Invertible unital bimodules over rings with local units, and related exact sequences of groups

TL;DR

This paper explores the structure of invertible bimodules over rings with local units, constructing exact sequences that relate the Picard groups of the rings involved, inspired by Miyashita's work.

Contribution

It introduces new exact sequences connecting Picard groups of rings with local units, extending previous theories by Miyashita.

Findings

Four exact sequences relating Picard groups of rings with local units

Extension of Miyashita's work to rings with local units

Enhanced understanding of bimodule structures over such rings

Abstract

Given an extension $R \subseteq S$ of rings with same set of local units, inspired by the works of Miyashita, we construct four exact sequences of groups relating Picard's groups of $R$ and $S$.