Given an abelian category C of finite length (i.e., every object has a Jordan-Holder filtration), let $A$ be the set of isomorphism classes of simple objects. there's an obvious map
$\mathbb Z^A \to K_0(C)$
given by sending a vector in $\mathbb Z^A$ (the free abelian group on A) to the obvious class in the Grothendieck group. The map is a surjection, obviously, but what's the best way to show it's an injection?
That is, given some equation in the Grothendieck group
$\sum_i n_i [S_i] = 0$
where $S_i$ is a simple object and $n_i \in \mathbb Z$, how do I show that all the $n_i$ must be zero?