Let $A$ be an essentially small abelian category, and $D(A)$ it derived category.
Does $K\_{0}(D(A)) = 0$?
Thank you!



Grothendieck group of derived category
Yes, it's always zero, assuming $D(A)$ means the unbounded derived category.


My complexes will be cochain complexes, and $X[1]$ will be $X$ shifted down in degree.


First suppose $X$ is bounded below, and let $X'=\bigoplus\_{n\geq0}X[-2n]$. Even if $A$ doesn't have infinite direct sums, this is still an object of $D(A)$, since in each degree it only involves a finite direct sum. Then there is a triangle
$$X'[-2]\to X'\to X\stackrel{0}{\to} X'[-1]$$
and so the class $[X]$ of $X$ in the Grothendieck group is zero.


A similar argument, using shifts in the opposite direction, works if $X$ is bounded above.


Finally, any $X$ fits in a triangle with a bounded below and a bounded above complex, so $[X]=0$ for all $X$.


