K-theory K0 and projective modules Concept
K0 of a ring
Summary
The group of a ring is the Grothendieck group of finitely generated projective modules over under direct sum. It is the first group of algebraic K-theory. It measures how far finitely generated projective modules are from being free. For a field or a local ring it is , and for a Dedekind domain it is plus the ideal class group.
Where this leads
15 pages build on this one.
- K0 of a Dedekind domain
- Reduced K0
- Plus equals Q theorem
- Quillen's localization theorem
- Mayer–Vietoris sequence in K-theory
- Excision in algebraic K-theory
- Resolution theorem
- Negative K-theory
- K-groups of rings of integers
- Whitehead group of a free abelian group
- Lower K-groups
- Homotopy invariance of K-theory for regular rings
- Bass–Heller–Swan theorem
- Relative K-groups
- Chern–Connes character
Further reading
About this page
| Tag | Q14JS2MA1C |
|---|---|
| URI | m2://Q14JS2MA1C |
| Kind | concept |
| Writing | W0 stub |
| MSC | 19A49, 13D15, 16E20 |
| Specializes | Grothendieck group |
| Changed | 3 November 2026 |
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- K0 of a Dedekind domain
- Reduced K0
- Plus equals Q theorem
- Mayer–Vietoris sequence in K-theory
- Excision in algebraic K-theory
- Resolution theorem
- Negative K-theory
- K-groups of rings of integers
- Lower K-groups
- Bass–Heller–Swan theorem
- Relative K-groups
- Chern–Connes character