Mathematics – Commutative Algebra
Scientific paper
2000-07-11
Mathematics
Commutative Algebra
9 pages
Scientific paper
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
Bruns Winfried
Vetter Udo
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