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

How this entity was disambiguated

This entity first appeared as the object of triple T2290622 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Grothendieck–Riemann–Roch theorem
Context triple: [Alexander Grothendieck, knownFor, Grothendieck–Riemann–Roch theorem]

A. Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
B. Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
C. Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
D. Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
E. Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Grothendieck–Riemann–Roch theorem
Target entity description: 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.

A. Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
B. Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
C. Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
D. Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
E. Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
F. None of above. chosen

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 ⓘ

How these facts were elicited

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 ⓘ

All labels observed (3)

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Statements (43)

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Referenced by (11)

How this entity was disambiguated

How these facts were elicited