[Paper Review] The Koszul complex in projective dimension one

Winfried Bruns, Udo Vetter|ArXiv.org|Jul 11, 2000

Commutative Algebra and Its Applications6 references3 citations

TL;DR

This paper investigates the homology of the Koszul complex associated with a linear form on a finitely generated module of projective dimension one over a Noetherian ring, focusing on the case where the first non-vanishing Fitting ideal has maximal possible grade. It establishes that the homology is grade-sensitive, with explicit descriptions of homology modules in terms of symmetric powers of the Ext module $ C = \operatorname{Ext}_R^1(M,R) $, and characterizes when the maximal grade $ r+1 $ is achieved via module-theoretic conditions on a kernel submodule.

ABSTRACT

Let $R$ be a noetherian ring and $M$ a finite $R$-module. With a linear form $χ$ on $M$ one associates the Koszul complex $K(χ)$. If $M$ is a free module, then the homology of $K(χ)$ is well-understood, and in particular it is grade sensitive with respect to $\Imχ$. 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 $r+1$, $r= ank M$. Then $h=\grade \Imχ\le r+1$ for all linear forms $χ$ on $M$, and it turns out that $H_{r-i}(K(χ))=0$ for all even $i

Motivation & Objective

• To understand the homology of the Koszul complex $ K(\chi) $ for a linear form $ \chi $ on a module $ M $ of projective dimension one.
• To determine when the grade of $ \operatorname{Im}\chi $ achieves its maximal possible value $ r+1 $, where $ r = \operatorname{rank}M $.
• To characterize the structure of the homology modules $ H_{r-i}(K(\chi)) $ in terms of symmetric powers of $ C = \operatorname{Ext}_R^1(M,R) $, particularly when $ \chi $ has maximal grade.
• To provide a module-theoretic characterization of when $ \operatorname{grade}\operatorname{Im}\chi = r+1 $, via existence of a special reflexive submodule $ U \subset M $ of rank $ r-1 $.

Proposed method

• Uses a Koszul bicomplex construction from a dualized presentation $ 0 \to F \xrightarrow{\psi} G \to M \to 0 $, with $ \psi $ an $ R $-homomorphism between free modules.
• Constructs the Koszul complex via antiderivations $ \partial_\psi $ and $ d_\varphi $, forming a bicomplex $ \mathcal{K} $ with terms $ M^{p,q} = \bigwedge^{p-q}G \otimes \operatorname{S}^q(F) $.
• Analyzes row and column homologies $ H_\varphi^{p,q} $ and $ H_\psi^{p,q} $, and relates them to the homology of $ K(\chi) $ via induced maps on kernels.
• Applies results from [BV1] and [HM], and uses duality via $ \psi^* $, linking the structure of $ M $ to that of $ C = \operatorname{Ext}_R^1(M,R) $.
• Employs a self-dual construction via a skew-symmetric isomorphism $ \sigma: U \to U^* $, with $ \sigma^* = -\sigma $, to realize maximal grade in special cases.
• Reduces the problem to analyzing the existence of a reflexive, orientable submodule $ U \subset M $ of rank $ r-1 $ satisfying local freeness conditions at primes of grade $ \leq r $.

Experimental results

Research questions

• RQ1 When does the grade of $ \operatorname{Im}\chi $ for a linear form $ \chi $ on $ M $ of projective dimension one achieve the maximal value $ r+1 $?
• RQ2 What is the precise structure of the homology modules $ H_{r-i}(K(\chi)) $ of the Koszul complex when $ \chi $ has grade $ h < r+1 $?
• RQ3 Under what module-theoretic conditions does $ \operatorname{grade}\operatorname{Im}\chi = r+1 $ hold, and how can such forms be constructed explicitly?
• RQ4 How does the symmetric power structure of $ C = \operatorname{Ext}_R^1(M,R) $ relate to the homology of $ K(\chi) $?
• RQ5 Can the maximal grade condition be realized via a self-dual, skew-symmetric submodule $ U \subset M $, and what are the constraints on $ r $ and $ \operatorname{rank}F $?

Key findings

• For a module $ M $ of projective dimension one with Fitting ideal of grade $ r+1 $, the homology of $ K(\chi) $ satisfies $ H_{r-i}(K(\chi)) = 0 $ for all even $ i < h $, where $ h = \operatorname{grade}\operatorname{Im}\chi $.
• For odd $ i < h $, $ H_{r-i}(K(\chi)) \cong \operatorname{S}^{(i-1)/2}(C) $, where $ C = \operatorname{Ext}_R^1(M,R) $, showing a symmetric power structure in odd degrees.
• If $ h \leq r $, then $ H_{r-h}(K(\chi)) $ is neither zero nor isomorphic to a symmetric power of $ C $, confirming the grade-sensitivity of the complex.
• The maximal grade $ r+1 $ for $ \operatorname{Im}\chi $ occurs only in two cases: $ r = 1 $, or $ \operatorname{rank}F = 1 $ with $ r $ odd.
• The existence of a linear form $ \chi $ with $ \operatorname{grade}\operatorname{Im}\chi = r+1 $ is equivalent to the existence of a reflexive, orientable submodule $ U \subset M $ of rank $ r-1 $, locally free at primes of grade $ \leq r $.
• A self-dual, skew-symmetric isomorphism $ \sigma: U \to U^* $ with $ \sigma^* = -\sigma $ can be constructed explicitly when $ \operatorname{rank}F = 1 $ and $ r $ is odd, via a suitable choice of $ \chi $.