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 formula_1 or formula_2. The theorem was first proved by Hyman Bass for formula_3 and was later extended to higher "K"-groups by Daniel Quillen.
Description.
Let formula_4 be the algebraic K-theory of the category of finitely generated modules over a noetherian ring "R"; explicitly, we can take formula_5, where formula_6 is given by Quillen's Q-construction. If "R" is a regular ring (i.e., has finite global dimension), then formula_7 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:
The proof of the theorem uses the Q-construction. There is also a version of the theorem for the singular case (for formula_10); this is the version proved in Grayson's paper.