Definition 13.28.1. Let \mathcal{D} be a triangulated category. We denote K_0(\mathcal{D}) the zeroth K-group of \mathcal{D}. It is the abelian group constructed as follows. Take the free abelian group on the objects on \mathcal{D} and for every distinguished triangle X \to Y \to Z impose the relation [Y] - [X] - [Z] = 0.
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