A Note on the Grothendieck Group of an Additive Category

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.

9 pages

References in corpus (2)

• Completions of Grothendieck groups
• A Categorification of Quantum sl_3 Projectors and the sl_3 Reshetikhin-Turaev Invariant of Tangles

Cited by in corpus (3)

• A Categorification of Quantum sl_3 Projectors and the sl_3 Reshetikhin-Turaev Invariant of Tangles
• Mixed perverse sheaves on flag varieties of Coxeter groups
• Fourier-Mukai functors and perfect complexes on dual numbers