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We reformulate the ranks that appear in the dimension formula for the linear space of periods of a 1-motive established by Huber and W\"ustholz in a more conceptual and categorical way, as dimensions of Ext¹ 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 1-motives.
Nicola Nesa (2024) studied this question.