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We discuss the relationship between different notions of "integrality" inmotivic cohomology/K-theory which arise in the Beilinson and Bloch-Katoconjectures, and prove their equivalence in some cases for products of curves(used in the authors' previous paper in this series), as well as obtaining ageneral result, first proved by Jannsen (unpublished), which reduces theirequivalence to standard conjectures in arithmetic algebraic geometry.

Integral elements of K-theory and products of modular curves II