Av-theory of categories

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Quillen has constructed a A-theory K*C for nice categories, one of which is the category of projective A-modules. We construct a theory KV* C for the nice categories parametrized by rings. When applied to projective modules we recover the Karoubi-Villamayor A-theory KV*(R). As an application, we show that the Cartan map from K*(R) to G*(R) factors through the groups KVJ*(R). We also compute KV* for the categories of faithful projectives and Azumaya algebras, generalizing results of Bass.

Original languageEnglish (US)
Pages (from-to)365-621
Number of pages257
JournalTransactions of the American Mathematical Society
Issue number2
StatePublished - 1981
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Mathematics(all)


  • Exact category
  • Infinite loop space
  • Karoubi-Villamayor A-theory
  • Symmetric monoidal category


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