On blended extensions in filtered abelian categories and motives with maximal unipotent radicals

Abstract

Grothendieck’s theory of blended extensions (extensions panachées) provides a natural framework to study 3-step filtrations in abelian categories. We give a generalization of this theory that is suitable for filtrations with an arbitrary finite number of steps. We use this generalization to study two natural classification problems for objects with a fixed associated graded in an abelian category equipped with a filtration similar to the weight filtration on rational mixed Hodge structures. We then give an application to the study of mixed motives with a given associated graded and maximal unipotent radicals of motivic Galois groups. We prove a homological classification result for the isomorphism classes of such motives when the given associated graded is “graded-independent”, a condition defined in the paper. The special case of this result for motives with 3 weights was proved with K. Murty in [Algebra Number Theory 17 (2023), no. 1, 165–215] under some extra hypotheses.