Adequate equivalence relations and Pontryagin products

Abstract

Let A be an abelian variety over a field k. We consider CH 0 ( A ) as a ring under Pontryagin product and relate powers of the ideal I ⊆ CH 0 ( A ) of degree zero elements to powers of the algebraic equivalence relation. We also consider a filtration F 0 ⊇ F 1 ⊇ … on the Chow groups of varieties of the form T × k A (defined using Pontryagin products on A × k A considered as an A-scheme via projection on the first factor) and prove that F r coincides with the r-fold product ( F 1 ) * r as adequate equivalence relations on the category of all such varieties.