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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,…,zi−1, 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.
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