The Koszul complex in projective dimension one (Q2717164)

This note deals with the homology of the Koszul complex \({\mathbf K}(\chi)\), where \(\chi\) is a linear form on a finite \(R\)-module \(M\) (\(R\) a noetherian ring), of projective dimension one and with first non vanishing Fitting ideal of grade \(r+1\) \((r=\text{rank} M)\). All the modules \(H_u({\mathbf K}(\chi))\) are described, for \(u=r-i\), \(i<h=\text{grade Im}(\chi)\). The results proved in this paper are a generalization of the well-known ``grade sensitivity'' of the Koszul homology, when \(M\) is free; moreover, they generalize some results of J. Migliore, U. Nagel and C. Peterson, proved when \(R\) is a Gorenstein ring.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].