2005/10/27 by Bogdan Ichim, Ichim, Bogdan, Udo Vetter +1
Mathematics · #13D25 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AC #msc:13D25
paper · pdf · doi:10.48550/arxiv.math/0510608
arxiv created 2005/10/27 · openalex publication_date 2005/10/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a finite module over a noetherian ring R with a free resolution of length 1. We consider the generalized Koszul complexes Cλ(t) associated with a map λ:M\toH into a finite free R-module H (see [IV], section 3), and investigate the homology of Cλ(t) in the special setup when \grade IM=\rank M=dim R. (IM is the first non-vanishing Fitting ideal of M.) In this case the (interesting) homology of Cλ(t) has finite length, and we deduce some length formulas. As an application we give a short algebraic proof of an old theorem due to Greuel (see [G], Proposition 2.5). We refer to [HM] where one can find another proof by similar methods.