ChromoCats · The geometry of chromatic categories

The geometry of chromatic categories

In modern mathematics, objects of a similar kind are often studied not in isolation, but together with the relations and symmetries between them. A collections of objects of a similar type together with their relations---the morphisms between them---forms what is called a category. Categories usually provide a useful and unifying perspective; in addition, they often come equipped with additional operations that allow the construction of new objects from old ones. There is a particularly well-behaved class of categories with many such operations, which here we will refer to as a chromatic category. Prominent examples of chromatic categories which are of particular current interest include: (a) Algebraic geometry: Derived categories of (quasi-coherent or ind-coherent) sheaves on certain (derived) schemes or stacks. (b) Representation theory: The stable module category of linear representations of a finite group over a field of characteristic p. (c) Topology: The stable homotopy category and its equivariant or motivic variants. The guiding perspective taken in this project is that we should view such chromatic categories through a prism: As chromatic dispersion splits white light into its spectrum consisting of different colors, any given chromatic category C should decompose over a space Spc(C)---its so-called Balmer spectrum---into local or `monochromatic' categories C_p of `color' p in Spc(C). This point of view not only allows to study seemingly unrelated problems and phenomena in different areas of mathematics uniformly, but also leads to specific geometrically-inspired questions about chromatic categories: (1) Local structure: What can be said about the structure of the local categories C_p? How to compute important invariants of these categories, like Picard groups or dualizing objects, for example using descent-theoretic techniques? (2) Local-to-global principles: What is the Balmer spectrum Spc(C) for important examples of C? How do the local categories (C_p) reassemble over the space Spc(C) to reconstruct C? (3) Asymptotic behavior: Is there a notion of compactification of a chromatic category C, by adding appropriate boundary points? How can we describe these boundary points and what kind of information about C do they contain? The goal of this project is two-fold: Firstly, develop a rigorous framework in which we can decompose and study chromatic categories geometrically, and secondly approach the above questions, in particular pertaining to outstanding conjectures in the respective areas.

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