Grothendieck ring
E884927
The Grothendieck ring is an algebraic structure formed from isomorphism classes of objects (such as varieties or modules), where addition comes from direct sum or disjoint union and multiplication from tensor product or Cartesian product, encoding their relations in a universal way.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure
ⓘ
ring ⓘ |
| additionInducedBy |
direct sum
ⓘ
disjoint union ⓘ |
| associatedWith | Grothendieck group completion process NERFINISHED ⓘ |
| assumes | isomorphism as notion of sameness of objects ⓘ |
| belongsTo |
abstract algebra
ⓘ
homological algebra ⓘ |
| captures |
additive invariants such as Euler characteristic
ⓘ
multiplicative invariants such as Hodge–Deligne polynomials ⓘ |
| constructedFrom | isomorphism classes of objects in a category ⓘ |
| context | categories with finite direct sums and tensor products ⓘ |
| definedByRelation | [X] = [Y] + [X \ Y] for suitable decompositions ⓘ |
| elementRepresents | formal difference or combination of isomorphism classes ⓘ |
| encodes | universal additive and multiplicative relations between isomorphism classes ⓘ |
| formalizes |
additivity of invariants under decompositions
ⓘ
multiplicativity of invariants under products ⓘ |
| generalizes | Grothendieck group by adding a compatible multiplication ⓘ |
| hasApplication |
character theory of representations
ⓘ
enumerative geometry ⓘ motivic integration ⓘ |
| hasConstructionStep |
define multiplication via monoidal product
GENERATED
ⓘ
mod out by relations expressing additivity GENERATED ⓘ take free abelian group on isomorphism classes GENERATED ⓘ |
| hasDefinition | ring constructed from isomorphism classes of objects with operations induced by sum and product ⓘ |
| hasExample |
Grothendieck ring of coherent sheaves
NERFINISHED
ⓘ
Grothendieck ring of motives NERFINISHED ⓘ Grothendieck ring of representations NERFINISHED ⓘ Grothendieck ring of varieties NERFINISHED ⓘ |
| hasOperation |
addition
ⓘ
multiplication ⓘ |
| hasProperty | functorial with respect to exact or compatible functors between categories ⓘ |
| hasUniversalProperty | initial ring receiving additive and multiplicative invariants of objects ⓘ |
| multiplicationInducedBy |
Cartesian product
ⓘ
tensor product ⓘ |
| namedAfter | Alexander Grothendieck NERFINISHED ⓘ |
| relatedTo | Grothendieck group NERFINISHED ⓘ |
| requires |
finite coproducts or direct sums
ⓘ
symmetric monoidal structure on the category ⓘ |
| toolFor | studying equivalence classes of geometric or algebraic objects ⓘ |
| usedIn |
algebraic K-theory
NERFINISHED
ⓘ
algebraic geometry ⓘ category theory ⓘ representation theory ⓘ |
| usedToCompare | different cohomological invariants ⓘ |
| usedToDefine | motivic measures ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.