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 R[t] or R[t,t−1]. The theorem was first proved by Hyman Bass for K0,K1 and was later extended to higher K-groups by Daniel Quillen.

## Contents

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

## Description

Let Gi(R) be the algebraic K-theory of the category of finitely generated modules over a noetherian ring R; explicitly, we can take Gi(R)=πi(B+f-gen-ModR), where B+=ΩBQ is given by Quillen's Q-construction. If R is a regular ring (i.e., has finite global dimension), then Gi(R)=Ki(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) Gi(R[t])=Gi(R),i≥0.
* (ii) Gi(R[t,t−1])=Gi(R)⊕Gi−1(R),i≥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 Ki); this is the version proved in Grayson's paper.

## See also

* Basic theorems in algebraic K-theory

## Notes

1. ↑ By definition, Ki(R)=πi(B+proj-ModR),i≥0.
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. Graduate Studies in Mathematics 145. doi:10.1090/gsm/145. ISBN 978-0-8218-9132-2. https://www.math.rutgers.edu/~weibel/Kbook.html.

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