Length Formulas for the Homology of Generalized Koszul Complexes
Abstract
Let
M
be a finite module over a noetherian ring
R
with a free resolution of length 1. We consider the generalized Koszul complexes
C
λ
¯
(t)
associated with a map
λ
¯
:M→H
into a finite free
R
-module
H
(see [IV], section 3), and investigate the homology of
C
λ
¯
(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
C
λ
¯
(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.