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 or. The theorem was first proved by Hyman Bass for and was later extended to higher K-groups by Daniel Quillen.

Description

Let be the algebraic K-theory of the category of finitely generated modules over a noetherian ring R; explicitly, we can take, where is given by Quillen's Q-construction. If R is a regular ring, then 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.
For a noetherian ring R, the fundamental theorem states:

The proof of the theorem uses the Q-construction. There is also a version of the theorem for the singular case ; this is the version proved in Grayson's paper.