Definition 12.11.1. Let $\mathcal{A}$ be an abelian category. We denote $K_0(\mathcal{A})$ the zeroth $K$-group of $\mathcal{A}$. It is the abelian group constructed as follows. Take the free abelian group on the objects on $\mathcal{A}$ and for every short exact sequence $0 \to A \to B \to C \to 0$ impose the relation $[B] - [A] - [C] = 0$.
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