In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to R[t] or \( R[t,t^{{-1}}] \). The theorem was first proved by Hyman Bass for \( K_{0},K_{1} \) and was later extended to higher K-groups by Daniel Quillen.
Description
Let \(G_{i}(R) \) be the algebraic K-theory of the category of finitely generated modules over a noetherian ring R; explicitly, we can take \( G_{i}(R)=\pi _{i}(B^{+}{\text{f-gen-Mod}}_{R}) \), where \( B^{+}=\Omega BQ \) is given by Quillen's Q-construction. If R is a regular ring (i.e., has finite global dimension), then \( G_{i}(R)=K_{i}(R) \) , 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) \( G_{i}(R[t])=G_{i}(R),\,i\geq 0.
(ii) \( G_{i}(R[t,t^{{-1}}])=G_{i}(R)\oplus G_{{i-1}}(R),\,i\geq 0,\,G_{{-1}}(R)=0. \)
The proof of the theorem uses the Q-construction. There is also a version of the theorem for the singular case (for \( K_{i} \)); this is the version proved in Grayson's paper.
See also
basic theorems in algebraic K-theory
Notes
By definition, \( K_{i}(R)=\pi _{i}(B^{+}{\text{proj-Mod}}_{R}),\,i\geq 0 \) .
Weibel 2013, Ch. V. Theorem 3.3 and Theorem 6.2
References
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"
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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