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 ] {\displaystyle R[t]} or R [ t , t − 1 ] {\displaystyle R[t,t^{-1}]} . The theorem was first proved by Hyman Bass for K 0 , K 1 {\displaystyle K_{0},K_{1}} and was later extended to higher K-groups by Daniel Quillen.
Description Let G i ( R ) {\displaystyle 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 ) = π i ( B + f-gen-Mod R ) {\displaystyle G_{i}(R)=\pi _{i}(B^{+}{\text{f-gen-Mod}}_{R})} , where B + = Ω B Q {\displaystyle 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 ) , {\displaystyle G_{i}(R)=K_{i}(R),} 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). For a noetherian ring R, the fundamental theorem states:
(i) G i ( R [ t ] ) = G i ( R ) , i ≥ 0 {\displaystyle G_{i}(R[t])=G_{i}(R),\,i\geq 0} . (ii) G i ( R [ t , t − 1 ] ) = G i ( R ) ⊕ G i − 1 ( R ) , i ≥ 0 , G − 1 ( R ) = 0 {\displaystyle 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 {\displaystyle K_{i}} ); this is the version proved in Grayson's paper.
See also Basic theorems in algebraic K-theory
Notes
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 Weibel, Charles (2013). "The K-book: An introduction to algebraic K-theory". Graduate Studies in Math. Graduate Studies in Mathematics. 145. doi:10.1090/gsm/145. ISBN 978-0-8218-9132-2.{{cite journal}}: CS1 maint: periodical has ISBN (link)