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

↑ By definition, $K_{i}(R)=\pi _{i}(B^{+}{\text{proj-Mod}}_{R}),\,i\geq 0$ .
↑ 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"

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.