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Fundamental theorem of algebraic K-theory

{{DISPLAYTITLE: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 K_0, K_1 and was later extended to higher K-groups by Daniel Quillen.

Description

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.By definition, K_i(R) = \pi_i(B^+\text{proj-Mod}_R), \, i \ge 0. 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:{{harvnb|Weibel|2013|loc=Ch. V. Theorem 3.3 and Theorem 6.2}}

  • (i) G_i(R[t]) = G_i(R), \, i \ge 0.
  • (ii) 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 K_i); this is the version proved in Grayson's paper.

See also

Notes

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References

  • Daniel Grayson, [http://www.math.uiuc.edu/~dan/Papers/HigherAlgKThyII.pdf Higher algebraic K-theory II [after Daniel Quillen]], 1976
  • {{citation | last=Srinivas | first=V. | title=Algebraic K-theory | edition=Paperback reprint of the 1996 2nd | series=Modern Birkhäuser Classics | location=Boston, MA | publisher=Birkhäuser | year=2008 | isbn=978-0-8176-4736-0 | zbl=1125.19300 }}
  • {{cite journal |first=Charles |last=Weibel |url=http://www.math.rutgers.edu/~weibel/Kbook.html |title=The K-book: An introduction to algebraic K-theory |journal=Graduate Studies in Math |series=Graduate Studies in Mathematics |volume=145 |year=2013|doi=10.1090/gsm/145 |isbn=978-0-8218-9132-2 |url-access=subscription }}

Category:Algebraic K-theory

Category:Theorems in algebraic topology

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