Английская Википедия:Fundamental theorem of algebraic K-theory

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Шаблон:Short description 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>R[t]</math> or <math>R[t, t^{-1}]</math>. The theorem was first proved by Hyman Bass for <math>K_0, K_1</math> and was later extended to higher K-groups by Daniel Quillen.

Description

Let <math>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>G_i(R) = \pi_i(B^+\text{f-gen-Mod}_R)</math>, where <math>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>G_i(R) = K_i(R),</math> the i-th K-group of R. 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) <math>G_i(R[t]) = G_i(R), \, i \ge 0</math>.
• (ii) <math>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>K_i</math>); 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
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