Grothendieck–Lefschetz trace formula

E904007

mathematical theorem result in algebraic geometry

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.

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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 ⓘ

Referenced by (1)

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

Lefschetz fixed-point theorem → relatedTo → Grothendieck–Lefschetz trace formula ⓘ