Grothendieck–Lefschetz trace formula
E904007
The Grothendieck–Lefschetz trace formula is a fundamental result in algebraic geometry that expresses the number of rational points of a variety over a finite field in terms of traces of Frobenius acting on its étale cohomology groups.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Grothendieck–Lefschetz trace formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11085978 — 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–Lefschetz trace formula Context triple: [Lefschetz fixed-point theorem, relatedTo, Grothendieck–Lefschetz trace formula]
-
A.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
B.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
C.
Grothendieck–Riemann–Roch theorem
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.
-
D.
Serre’s cohomological methods in algebraic geometry
Serre’s cohomological methods in algebraic geometry are foundational techniques that use sheaf cohomology to relate and study algebraic and analytic geometry, profoundly influencing modern algebraic geometry and complex geometry.
-
E.
Standard Conjectures on Algebraic Cycles
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
- 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–Lefschetz trace formula Target entity description: The Grothendieck–Lefschetz trace formula is a fundamental result in algebraic geometry that expresses the number of rational points of a variety over a finite field in terms of traces of Frobenius acting on its étale cohomology groups.
-
A.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
B.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
C.
Grothendieck–Riemann–Roch theorem
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.
-
D.
Serre’s cohomological methods in algebraic geometry
Serre’s cohomological methods in algebraic geometry are foundational techniques that use sheaf cohomology to relate and study algebraic and analytic geometry, profoundly influencing modern algebraic geometry and complex geometry.
-
E.
Standard Conjectures on Algebraic Cycles
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in algebraic geometry ⓘ |
| appliesTo | variety over a finite field ⓘ |
| assumes |
continuous action of Frobenius on cohomology
ⓘ
variety of finite type over a finite field ⓘ |
| category | cohomological fixed-point theorem ⓘ |
| cohomologicalDegree | alternating sum over all i ≥ 0 ⓘ |
| context |
derived category of ℓ-adic sheaves
ⓘ
schemes over finite fields ⓘ |
| domain | finite fields ⓘ |
| expresses | number of F_q-rational points as an alternating sum of traces of Frobenius ⓘ |
| field |
algebraic geometry
ⓘ
arithmetic geometry ⓘ |
| framework | Grothendieck’s theory of étale cohomology ⓘ |
| generalizationOf |
Lefschetz fixed-point theorem
NERFINISHED
ⓘ
Lefschetz trace formula NERFINISHED ⓘ |
| hasVariant |
relative trace formula for morphisms
ⓘ
version for non-proper varieties using compact support ⓘ |
| holdsFor | smooth projective varieties over finite fields ⓘ |
| inspired | later trace formulas in arithmetic geometry ⓘ |
| involves |
Weil cohomology theory
NERFINISHED
ⓘ
compactly supported étale cohomology ⓘ ℓ-adic cohomology ⓘ |
| isPartOf | Grothendieck’s program for the Weil conjectures ⓘ |
| motivationFor | development of ℓ-adic cohomology ⓘ |
| namedAfter |
Alexander Grothendieck
NERFINISHED
ⓘ
Solomon Lefschetz NERFINISHED ⓘ |
| output | equality between point count and cohomological trace sum ⓘ |
| relatedTo |
Hasse–Weil zeta function
NERFINISHED
ⓘ
Weil conjectures on zeta functions of varieties NERFINISHED ⓘ |
| relates |
number of rational points
ⓘ
traces of Frobenius on étale cohomology ⓘ |
| requires |
finiteness of étale cohomology groups
ⓘ
trace class action of Frobenius on cohomology ⓘ |
| statedInTermsOf |
action of geometric Frobenius on cohomology
ⓘ
fixed points of Frobenius on the variety ⓘ |
| toolFor |
Weil conjectures
NERFINISHED
ⓘ
arithmetic applications of cohomology ⓘ counting points on varieties over finite fields ⓘ |
| type | cohomological trace formula ⓘ |
| usedIn |
proofs of rationality of zeta functions of varieties over finite fields
ⓘ
study of eigenvalues of Frobenius ⓘ |
| usesConcept |
Frobenius endomorphism
NERFINISHED
ⓘ
trace of an endomorphism ⓘ étale cohomology ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Grothendieck–Lefschetz trace formula Description of subject: The Grothendieck–Lefschetz trace formula is a fundamental result in algebraic geometry that expresses the number of rational points of a variety over a finite field in terms of traces of Frobenius acting on its étale cohomology groups.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.