The Koszul Complex in Projective Dimension One

Let R be a noetherian ring and M a finite R-module. With a linear form χ on M one associates the Koszul complex K(χ). If M is a free module, then the homology of K(χ) is well-understood, and in particular it is grade sensitive with respect to Imχ. In this note we investigate the case of a module M of projective dimension 1 (more precisely, M has a free resolution of length 1) for which the first nonvanishing Fitting ideal IM has the maximally possible grade r + 1, r = rankM . Then h = grade Imχ ≤ r + 1 for all linear forms χ on M , and it turns out that Hr−i(K(χ)) = 0 for all even i < h and Hr−i(K(χ)) ∼= S(i−1)/2(C) for all odd i < h where S denotes symmetric power and C = ExtR(M,R), in other words, C = Cokψ∗ for a presentation