Cosilting modules arising from cotilting objects

TL;DR

This paper explores the relationship between cosilting modules over a ring and cotilting objects in a Grothendieck category, providing characterizations and conditions linking these concepts.

Contribution

It establishes a correspondence between cosilting modules and cotilting objects within certain subcategories, extending the understanding of their structural connections.

Findings

Every cosilting right R-module T can be viewed as a cotilting object in a specific subcategory.

Under certain conditions, cotilting objects in these subcategories are also cosilting modules.

The paper characterizes cosilting modules via cotilting objects in subcategories generated by quotients R/I.

Abstract

Let $R$ be a ring. In this paper, we study the characterization of cosilting modules and establish a relation between cosilting modules and cotilting objects in a Grothendieck category. We proved that each cosilting right $R$-module $T$ can be described as a cotilting object in $\sigma[R/I]$, where $I$ is a right ideal of $R$ determined by $T$ and $\sigma[R/I]$ is the full subcategory of right $R$-modules, consisting of submodules of $R/I$-generated modules. Conversely, under some suitable conditions, if $T$ is a cotilting object in $\sigma[R/I]$, then $T$ is cosilting.