Fundamental theorem of algebraic K-theory

Template: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 $R[t]$ 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 $B^{+}=\mathrm{\Omega }BQ$ 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) $G_i(R[t])=G_i(R),i\ge 0$.
• (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

• Basic theorems in algebraic K-theory

References

• Daniel Grayson, Higher algebraic K-theory II [after Daniel Quillen], 1976
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