The equivalence of several conjectures on independence of ℓ

Remy van Dobben de Bruyn

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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.

Original languageEnglish (US)
Article number16
JournalEpijournal de Geometrie Algebrique
StatePublished - Nov 30 2020

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Geometry and Topology


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


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