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

TL;DR

This paper generalizes Grothendieck's theory of blended extensions to filtrations with multiple steps, applying it to classify motives with maximal unipotent radicals and fixed associated graded structures.

Contribution

It introduces a new framework for blended extensions with multiple filtration steps and applies it to classify certain mixed motives and their Galois groups.

Findings

Homological classification of motives with graded-independent associated graded

Extension of blended extension theory to arbitrary finite filtrations

Application to motives with maximal unipotent radicals

Abstract

Grothendieck's theory of blended extensions (extensions panach\'ees) gives 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 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 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 earlier with K. Murty under some extra hypotheses.