Let $A$ be an essentially small abelian category, and $D(A)$ its derived category. Does $K_{0}(D(A)) = 0$? Thank you!
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3$\begingroup$ This is true if your category has infinite sums (by the Eilenberg swindle), but clearly not in general. $\endgroup$– Tom BachmannCommented Feb 15, 2017 at 9:10
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3$\begingroup$ This example is probably too obvious, but let me state it anyway. Let $A$ is the category of finite dimensional vector spaces over a field. Let $D(A)=D^b(A)$ denote the bounded derived category. Then $K_0(D(A))=\mathbb{Z}$. $\endgroup$– asvCommented Feb 15, 2017 at 13:24
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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 $X$.
answered Feb 15, 2017 at 10:28