I am reading these notes, and do not quite understand how in the proof of Lemma 9.1, the $ \Delta$-equivariant exact sequence is written.
For the specific case I am concerned about, I assume that $ p $ splits into two parts in $K $, so that $ F/K $ is a cyclic extension of degree $ p - 1 $, $ \Delta = \mathrm{Gal}(F/K)$. Dirichlet's unit theorem tells us that $ \mathcal{O}_{F}^{\times}\otimes \mathbb{Q} $ is of dimension $ p-2 $. So the dimensions do add up. But how exactly are the maps defined?