Fundamental Theorem Of Algebraic K-theory

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 /math> or R , t^/math>. 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 $G_i(R) = \pi_i(B^+\text_R)$, 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''. 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 = G_i(R), \, i \ge 0.
*(ii) G_i(R , t^ = G_i(R) \oplus G_(R), \, i \ge 0, \, G_(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

Notes

References

*Daniel Grayson
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Algebraic K-theory
Theorems in algebraic topology

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