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 ${\displaystyle R[t]}$ or ${\displaystyle R[t,t^{-1}]}$. The theorem was first proved by Hyman Bass for ${\displaystyle K_{0},K_{1}}$ and was later extended to higher K-groups by Daniel Quillen.

Description

Let ${\displaystyle G_{i}(R)}$ be the algebraic K-theory of the category of finitely generated modules over a noetherian ring R; explicitly, we can take ${\displaystyle G_{i}(R)=\pi _{i}(B^{+}{\text{f-gen-Mod}}_{R})}$, where ${\displaystyle B^{+}=\Omega BQ}$ is given by Quillen's Q-construction. If R is a regular ring (i.e., has finite global dimension), then ${\displaystyle G_{i}(R)=K_{i}(R),}$ 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) ${\displaystyle G_{i}(R[t])=G_{i}(R),\,i\geq 0}$.
• (ii) ${\displaystyle G_{i}(R[t,t^{-1}])=G_{i}(R)\oplus G_{i-1}(R),\,i\geq 0,\,G_{-1}(R)=0}$.

The proof of the theorem uses the Q-construction. There is also a version of the theorem for the singular case (for ${\displaystyle K_{i}}$); this is the version proved in Grayson's paper.

See also

• basic theorems in algebraic K-theory

Notes

1. By definition, ${\displaystyle K_{i}(R)=\pi _{i}(B^{+}{\text{proj-Mod}}_{R}),\,i\geq 0}$.
2. Weibel 2013, Ch. V. Theorem 3.3 and Theorem 6.2 harvnb error: no target: CITEREFWeibel2013 (help)

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"