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    • CommentRowNumber1.
    • CommentAuthorIngoD
    • CommentTimeSep 10th 2021
    • (edited Sep 10th 2021)
    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?