  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 $\grade I_M=\rank
M=\dim R$. ($I_M$ is the first non-vanishing Fitting ideal of $M$.) 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.
