Support varieties for finite tensor categories: the tensor product property

TL;DR

This paper investigates the tensor product property for support varieties in finite tensor categories, establishing conditions under which it holds and applying results to specific algebraic structures.

Contribution

It proves the tensor product property holds for support varieties if and only if it holds for indecomposable periodic objects, with applications to Hopf algebras and symmetric categories.

Findings

Tensor product property holds iff it holds for indecomposable periodic objects.

The property is valid for all objects in symmetric finite tensor categories over characteristic zero fields.

Application to skew group algebras demonstrates the property in specific algebraic contexts.

Abstract

We show that in a finite tensor category, the tensor product property holds for support varieties if and only if it holds between indecomposable periodic objects. We apply this to certain Hopf algebras in the form of skew group algebras. In particular, we show that the tensor product property holds for all objects in a symmetric finite tensor category over an algebraically closed field of characteristic zero.