I have a reference request on following comment I found in Ncatlab article on Karoubian categories & envelops. It states:

$$\text{ "The Karoubian envelope is also used in the construction of the } \\ \text{ category of pure motives, and in K-theory." } $$

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 Karoubian envelope is involved, the quoted sentence refers.

The first naive 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 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 are there more general constuructions in $K$-theory where Karoubian envelopes are involved?

For example another Ncatlab article on more general development of $K$-theory nowhere explains where Karoubian envelope is explicitely used as technical tool for certain constructions. Therefore I would like to know if there are some recommendable papers on development of $K$-theory where such konstruction involving Karoubian envelopes are discussed.

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.

I have a reference request on following comment I found in Ncatlab article on Karoubian categories & envelops. It states:

$$\text{ "The Karoubian envelope is also used in the construction of the } \\ \text{ category of pure motives, and in K-theory." } $$

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 $(Sm/k)$ of smooth curves over $k$ to it's Karoubian closure

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

The first naive 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 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 are there more general constuructions in $K$-theory where Karoubian envelopes are involved?

For example another Ncatlab article on more general development of $K$-theory nowhere explains where Karoubian envelope is explicitely used as technical tool for certain constructions. Therefore I would like to know if there are some recommendable papers on development of $K$-theory where such konstruction involving Karoubian envelopes are discussed.

I have a reference request on following comment I found in Ncatlab article on Karoubian categories & envelops. It states:

$$\text{ "The Karoubian envelope is also used in the construction of the } \\ \text{ category of pure motives, and in K-theory." } $$

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 Karoubian envelope is involved, the quoted sentence refers.

The first naive 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 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 are there more general constuructions in $K$-theory where Karoubian envelopes are involved?

For example another Ncatlab article on more general development of $K$-theory nowhere explains where Karoubian envelope is explicitely used as technical tool for certain constructions. Therefore I would like to know if there are some recommendable papers on development of $K$-theory where such konstruction involving Karoubian envelopes are discussed.