Length Formulas for the Homology of Generalized 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 ${\mathcal{C}}_{\bar{\lambda }}(t)$ associated with a map $\bar{\lambda }:M\to \mathcal{H}$ into a finite free $R$-module $\mathcal{H}$ (see [IV], section 3), and investigate the homology of ${\mathcal{C}}_{\bar{\lambda }}(t)$ in the special setup when $\grade I_M=\rank M=\dim R$\mathrm{.}($I_M$isthefirstnon-vanishingFittingidealof$M$.) In this case the (interesting) homology of ${\mathcal{C}}_{\bar{\lambda }}(t)$ 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.