2006/01/23 by Sean Sather-Wagstaff, Sather-Wagstaff, Sean, Diana White +1
Mathematics · #13D02 #13D05 (Primary) #13D07 #13D25 #13H10 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras #math.AC #msc:13D02 #msc:13D05 #msc:13D07 #msc:13D25 #msc:13H10
paper · pdf · doi:10.48550/arxiv.math/0601538
20 pages, uses xypic, minor changes to final version, to appear in Math. Scand
openalex publication_date 2006/01/23 · arxiv created 2007/12/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We extend Auslander and Buchsbaum's Euler characteristic from the category of finitely generated modules of finite projective dimension to the category of modules of finite G-dimension using Avramov and Martsinkovsky's notion of relative Betti numbers. We prove analogues of some properties of the classical invariant and provide examples showing that other properties do not translate to the new context. One unexpected property is in the characterization of the extremal behavior of this invariant: the vanishing of the Euler characteristic of a module M of finite G-dimension implies the finiteness of the projective dimension of M. We include two applications of the Euler characteristic as well as several explicit calculations.