Tannakian fundamental groups of blended extensions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909258874880000 |
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| author | Eskandari, Payman |
| author_facet | Eskandari, Payman |
| contents | Let $A_1, A_2,A_3$ be semisimple objects in a neutral tannakian category over a field of characteristic zero. Let $L$ be an extension of $A_2$ by $A_1$, and $N$ an extension of $A_3$ by $A_2$. Let $M$ be a blended extension (extension panachée) of $N$ by $L$. Under very mild and natural hypotheses, we study the unipotent radical of the tannakian fundamental group of $M$. Examples where our results apply include the unipotent radicals of motivic Galois groups of any mixed motive with three weights.
As an application, we give a proof of the unipotent part of the Hodge-Nori conjecture for 1-motives (which is now a theorem of André in the setting of Nori motives) in the setting of any tannakian category of motives where the group $Ext^1(1,\mathbb{Q}(1))$ is as expected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01379 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tannakian fundamental groups of blended extensions Eskandari, Payman Algebraic Geometry Number Theory 18M25, 14Fxx, 14C30, 19E15, 11Gxx Let $A_1, A_2,A_3$ be semisimple objects in a neutral tannakian category over a field of characteristic zero. Let $L$ be an extension of $A_2$ by $A_1$, and $N$ an extension of $A_3$ by $A_2$. Let $M$ be a blended extension (extension panachée) of $N$ by $L$. Under very mild and natural hypotheses, we study the unipotent radical of the tannakian fundamental group of $M$. Examples where our results apply include the unipotent radicals of motivic Galois groups of any mixed motive with three weights. As an application, we give a proof of the unipotent part of the Hodge-Nori conjecture for 1-motives (which is now a theorem of André in the setting of Nori motives) in the setting of any tannakian category of motives where the group $Ext^1(1,\mathbb{Q}(1))$ is as expected. |
| title | Tannakian fundamental groups of blended extensions |
| topic | Algebraic Geometry Number Theory 18M25, 14Fxx, 14C30, 19E15, 11Gxx |
| url | https://arxiv.org/abs/2407.01379 |