Author: IngoD Format: TextI have a question about a statement in [[Karoubian category]] (4. Examples):
The Karoubian envelope is also used in the construction of the category of pure motives,
and in K-theory.
Although there is a lot of online notes/courses available where is precisely explaned how taking Karoubian envelope
is involved in the construction of (pure) motives, there seems to be a serious lack of sources where is explained how
the Karoubian envolope is involved in construtions in K-theory.
(appart from the 'basic' construction of algebraic K-theory K_0 (A) for a ring A as K_0 (P_A), where P_A
is the category of finitely generated A-modules, where P_A can also be recognized as Karoubi completion of the
category F_A of finite generated free A-modules.
Nevertheless this construction of algebraic K_0 might be considered as a 'toy' example.
Is there in the quoted sentence above also referred to certain constructions in K-theory in more general setting (eg for K-groups of exact or Waldhausen-categories)
which make use of the Karoubian completion?
I have a question about a statement in Karoubian category (4. Examples):
The Karoubian envelope is also used in the construction of the category of pure motives, and in K-theory.
Although there is a lot of online notes/courses available where is precisely explaned how taking Karoubian envelope is involved in the construction of (pure) motives, there seems to be a serious lack of sources where is explained how the Karoubian envolope is involved in construtions in K-theory. (appart from the 'basic' construction of algebraic K-theory K_0 (A) for a ring A as K_0 (P_A), where P_A is the category of finitely generated A-modules, where P_A can also be recognized as Karoubi completion of the category F_A of finite generated free A-modules.
Nevertheless this construction of algebraic K_0 might be considered as a 'toy' example. Is there in the quoted sentence above also referred to certain constructions in K-theory in more general setting (eg for K-groups of exact or Waldhausen-categories) which make use of the Karoubian completion?