I have a reference request on following comment I found in
[nLab article](https://ncatlab.org/nlab/show/Karoubian+category) on Karoubian categories & envelopes. It states:



> 
> The Karoubian envelope is also used in the construction of the
> category of pure motives, and in K-theory.
> 
> 
> 


Almost every introduction to motives containing the basic
constructions (eg Manin's *"Correspondences, motifs and monoidal
transformations"* as 'standard' reference) includes the
fundamental part where one passes from the category $(\mathsf{Sm}/k)$ of smooth curves over $k$
to its Karoubian closure.


On the other hand I'm not sure to which construction in
K-theory where the Karoubian envelope is involved, the quoted
sentence refers.


The first naïve observation is that for a ring $R$ the $K\_0(R)$
in algebraic K-theory is obtained as a certain quotient group
("Grothendieck group") of the set of projective $R$-modules.
This set of projective $R$-modules can be reinterpreted as the
Karoubian envelope of the set of free $R$-modules.


Question: Is this the only explicit usage of Karoubian envelope
in constructions in K-theory or are there more general cases?
I'm quite not sure if the remark above only refers to this 'baby' case
with $K\_0(R)$, or there are more general constructions in K-theory where
Karoubian envelopes are involved.


For example [another nLab article](https://ncatlab.org/nlab/show/K-theory) on the more general development
of K-theory nowhere explains where Karoubian envelope is explicitly used
as a technical tool for certain constructions. Therefore I would like to know
if there are some recommendable papers on development of K-theory
where such constructions involving Karoubian envelopes are discussed.



Idempotent completions in K-theory
In Schlichting's paper [Negative K-theory of derived categories](https://link.springer.com/article/10.1007/s00209-005-0889-3). Math. Z. 253, 97–134 (2006), a definition of negative $K$-theory of triangulated categories is given. These are abelian group valued functors $\mathbf{K}\_{i}$ with $i\leq 0$.


These groups agree with the known negative $K$-theories, for example in the Thomason-Trobaugh paper mentioned by Prof. Grayson. However they use idempotents in an essential way.


In particular, if $E$ is an exact category then the bounded derived category $D^b(E)$ is triangulated and $\mathbf{K}\_{0}(E)$ is defined to be the usual $K\_0$ of $\widetilde{E}$, the idempotent completion of $E$.


If $E$ is idempotent complete, then the unbounded derived category $D(E)$ exists, and the $\mathbf{K}\_{-1}(E)$ turns out to be the quotient of the abelian monoid (under direct sum) of isomorphism classes of idempotents in $D(E)$ by the submonoid of split idempotents. In particular $K\_{-1}(E)=0$ if and only if $D(E)$ is idempotent complete.


More generally, if $A$ is an idempotent complete triangulated category then $\mathbf{K}\_{-1}(A)$ is zero if and only if the Verdier quotient $B/A$ is idempotent complete for all full triangulated embeddings $A\to B$ with $B$ idempotent complete.


Incidentally, Schlichting conjectured that whenever $A$ is a small abelian category, then $\mathbf{K}\_{i}(A)=0$ for all $i<0$.


This conjecture was generalised to stable $\infty$-categories by Antieau, Gepner, and Heller in [K-theoretic obstructions to bounded t-structures.](https://link.springer.com/article/10.1007/s00222-018-00847-0) Invent. math. 216, 241–300 (2019), wherein they conjecture that $\mathbf{K}\_{i}(A)=0$ for $i<0$ whenever $A$ is a stable $\infty$-category with a bounded $t$-structure.


Neeman gave a very simple and elegant counter-example to both these conjectures in a recent preprint <https://arxiv.org/abs/2006.16536>.


