Fundamental theorem of algebraic K-theory
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. This is an immediate consequence of the resolution theorem, which compares the K-theories of two different categories (with inclusion relation.)
- (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.
- By definition, .
- Weibel 2013, Ch. V. Theorem 3.3 and Theorem 6.2
- 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
- C. Weibel "The K-book: An introduction to algebraic K-theory"
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