Some conditions on the homology groups of the Koszul complex (Q1057326)

Let \(A\) be a noetherian ring, \(\bar z=z_ 1,...,z_ n\) a sequence of elements in the Jacobson radical of \(A\), \(K_*(\underline z;A)\) the Koszul complex with respect to \(\bar z\) (generated by \(e_ 1,...,e_ n)\) and \(T_ i^{(n,d)}\), \(1\leq i\leq d\leq n\), the submodule of \(K_ i(\underline z;A)\), generated by all elements of the form \(e_{j_ 1}\wedge...\wedge e_{j_ i}\), \(1\leq j_ 1<...<j_ i\leq n\), \(d<j_ i\). The authors call \(\bar z\) a \((d,i)\)-sequence if the homology group \(H_ i(K(z;A))\) is generated by the image of \(T_ i^{(n,d)}\). They investigate the properties of \((d,i)\)-sequences and the connections of this concept with that of regular sequences and other generalizations of them. Let \(\bar z\) be a \((d,i)\)-sequence. The authors give conditions for \(\bar z_ 1,...,\bar z_ h\) to be a \((d,i)\)-sequence in \({}_ h\bar A:=A/(z_{h+1},...,z_ n)\), \(d\leq h\leq n\). As an application they can show e.g.: Suppose \(z_{d+1},...,z_ n\) is a regular sequence. Then \(z_ 1,...,z_ n\) is a (d,i)-sequence iff \(depth(\bar z_ 1,...,\bar z_ d)\geq d-i+1\) in \({}_ d\bar A\).