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 = z_{1},…,z_{n}: the concept of a (d,i)sequence is
expressed in terms of the structure of H_{1}(K(z,A)); in particular, it turns
out that z is an (n,i)sequence iff H_{1}(K(z,A)) = 0, and such a condition
implies z is a (d,i)sequence for any d ≤ n. If z_{1},…,z_{h} is a (d,i)sequence in
_{h}A = A∕(z_{h+1},…,z_{n}), d ≤ h ≤ n, then z is seen to be a (d,i)sequence in A; so, in
particular, if H_{1}(K(z;_{d}A)) = 0 in _{d}A, 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 z_{1},…,z_{d} is regular in _{d}A. For i > 1, examples show
that z is a (d,i)sequence is a condition strictly weaker than z_{1},…,z_{h} is a
(d,i)sequence in _{h}A, and we investigate the relationship between those two
properties. In fact, their equivalence allows us to read the depth of a quotient
ring A∕(z_{h+1},…,z_{n}) 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 depthsensitivity of the
Koszul complex. A characteristic condition for their equivalence is a kind of
weak surjectivity of a natural map acting between syz^{i+1}(K(z;A)) and
syz^{i+1}(K(z;_{h}A)).
From an algebraic form of that weak surjectivity we get some sufficient
conditions, in terms of weak regularity of the sequence z_{h+1},…,z_{n}. For instance, if
z_{h+1},…,z_{n} is a dsequence, or a relative regular sequence, or less, if z_{h+1},…,z_{n} is a
relative regular Asequence with respect to a convenient set of ideals, then z is a
(d,i)sequence in A implies z_{1}.…,z_{h} is a (d,i)sequence in _{h}A.
Moreover, if z is a (d,i)sequence and z_{d+1},…,z_{n} is a regular sequence, then
H_{1}(K(z;A)) = 0, while this vanishing implies that it is possible to find x_{1},…,x_{n} in
I = (z_{1},…,z_{n}) such that z_{1},…,z_{i−1}, x_{1},…,x_{n} is a (d,i)sequence and x_{d+1},…,x_{n} 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.
