# 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 ${\displaystyle R[t]}$ or ${\displaystyle R[t,t^{-1}]}$. The theorem was first proved by Bass for ${\displaystyle K_{0},K_{1}}$ and was later extended to higher K-groups by Quillen.

Let ${\displaystyle G_{i}(R)}$ be the algebraic K-theory of the category of finitely generated modules over a noetherian ring R; explicitly, we can take ${\displaystyle G_{i}(R)=\pi _{i}(B^{+}{\text{f-gen-Mod}}_{R})}$, where ${\displaystyle B^{+}=\Omega BQ}$ is given by Quillen's Q-construction. If R is a regular ring (i.e., has finite global dimension), then ${\displaystyle G_{i}(R)=K_{i}(R),}$ 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) ${\displaystyle G_{i}(R[t])=G_{i}(R),\,i\geq 0}$.
• (ii) ${\displaystyle G_{i}(R[t,t^{-1}])=G_{i}(R)\oplus G_{i-1}(R),\,i\geq 0,\,G_{-1}(R)=0}$.

The proof of the theorem uses the Q-construction. There is also a version of the theorem for the singular case (for ${\displaystyle K_{i}}$); this is the version proved in Grayson's paper.

## References

1. ^ By definition, ${\displaystyle K_{i}(R)=\pi _{i}(B^{+}{\text{proj-Mod}}_{R}),\,i\geq 0}$.
2. ^ Weibel 2013, Ch. V. Theorem 3.3 and Theorem 6.2