A remark on $\operatorname{Ext}$ groups for motives with maximal unipotent radicals

Payman Eskandari

doi:10.5802/crmath.817

Article de recherche - Géométrie algébrique, Théorie des nombres

A remark on $\operatorname{Ext}$ groups for motives with maximal unipotent radicals
[Remarque sur les groupes $\operatorname{Ext}$ pour les motifs avec des radicaux unipotents maximaux]

Payman Eskandari ¹

¹Department of Mathematics and Statistics, University of Winnipeg, Winnipeg MB, Canada

Comptes Rendus. Mathématique, Volume 364 (2026), pp. 127-136

Abstract (VO)
Résumé

Let $\mathbf{T}$ be a neutral tannakian category over a field of characteristic 0. Let $M$ be an object of $\mathbf{T}$ with a filtration $0=F_0M\subsetneq F_1M\subsetneq \dots \subsetneq F_kM=M$, such that each successive quotient $F_iM/F_{i-1}M$ is semisimple. Assume that the unipotent radical of the tannakian fundamental group of $M$ is as large as it is permitted under the constraints imposed by the filtration $(F_\bullet M)$. In this note, we first describe the $\operatorname{Ext}^1$ groups in the tannakian subcategory of $\mathbf{T}$ generated by $M$. We then give two applications for motives, one involving 1-motives and another involving mixed Tate motives, leading to some implications of Grothendieck’s period conjecture.

Soit $\mathbf{T}$ une catégorie tannakienne neutre sur un corps de caractéristique 0. Soit $M$ un objet de T avec une filtration $0=F_0M\subsetneq F_1M\subsetneq \dots \subsetneq F_kM=M$, telle que chaque quotient successif $F_iM/F_{i-1}M$ soit semi-simple. Supposons que le radical unipotent du groupe fondamental tannakien de $M$ soit aussi grand que le permettent les contraintes imposées par la filtration $(F_\bullet M)$. Dans cette note, nous décrivons d’abord les groupes $\operatorname{Ext}^1$ dans la sous-catégorie tannakienne de $\mathbf{T}$ générée par $M$. Nous donnons ensuite deux applications pour les motifs, l’une impliquant les 1-motifs et l’autre impliquant les motifs mixtes de Tate, ce qui conduit à certaines implications de la conjecture de période de Grothendieck.

Reçu le : 2025-06-24
Accepté le : 2026-01-09
Publié le : 2026-03-13

DOI : 10.5802/crmath.817

Classification : 19E15, 14C15, 18M25, 11G99

Affiliations des auteurs :

Payman Eskandari ¹

¹ Department of Mathematics and Statistics, University of Winnipeg, Winnipeg MB, Canada

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Payman Eskandari},
     title = {A remark on $\operatorname{Ext}$ groups for motives with maximal unipotent radicals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {127--136},
     year = {2026},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {364},
     doi = {10.5802/crmath.817},
     language = {en},
}

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Payman Eskandari. A remark on $\operatorname{Ext}$ groups for motives with maximal unipotent radicals. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 127-136. doi: 10.5802/crmath.817

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