A remark on Ext groups for motives with maximal unipotent radicals

Let T be a neutral tannakian category over a field of characteristic 0. Let M be an object of T with a filtration 0 = F 0 M ⊊ F 1 M ⊊ ⋯ ⊊ F k M = M , such that each successive quotient F i M / 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 • M ) . In this note, we first describe the Ext 1 groups in the tannakian subcategory of 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.