K-theory of triangulated categories 3 3/4: 
A direct proof of the theorem of the heart

Amnon Neeman

Let $\mathcal T$ be a triangulated categories, and assume it
admits at least one model. In this article, we define a K-theory
for $\mathcal T$. The main theorem is that, given any bounded
t-structure on $\mathcal T$, the K-theory of the heart agrees
with the K-theory of $\mathcal T$. An immediate consequence is
that, if two abelian categories occur as hearts of a triangulated
category for two different t-structures, then their K-theories
must be isomorphic.

The proof was sketched in previous articles in this series. The
virtue of this article is in the careful detail in which it was
written down.