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 Bass for K 0 , K 1 and was later extended to higher K-groups by Quillen.
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 ) = π i ( B + f-gen-Mod R ) , where B + = Ω B Q 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. 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 .(ii) G i ( R [ t , t − 1 ] ) = G i ( R ) ⊕ G i − 1 ( R ) , i ≥ 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.
