2023/07/28 by Eskandari, Payman · 2 citations
#11F67 #11M32 #14F42 #14G10 (secondary) #18M25 (primary) #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2307.15487
Grothendieck's theory of blended extensions (extensions panachées) 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.