The Koszul complex in projective dimension one

TL;DR

This paper studies the homology of Koszul complexes associated with modules of projective dimension one, revealing grade sensitivity and specific conditions under which the complex's homology behaves in particular ways.

Contribution

It characterizes the homology of Koszul complexes for modules of projective dimension one with maximal grade Fitting ideals, highlighting grade sensitivity and special cases.

Findings

Homology vanishes in certain degrees depending on grade

Homology is isomorphic to symmetric powers of a specific module

Maximal grade occurs only in two specific cases

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 $i<h$ 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, $C=\Cok\psi^*$ for a presentation $$ 0\to F\stackrel{\psi}{\to} G \to M\to 0. $$ Moreover, if $h\le r$, 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 $\grade \Im\chi=r+1$ can only occur in two extreme cases: (i) $r=1$ or (ii) $\rank F=1$ and $r$ is odd.