Winfried Bruns
,
Udo Vetter
The paper is published:
J. Herzog, G. Restuccia (Hrsg.), Geometric and combinatorial aspects of commutative algebra, Marcel Dekker, 2001, 89 - 97
MSC 2000
- 13D25 Complexes
-
13C13 Other special types
Abstract
Let $R$ be a noetherian ring and $M$ a finite $R$-module. With a linear form
$\chi$ on $M$ one associates the Koszul complex $K(\chi)$. If $M$ is a free
module, then the homology of $K(\chi)$ is well-understood, and in particular it
is grade sensitive with respect to $\Im\chi$.
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 non-vanishing Fitting ideal $\I_M$ has the maximally possible grade
$r+1$, $r=\rank M$. Then $h=\grade \Im\chi\le r+1$ for all linear forms $\chi$
on $M$, and it turns out that $H_{r-i}(K(\chi))=0$ for all even $i(i) $r=1$ or (ii) $\rank F=1$ and $r$ is odd.
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