# Separable commutative rings in the stable module category of cyclic groups

## TL;DR

This paper characterizes separable commutative ring-objects in the stable module category of cyclic p-groups, showing they correspond exactly to subgroup structures, and explores the tensor-closure of the Kelly radical in these categories.

## Contribution

It provides a complete classification of separable commutative rings in the stable module category of cyclic p-groups and describes the tensor-closure of the Kelly radical.

## Key findings

- Separable commutative rings correspond to subgroups of cyclic p-groups.
- The tensor-closure of the Kelly radical is explicitly described.
- Classification applies to stable module categories of finite groups.

## Abstract

We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group.