The equivalence of several conjectures on independence of ℓ

Remy van Dobben de Bruyn

doi:10.46298/EPIGA.2020.VOLUME4.5570

The equivalence of several conjectures on independence of ℓ

Remy van Dobben de Bruyn

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We consider several conjectures on the independence of ` of the étale cohomology of (singular, open) varieties over F^¯_p. The main result is that independence of ` of the Betti numbers hⁱ_c(X,Q_`) for arbitrary varieties is equivalent to independence of ` of homological equivalence ∼hom_,` for cycles on smooth projective varieties. We give several other equivalent statements. As a surprising consequence, we prove that independence of ` of Betti numbers for smooth quasi-projective varieties implies the same result for arbitrary separated finite type k-schemes.

Original language	English (US)
Article number	16
Journal	Epijournal de Geometrie Algebrique
Volume	4
DOIs	https://doi.org/10.46298/EPIGA.2020.VOLUME4.5570
State	Published - Nov 30 2020

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Geometry and Topology

Keywords

Algebraic geometry
Arithmetic geometry
Independence of `
Mathematics
Motives
Positive characteristic
étale cohomology

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10.46298/EPIGA.2020.VOLUME4.5570

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abstract = "We consider several conjectures on the independence of ` of the {\'e}tale cohomology of (singular, open) varieties over F¯p. The main result is that independence of ` of the Betti numbers hic(X,Q`) for arbitrary varieties is equivalent to independence of ` of homological equivalence ∼hom,` for cycles on smooth projective varieties. We give several other equivalent statements. As a surprising consequence, we prove that independence of ` of Betti numbers for smooth quasi-projective varieties implies the same result for arbitrary separated finite type k-schemes.",

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N2 - We consider several conjectures on the independence of ` of the étale cohomology of (singular, open) varieties over F¯p. The main result is that independence of ` of the Betti numbers hic(X,Q`) for arbitrary varieties is equivalent to independence of ` of homological equivalence ∼hom,` for cycles on smooth projective varieties. We give several other equivalent statements. As a surprising consequence, we prove that independence of ` of Betti numbers for smooth quasi-projective varieties implies the same result for arbitrary separated finite type k-schemes.

AB - We consider several conjectures on the independence of ` of the étale cohomology of (singular, open) varieties over F¯p. The main result is that independence of ` of the Betti numbers hic(X,Q`) for arbitrary varieties is equivalent to independence of ` of homological equivalence ∼hom,` for cycles on smooth projective varieties. We give several other equivalent statements. As a surprising consequence, we prove that independence of ` of Betti numbers for smooth quasi-projective varieties implies the same result for arbitrary separated finite type k-schemes.

KW - Algebraic geometry

KW - Arithmetic geometry

KW - Independence of `

KW - Mathematics

KW - Motives

KW - Positive characteristic

KW - étale cohomology

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JF - Epijournal de Geometrie Algebrique

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ER -

The equivalence of several conjectures on independence of ℓ

Abstract

All Science Journal Classification (ASJC) codes

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