Grothendieck group

E254131

abelian group algebraic construction group completion

The Grothendieck group is an algebraic construction that formally turns a commutative monoid (often arising from isomorphism classes of objects like vector bundles or modules) into an abelian group, playing a central role in K-theory and modern algebraic geometry.

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All labels observed (5)

Label	Occurrences
Grothendieck group canonical	2
Grothendieck K0-group	1
Grothendieck group of coherent sheaves	1
Grothendieck group of varieties	1
Grothendieck group of vector bundles	1

Statements (52)

Predicate	Object
instanceOf	abelian group ⓘ algebraic construction ⓘ group completion ⓘ
alsoKnownAs	group completion of a commutative monoid ⓘ
appliedTo	isomorphism classes of coherent sheaves ⓘ isomorphism classes of modules ⓘ isomorphism classes of objects in an abelian category ⓘ isomorphism classes of objects in an exact category ⓘ isomorphism classes of representations ⓘ isomorphism classes of vector bundles ⓘ
constructionMethod	formal differences of monoid elements ⓘ quotient of free abelian group on the monoid ⓘ
definesElement	formal difference [a] - [b] of monoid elements ⓘ
generalizes	construction of K0 in K-theory ⓘ construction of Z from N ⓘ difference of integers from natural numbers ⓘ
hasCanonicalMapFrom	underlying commutative monoid ⓘ
hasCanonicalMapType	monoid homomorphism ⓘ
hasDomain	K-theory ⓘ algebra ⓘ algebraic geometry ⓘ category theory ⓘ
hasHistoricalContext	introduced in the development of Grothendieck’s K-theory ⓘ
hasKeyOperation	addition induced by monoid operation ⓘ
hasKeyRelation	[a] + [b] = [a + b] in the group ⓘ
hasProperty	universal for monoid homomorphisms into abelian groups ⓘ
inputStructure	commutative monoid ⓘ
mathematicsSubjectClassification	14-XX algebraic geometry ⓘ 18-XX category theory ⓘ 19-XX K-theory ⓘ
namedAfter	Alexander Grothendieck ⓘ
outputStructure	abelian group ⓘ
relatedConcept	Grothendieck group self-linksurface differs ⓘ surface form: Grothendieck K0-group Grothendieck group of a category ⓘ Grothendieck group of a scheme ⓘ Grothendieck group of an abelian category ⓘ Grothendieck group of an exact category ⓘ Grothendieck group self-linksurface differs ⓘ surface form: Grothendieck group of coherent sheaves Grothendieck group self-linksurface differs ⓘ surface form: Grothendieck group of vector bundles Grothendieck ring ⓘ
satisfies	universal property of group completion ⓘ
universalProperty	every monoid homomorphism to an abelian group factors uniquely through it ⓘ
usedIn	K-theory ⓘ surface form: algebraic K-theory algebraic geometry ⓘ homological algebra ⓘ number theory ⓘ operator algebras ⓘ representation theory ⓘ K-theory ⓘ surface form: topological K-theory
usedToDefine	Grothendieck group self-linksurface differs ⓘ surface form: Grothendieck group of varieties K0 of a ring ⓘ K0 of a scheme ⓘ

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Alexander Grothendieck → notableConcept → Grothendieck group ⓘ

Grothendieck group → relatedConcept → Grothendieck group self-linksurface differs ⓘ

this entity surface form: Grothendieck K0-group

Grothendieck group → relatedConcept → Grothendieck group self-linksurface differs ⓘ

this entity surface form: Grothendieck group of vector bundles

Grothendieck group → relatedConcept → Grothendieck group self-linksurface differs ⓘ

this entity surface form: Grothendieck group of coherent sheaves

Grothendieck group → usedToDefine → Grothendieck group self-linksurface differs ⓘ

this entity surface form: Grothendieck group of varieties

K-theory → relatedTo → Grothendieck group ⓘ