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We consider a Deligne–Mumford stack 
$X$ which is the quotient of an affine scheme 
$\operatorname {Spec}A$ by the action of a finite group 
$G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on 
$X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring 
$H^*(G,A)$.
We study the Balmer spectrum of the category of finite 
$G$-spectra for a compact Lie group 
$G$, extending the work for finite 
$G$ by Strickland, Balmer–Sanders, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of 
$G$. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes 
$p$.
