A Note on the Grothendieck Group of an Additive Category

2011/09/09 by David E. V. Rose, Rose, David E. V. · 1 citation
Mathematics · #18E05 #18E30 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.CT #math.QA #msc:18E05 #msc:18E30

paper · pdf · doi:10.48550/arxiv.1109.2040

9 pages

arxiv created 2011/09/09 · openalex publication_date 2011/09/09 · arxiv updated 2011/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

There are two abelian groups which can naturally be associated to an additive category A: the split Grothendieck group of A and the triangulated Grothendieck group of the homotopy category of (bounded) complexes in A. We prove that these groups are isomorphic. Along the way, we deduce that the `Euler characteristic' of a complex in A is invariant under homotopy equivalence.

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