Grothendieck–Riemann–Roch theorem

E254119

Riemann–Roch type theorem theorem in algebraic geometry

The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.

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All labels observed (3)

Label	Occurrences
Grothendieck–Riemann–Roch theorem canonical	9
Grothendieck–Riemann–Roch integrand	1
Riemann–Roch theorem for higher-dimensional varieties	1

Statements (43)

Predicate	Object
instanceOf	Riemann–Roch type theorem ⓘ theorem in algebraic geometry ⓘ
appearsIn	FGA (Fondements de la géométrie algébrique) ⓘ Éléments de géométrie algébrique ⓘ
appliesTo	proper morphisms of schemes of finite type ⓘ proper morphisms of smooth varieties ⓘ
domain	schemes ⓘ varieties ⓘ
expresses	compatibility of Chern character with pushforward ⓘ equality between K-theoretic and cohomological pushforwards up to Todd class ⓘ
field	algebraic geometry ⓘ
generalizes	Hirzebruch–Riemann–Roch theorem ⓘ Riemann–Roch theorem ⓘ
hasConsequence	Hirzebruch–Riemann–Roch theorem ⓘ surface form: Hirzebruch–Riemann–Roch for smooth projective varieties Lefschetz–Riemann–Roch type formulas ⓘ Riemann–Roch theorem ⓘ surface form: Riemann–Roch for curves
hasFormulation	Grothendieck’s original formulation in the language of schemes ⓘ formulation using Chow groups ⓘ formulation using algebraic K-theory ⓘ
historicalPeriod	mid 20th century ⓘ
involvesConcept	Chern character ⓘ Todd class ⓘ characteristic classes ⓘ coherent sheaf ⓘ proper morphism of schemes ⓘ pushforward in K-theory ⓘ pushforward in cohomology ⓘ vector bundle ⓘ
mathematicsSubjectClassification	14C40 ⓘ 19E08 ⓘ
namedAfter	Alexander Grothendieck ⓘ
provedBy	Alexander Grothendieck ⓘ
relatedTo	Atiyah–Singer index theorem ⓘ Hirzebruch–Riemann–Roch theorem ⓘ
relates	K-theory ⓘ surface form: algebraic K-theory cohomology ⓘ
requires	Chern classes ⓘ Chow groups or cycle classes ⓘ cohomology theory ⓘ
usedIn	enumerative geometry ⓘ index theorems in algebraic geometry ⓘ intersection theory ⓘ study of moduli spaces ⓘ

Referenced by (11)

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

Alexander Grothendieck → knownFor → Grothendieck–Riemann–Roch theorem ⓘ

Riemann–Roch theorem → generalizedBy → Grothendieck–Riemann–Roch theorem ⓘ

this entity surface form: Riemann–Roch theorem for higher-dimensional varieties

Chern classes → usedFor → Grothendieck–Riemann–Roch theorem ⓘ

SGA → mainTopic → Grothendieck–Riemann–Roch theorem ⓘ

subject surface form: SGA 6

Hirzebruch–Riemann–Roch theorem → inspired → Grothendieck–Riemann–Roch theorem ⓘ

Hirzebruch–Riemann–Roch theorem → relatedTo → Grothendieck–Riemann–Roch theorem ⓘ

Chern character → usedIn → Grothendieck–Riemann–Roch theorem ⓘ

Todd class → roleIn → Grothendieck–Riemann–Roch theorem ⓘ

Todd class → appearsAsFactorIn → Grothendieck–Riemann–Roch theorem ⓘ

this entity surface form: Grothendieck–Riemann–Roch integrand

families index theorem → relatedTo → Grothendieck–Riemann–Roch theorem ⓘ