Length Formulas for the Homology of General- Ized Koszul Complexes

Let M be a finite module over a noetherian ring R with a free resolution of length 1. We consider the generalized Koszul complexes Cλ̄(t) associated with a map λ̄ : M → H into a finite free R-module H (see [IV], section 3), and investigate the homology of Cλ̄(t) in the special setup when grade IM = rankM = dimR. (IM is the first non-vanishing Fitting ideal of M .) In this case the (interesting) homology of Cλ̄(t) has finite length, and we deduce some length formulas. As an application we give a short algebraic proof of an old theorem due to Greuel (see [G], Proposition 2.5). We refer to [HM] where one can find another proof by similar methods.