On blended extensions in filtered abelian categories and motives with maximal unipotent radicals
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916801218084864 |
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| author | Eskandari, Payman |
| author_facet | Eskandari, Payman |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_15487 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On blended extensions in filtered abelian categories and motives with maximal unipotent radicals Eskandari, Payman Algebraic Geometry K-Theory and Homology Number Theory Representation Theory 14F42, 18M25 (primary), 11F67, 11M32, 14G10 (secondary) 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. |
| title | On blended extensions in filtered abelian categories and motives with maximal unipotent radicals |
| topic | Algebraic Geometry K-Theory and Homology Number Theory Representation Theory 14F42, 18M25 (primary), 11F67, 11M32, 14G10 (secondary) |
| url | https://arxiv.org/abs/2307.15487 |