Ranks of 1-motives as dimensions of Ext¹ vector spaces

In this paper, we reformulate the ranks that appear in the dimension formula for the linear space of periods of a [Formula: see text]-motive established by Huber and Wüstholz in a more conceptual and categorical way, as dimensions of Ext 1 vector spaces. This constitutes the first step towards rewriting the dimension formula purely in general categorical terms, rather than through definitions and computations introduced ad hoc for [Formula: see text]-motives.