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. 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) $G_{i}(R[t])=G_{i}(R),\,i\geq 0$.
• (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.

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"