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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 Hyman Bass for and was later extended to higher K-groups by Daniel Quillen.

Description

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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.

See also

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Notes

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

References

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  • Daniel Grayson, Higher algebraic K-theory II [after Daniel Quillen], 1976
  • Srinivas, V. (2008), Algebraic K-theory, Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.), Boston, MA: Birkhäuser, ISBN 978-0-8176-4736-0, Zbl 1125.19300
  • Weibel, Charles (2013). "The K-book: An introduction to algebraic K-theory". Graduate Studies in Math. 145.