# 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 Bass for and was later extended to higher K-groups by Quillen.

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 (i.e., has finite global dimension), then 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]}

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

## References

- ↑ By definition, .
- ↑ Template:Harvnb

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