Fundamental theorem of algebraic K-theory

Short description: On the effects of changing the ring of K-groups

In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to [math]\displaystyle{ R[t] }[/math] or [math]\displaystyle{ R[t, t^{-1}] }[/math]. The theorem was first proved by Hyman Bass for [math]\displaystyle{ K_0, K_1 }[/math] and was later extended to higher K-groups by Daniel Quillen.

## Contents

* 1 Description
* 2 See also
* 3 Notes
* 4 References

## Description

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

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

## See also

* basic theorems in algebraic K-theory

## Notes

1. ↑ By definition, [math]\displaystyle{ K_i(R) = \pi_i(B^+\text{proj-Mod}_R), \, i \ge 0 }[/math].
2. ↑ Weibel 2013, Ch. V. Theorem 3.3 and Theorem 6.2

## 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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