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 $$R[t]$$ 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. 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) $$G_i(R[t]) = G_i(R), \, i \ge 0$$.
 * (ii) $$G_i(R[t, t^{-1}]) = G_i(R) \oplus G_{i-1}(R), \, i \ge 0, \, G_{-1}(R) = 0$$.

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.