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Separable commutative rings in the stable module category of cyclic groups

Paul Balmer, Jon F. Carlson

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.

Separable commutative rings in the stable module category of cyclic 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.

Paper Structure

This paper contains 4 sections, 17 theorems, 15 equations.

Table of Contents

  1. Separable ring-objects
  2. The Kelly radical and the tensor
  3. The case of the group of prime order
  4. The case of the general cyclic group

Key Result

Proposition 1.3

Let $A$ be a separable commutative ring-object in a tensor category $\mathscr{C}$. Then we have:

Theorems & Definitions (53)

  • Definition 1.1
  • Example 1.2
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Proposition 1.5
  • proof
  • Definition 2.1
  • Remark 2.2
  • Remark 2.5
  • ...and 43 more