# On support varieties and tensor products for finite dimensional algebras

## TL;DR

This paper investigates the tensor product property for support varieties over finite dimensional algebras and demonstrates that, in general, such a property does not hold, especially in the context of quantum complete intersections.

## Contribution

It proves that a general tensor product property for support varieties cannot exist over finite dimensional algebras, providing counterexamples involving quantum complete intersections.

## Key findings

- Support varieties do not satisfy the tensor product property in general.
- Counterexamples are constructed using quantum complete intersections.
- The variety of a tensor product can be larger than expected.

## Abstract

It has been asked whether there is a version of the tensor product property for support varieties over finite dimensional algebras defined in terms of Hochschild cohomology. We show that in general no such version can exist. In particular, we show that for certain quantum complete intersections, there are modules and bimodules for which the variety of the tensor product is not even contained in the variety of the one-sided module.