Fundamental theorem of algebraic K-theory

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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 Bass for K_0, K_1 and was later extended to higher K-groups by Quillen.

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.[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) 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.


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