Support varieties for finite tensor categories: the tensor product property

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 result to deduce the tensor product property for a large class of categories, those of modules for skew group algebras formed by exterior algebras with certain finite group actions. These include the symmetric finite tensor categories over algebraically closed fields of characteristic zero, thus giving a new proof of the tensor product property for these categories.