Idempotent completions in K-theory

Question

I have a reference request on following comment I found in nLab article 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 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.

Answer 1

In Schlichting's paper Negative K-theory of derived categories. 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. 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.

Answer 2

Thomason discusses the construction in (A.9.1) in his paper with Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. I don't know how he uses it in the body of the paper.

I use it in an essential way to derive a result for all exact categories in Corollary 6.5 of my paper Algebraic $K$-theory via binary complexes, J. Amer. Math. Soc. 25 (2012), 1149–1167.