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- W1618412048 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 $Imchi$. 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 $h=grade Imchile r+1$ for all linear forms $chi$ on $M$, and it turns out that $H_{r-i}(K(chi))=0$ for all even $i<h$ and $H_{r-i}(K(chi))iso SS^{(i-1)/2}(C)$ for all odd $i<h$ where $SS$ denotes symmetric power and $C=Ext_R^1(M,R)$, in other words, $C=Cokpsi^*$ for a presentation $$ 0to Fstackrel{psi}{to} G to Mto 0. $$ Moreover, if $hle r$, then $H_{r-h}(K(chi))$ is neither 0 nor isomorphic to a symmetric power of $C$, so that it is justified to say that $K(chi)$ is grade sensitive for the modules $M$ under consideration. We furthermore show that the maximally possible value $grade Imchi=r+1$ can only occur in two extreme cases: (i) $r=1$ or (ii) $rank F=1$ and $r$ is odd." @default.
- W1618412048 created "2016-06-24" @default.
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- W1618412048 date "2000-07-11" @default.
- W1618412048 modified "2023-09-27" @default.
- W1618412048 title "The Koszul complex in projective dimension one" @default.
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