Spectral theory of Green functors and other commutative 2-rings

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Let G be a finite group. Every Green functor for G – such as the Burnside, the cohomology, or the representation Green functor – captures some aspect of the representation theory of G. If the Green functor is commutative, as in the previous examples, then it admits a symmetric tensor product of its modules, and its derived category is a tensor triangulated category. Our first goal is to understand the extent to which this is like the derived category of a commutative ring. In particular we aim at proving a classification theorem for its thick subcategories of perfect complexes, in terms of subsets of a suitable topological space, the spectrum of the Green functor; this result should generalize the classification for commutative rings (the case G = 1) and Quillen stratification in modular representation theory. Our second goal lies in developing the proof methods themselves, i.e. new techniques of tensor triangular geometry adapted to Mackey and Green functors. We also wish to explore the natural context for this enterprise, namely presheaves over small symmetric tensor categories, or “commutative 2-rings”. Our third goal is to provide applications to G-equivariant stable homotopy (via the Burnside ring Green functor) and to G-equivariant KK-theory (via the representation ring Green functor).