Approximation of subcategories by abelian subcategories

TL;DR

This paper characterizes the category of $L_0$-complete modules over a commutative ring with a weakly proregular ideal as the minimal abelian subcategory containing all $I$-adically complete modules, highlighting a universal property.

Contribution

It establishes a universal property of the category of $L_0$-complete modules as the smallest replete exact abelian subcategory containing all $I$-adically complete modules.

Findings

The category of $L_0$-complete modules has a universal property.

It is the smallest replete exact abelian subcategory containing $I$-adically complete modules.

Provides a new perspective on the structure of module categories over rings with weakly proregular ideals.

Abstract

For a commutative ring $R$ and a weakly proregular ideal $I$, we prove a simple universal property of the category of $L_0$-complete $R$-modules: it is the smallest replete exact abelian subcategory of the category of $R$-modules which contains all the $I$-adically complete $R$-modules.