Connecting homomorphisms associated to Tate sequences

TL;DR

This paper provides explicit descriptions of connecting homomorphisms in Tate cohomology related to Tate sequences, advancing understanding of Galois module structures without assuming trivial class groups.

Contribution

It introduces explicit formulas for connecting homomorphisms in Tate cohomology associated with Ritter and Weiss's small S Tate sequence, expanding the theoretical framework.

Findings

Explicit descriptions of connecting homomorphisms under certain conditions

Enhanced understanding of Galois module structures

No assumption of vanishing S-class-group needed

Abstract

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 $S$.