Spectral theory of Green functors and other commutative 2-rings

设G是有限群。G的每一个Green函子--例如Burnside、上同调或表示Green函子--都包含G表示理论的某些方面。如果Green函子是可交换的，如前面的例子，那么它允许它的模的对称张量积，并且它的派生范畴是张量三角范畴。我们的第一个目标是了解这在多大程度上类似于交换环的派生范畴。特别地，我们利用适当的拓扑空间的子集，证明了完备复形的厚子范畴的一个分类定理，这一结果推广了模表示理论中的交换环(G=1)的分类和Quillen分层.我们的第二个目标是发展证明方法本身，即适用于Mackey和Green函子的张量三角几何新技术。我们还希望探索这个企业的自然背景，即在小的对称张量范畴上的预沉，或“交换2-环”。我们的第三个目标是提供G-等变稳定同伦(通过Burnside环Green函子)和G-等变KK-理论(通过表示环Green函子)的应用。

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).