Defect groups of tensor modules

We show that the Brauer map for the exterior tensor product of two G-algebras can be expressed as a tensor product of their Brauer maps. As a consequence, we prove that the tensor module of two connected modules for the group algebra is also connected and its defect group is the direct product of their defect groups. We also show that this process is compatible with the association defined by Barker (J. Algebra 168 (1994) 7287) when localizing to subgroups.