Geometry and Topology

A note on the Grothendieck group of an additive category

D. Rose

Department of Mathematics of University of Southern California, Los Angeles, USA

Abstract: There are two abelian groups which can naturally be associated to an additive category $\mathcal{A}$: the split Grothendieck group of $\mathcal{A}$ and the triangulated Grothendieck group of the homotopy category of (bounded) complexes in $\mathcal{A}$. We prove that these groups are isomorphic. Along the way, we deduce that the ‘Euler characteristic’ of a complex in $\mathcal{A}$ is invariant under homotopy equivalence, a result which has implications for (de)categorification.

Keywords: Grothendieck group, additive category, categorification.

UDC: 512.58
BBK: B152.4

Language: English