# Idempotent completions in K-theory

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.

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.

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.