Fundamental theorem of algebraic K-theory explained

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 ^{+ 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:

• (i)

G _{i(R[t])} = G _{i(R),} i \ge 0 .

• (ii)

G _{i(R[t,} t ^{-1} ]) = G _{i(R)} ⊕ 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

• Basic theorems in algebraic K-theory

References

• Daniel Grayson, Higher algebraic K-theory II [after Daniel Quillen], 1976
  • Charles . Weibel . The K-book: An introduction to algebraic K-theory . Graduate Studies in Math . 145 . 2013.

Notes and References

1. By definition, K _{i(R)} = \pi ^{+ proj-Mod} _{R),} i \ge 0 .