I will discuss a recent generalization of Kuhn's Blue-shift theorem about Tate cohomology. Combining this result with work of Arone, Dwyer, and Lesh we resolve a conjecture of Balmer and Sanders and classify the thick tensor ideals of compact genuine $A$-spectra, where $A$ is a finite abelian group. This is joint work with Tobias Barthel, Markus Hausmann, Niko Naumann, Thomas Nikolaus, and Nathaniel Stapleton.
