# Projective discrete modules over profinite groups

## TL;DR

This paper investigates the structure of discrete modules over infinite profinite groups, revealing they lack non-zero projective objects and do not satisfy certain categorical properties, with extensions to a generalized ring setting.

## Contribution

It establishes fundamental limitations on projective objects in categories of discrete modules over profinite groups and generalizes these results to rings with linear topology.

## Key findings

- No non-zero projective objects in the category of discrete modules over infinite profinite groups.
- The category does not satisfy Ab4* property.
- Results extend to modules over rings with linear topology.

## Abstract

We show that the category of discrete modules over an infinite profinite group has no non-zero projective objects and does not satisfy Ab4*. We also prove the same types of results in a generalized setting using a ring with linear topology.