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 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 $G_i(R) = \pi_i(B^+\text{f-gen-Mod}_R)$, 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) $G_i(R[t]) = G_i(R), \, i \ge 0$.
• (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.

References

1. By definition, $K_i(R) = \pi_i(B^+\text{proj-Mod}_R), \, i \ge 0$.
2. 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"