The étale site of the stable module category of a finite group

To any tensor-triangulated category $T$, there is a natural way to associate a site and a sheaf cohomology. The objects in this site are the commutative separable monoids in $T$. For instance, if $T$ is the derived category of quasi-coherent sheaves on a scheme $X$, then the site fits in between the classical étale site and the recently discovered pro-étale site on $X$.

In this talk, I will explain what separable monoids are and show how they pop up in various settings. For a finite group $G$, I will show that the compact separable monoids in both the derived and stable module category of $G$ correspond to $G$-sets. This allows us to describe the site associated to the derived and stable module category of $G$ and compute the corresponding sheaf cohomology.