# Injective Morita contexts (revisited)

## TL;DR

This paper revisits injective Morita contexts and introduces Morita semi-contexts and data, exploring their properties and applications in module theory, especially in localized or colocalized subcategories.

## Contribution

It extends the theory of Morita contexts by defining semi-contexts and data, and demonstrates their use in establishing equivalences between subcategories of modules.

## Key findings

- Injective Morita data can establish equivalences between localized subcategories.
- Morita $ ext{α}$-contexts relate to $ ext{*}$-modules and wide Morita contexts.
- New notions accommodate situations with limited trace ideal action.

## Abstract

This paper is an exposition of the so-called injective Morita contexts (in which the connecting bimodule morphisms are injective) and Morita $\alpha$contexts (in which the connecting bimodules enjoy some local projectivity in the sense of Zimmermann-Huisgen). Motivated by situations in which only one trace ideal is in action, or the compatibility between the bimodule morphisms is not needed, we introduce the notions of Morita semi-contexts and Morita data, and investigate them. Injective Morita data will be used (with the help of static and adstatic modules) to establish equivalences between some intersecting subcategories related to subcategories of modules that are localized or colocalized by trace ideals of a Morita datum. We end up with applications of Morita $\alpha$-contexts to $\ast$-modules and injective right wide Morita contexts.