Connecting homomorphisms associated to Tate sequences

Tate sequences are an important tool for tackling problems related to the (ill-understood) Galois structure of groups of $S$-units. The relatively recent Tate sequence "for small $S$" of Ritter and Weiss allows one to use the sequence without assuming the vanishing of the $S$-class-group, a significant advance in the theory. Associated to Ritter and Weiss's version of the sequence are connecting homomorphisms in Tate cohomology, involving the $S$-class-group, that do not exist in the earlier theory. In the present article, we give explicit descriptions of certain of these connecting homomorphisms under some assumptions on the set $S$.