Cohomology of twisted tensor products

TL;DR

This paper investigates the cohomology of twisted tensor products, providing explicit descriptions and conditions under which cohomological properties resemble those of ordinary tensor products, with applications to quantum complete intersections.

Contribution

It offers a detailed analysis of the Hochschild cohomology of twisted tensor products and characterizes when cohomology groups are finitely generated, extending classical results.

Findings

Explicit description of the Ext-algebra of tensor products

Conditions for Hochschild cohomology ring structure

Lower bounds for the representation dimension

Abstract

It is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies. We look at twisted tensor products, and investigate to which extend this is still true. We give an explicit description of the $\Ext$-algebra of the tensor product of two modules, and under certain additional conditions, describe an essential part of the Hochschild cohomology ring of a twisted tensor product. As an application, we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the Hochschild cohomology ring. Moreover, both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra.