Statements (28)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:application |
gptkb:C*-algebra_K-theory
motivic cohomology classification of vector bundles topological K-theory classification of projective modules |
| gptkbp:defines |
Grothendieck group of isomorphism classes of objects in an exact category
group completion of the monoid of isomorphism classes of finitely generated projective modules over a ring |
| gptkbp:example |
K_0(Z) = Z
K_0(field) = Z K_0 of a finite-dimensional semisimple algebra is free abelian of rank equal to the number of simple modules |
| gptkbp:field |
gptkb:algebraic_K-theory
|
| https://www.w3.org/2000/01/rdf-schema#label |
K 0 group
|
| gptkbp:introduced |
gptkb:Alexander_Grothendieck
|
| gptkbp:notation |
K_0(R) for a ring R
K_0(X) for a scheme X |
| gptkbp:property |
abelian group
functorial in the ring or category universal property for additive invariants |
| gptkbp:relatedTo |
gptkb:K_1_group
gptkb:Grothendieck_group projective module exact category |
| gptkbp:usedIn |
gptkb:algebraic_geometry
gptkb:topology operator algebras |
| gptkbp:bfsParent |
gptkb:K-theory
|
| gptkbp:bfsLayer |
5
|