Vol. 111, No. 1, 1984

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Some conditions on the homology groups of the Koszul complex

Carla Massaza and Alfio Ragusa

Vol. 111 (1984), No. 1, 137–161

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
Received: 30 October 1981
Revised: 10 July 1982
Published: 1 March 1984
Carla Massaza
Alfio Ragusa