 
Summary: Adequate equivalence relations and Pontryagin
products
Reza Akhtar
Abstract
Let A be an abelian variety over a eld 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 ltration F 0 F 1 : : : on the Chow groups of varieties of the
form T k A (dened using Pontryagin products on A k A considered as an
Ascheme via projection on the rst factor) and prove that F r coincides with
the rfold product (F 1 ) r as adequate equivalence relations on the category of
all such varieties.
Keywords: algebraic cycles, Pontryagin product, adequate equivalence relation
AMS classication codes: 14C15, 14C25
1 Introduction
Let k be a eld and V k the category of smooth projective varieties over k. We open
with a wellknown conjecture attributed to Bloch and Beilinson:
Conjecture 1.1. For every object X of V k there exists a descending ltration F on
CH j (X; Q) = CH j
(X)
