2000/07/11 by Winfried Bruns, Bruns, Winfried, Udo Vetter +1 · 1 citation
Mathematics · #13C13 #13D25 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AC #math.AG #msc:13C13 #msc:13D25
paper · pdf · doi:10.48550/arxiv.math/0007069
9 pages
arxiv created 2000/07/11 · openalex publication_date 2000/07/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \IM has the maximally possible grade r+1, r=\rank M. Then h=\grade \Imχ≤ r+1 for all linear forms χ on M, and it turns out that Hr-i(K(χ))=0 for all even i