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 ${\displaystyle B^{+}=\mathrm{\Omega }BQ}$ 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) ${\displaystyle G_i(R[t])=G_i(R),i\ge 0}$.
• (ii) ${\displaystyle G_i(R[t,t^{-1}])=G_i(R)\oplus G_{i-1}(R),i\ge 0,G_{-1}(R)=0}$.

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

See also

• basic theorems in algebraic K-theory

Notes

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 
• Weibel, Charles (2013). "The K-book: An introduction to algebraic K-theory". Graduate Studies in Math 145. http://www.math.rutgers.edu/~weibel/Kbook.html. 

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