Fundamental theorem of algebraic K-theory

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In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to or . The theorem was first proved by Bass for and was later extended to higher K-groups by Quillen.

Let be the algebraic K-theory of the category of finitely generated modules over a noetherian ring R; explicitly, we can take , where is given by Quillen's Q-construction. If R is a regular ring (i.e., has finite global dimension), then the i-th K-group of R.[1] This is an immediate consequence of the resolution theorem, which compares the K-theories of two different categories (with inclusion relation.)

For a noetherian ring R, the fundamental theorem states:[2]

  • (i) .
  • (ii) .

The proof of the theorem uses the Q-construction. There is also a version of the theorem for the singular case (for ); this is the version proved in Grayson's paper.

References[edit]

  1. ^ By definition, .
  2. ^ Weibel 2013, Ch. V. Theorem 3.3 and Theorem 6.2