Support varieties without the tensor product property

TL;DR

This paper demonstrates that in certain finite tensor categories over perfect fields, support varieties can exist without satisfying the tensor product property, highlighting limitations in the current theoretical framework.

Contribution

It constructs examples of finite tensor categories where support varieties lack the tensor product property, expanding understanding of their structural behavior.

Findings

Support varieties can exist without the tensor product property.

Embedding into larger categories can break the tensor product property.

The results apply to non-semisimple categories over perfect fields.

Abstract

We show that over a perfect field, every non-semisimple finite tensor category with finitely generated cohomology embeds into a larger such category where the tensor product property does not hold for support varieties.