance, if zh+1,,zn is a d-sequence, or a relative regular sequence, or less, if zh+1,,zn is a relative regular A-sequence with respect to a convenient set of ideals, then z is a (d,i)-sequence in A implies z1.,zh is a (d,i)-sequence in hA.

Moreover, if z is a (d,i)-sequence and zd+1,,zn is a regular sequence, then H1(K(z;A)) = 0, while this vanishing implies that it is possible to find x1,,xn in I = (z1,,zn) such that z1,,zi1, x1,,xn is a (d,i)-sequence and xd+1,,xn is a regular sequence.

In the last section we give an interpretation of our results in terms of the behaviour of some systems of linear equations.

Mathematical Subject Classification 2000
Primary: 13D25
Secondary: 13C15
Milestones
Received: 30 October 1981
Revised: 10 July 1982
Published: 1 March 1984
Authors
Carla Massaza
Alfio Ragusa