Statements (29)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:appliesTo |
abelian categories
exact categories monoids |
| gptkbp:built |
Given a commutative monoid M, the Grothendieck group K(M) is constructed by formally adding inverses.
|
| gptkbp:category |
abelian group
|
| gptkbp:defines |
The Grothendieck group of a commutative monoid is the universal group into which the monoid embeds.
|
| gptkbp:example |
The Grothendieck group of the monoid of isomorphism classes of vector bundles over a space is the K-theory group.
The Grothendieck group of the natural numbers under addition is the group of integers. |
| gptkbp:field |
gptkb:algebra
|
| https://www.w3.org/2000/01/rdf-schema#label |
Grothendieck group
|
| gptkbp:introducedIn |
1950s
|
| gptkbp:namedAfter |
gptkb:Alexander_Grothendieck
|
| gptkbp:property |
If M is a commutative monoid, then K(M) is an abelian group.
|
| gptkbp:relatedTo |
gptkb:Grothendieck_ring
gptkb:K_0_functor |
| gptkbp:usedFor |
classifying objects up to stable isomorphism
defining algebraic K-theory |
| gptkbp:usedIn |
gptkb:algebraic_geometry
gptkb:K-theory gptkb:category_theory |
| gptkbp:bfsParent |
gptkb:algebraic_geometry
gptkb:commutative_algebra gptkb:K-theory gptkb:Algebraic_geometry gptkb:Group_theory gptkb:Alexander_Grothendieck gptkb:k-theory |
| gptkbp:bfsLayer |
5
|