# The geometry of chromatic categories

## Periodic Reporting for period 1 - ChromoCats (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.
The outcome of the project so far comprises 13 articles freely available on the preprint server arXiv.org as well as the researcher's homepage, which have been submitted to peer-reviewed journals. Among these preprints, 9 were written during the 17 month duration of the fellowship, including 2 survey articles making the results and their context available to a wider mathematical audience. Moreover, 7 preprints have already been published or accepted for publication, with one paper appearing in a top 5 journal (Inventiones) and others in very good general journals (e.g. Compositio, Advances). The results contained in this work cover most of the goals set out in the project, taking into account the early termination (17 instead of 24 months). We single out three highlights:

(1) In joint work with several coauthors, we have studied the interactions between equivariant and chromatic homotopy theory by determining the blueshift behavior of geometric fixed point functors. This led to a resolution of the log_p-conjecture of Balmer--Sanders and consequently the computation of the Balmer spectrum of the G-equivariant stable homotopy category for any finite abelian group G. These results have subsequently been extended partially to all compact Lie groups G.
(2) With Schlank and Stapleton we develop a theory of ultraproducts of categories and apply it to construct compactifications of chromatic categories in such a way that the boundary controls generic information about the interior. This theory gave rise to a solution of the algebraicity problem in chromatic homotopy theory, by constructing and identifying the limit for p --> oo of the local stable homotopy category.
(3) In forthcoming work with Schlank and Stevenson, we develop a framework in which the motivating idea of the project can be made precise. More precisely, we construct an appropriate geometry in which every chromatic category C naturally and faithfully ``decomposes'' into a bundle of categories over its Balmer spectrum Spc(C), thereby recovering and extending Balmer's tensor-triangular geometry. This fully realizes the overarching goal of the project.

The fellow has presented the results of this work at 6 international conferences and in several seminars, besides further talks given by his collaborators.
In particular, the methods developed during the project have led to the resolution of the log_p-conjecture, the computation of several new Balmer spectra for important chromatic categories appearing in topology, (derived) algebra, and representation theory, and a better understanding of Picard groups in chromatic homotopy theory and modular representation theory. Moreover, we have obtained substantial progress on other open problems, as for example the chromatic splitting conjecture and its variations. Forthcoming work based on these results and currently being written up include the resolution of outstanding conjectures about endotrivial modules as well as the general construction of canonical decomposition of chromatic categories mentioned above.

The tools and results obtained during the project are likely going to give rise to many further developments in geometry, representation theory, and topology, contributing to the fruitful interaction between them and the unification of these areas of mathematics. The geometric point of view on chromatic categories has the potential to grow into a theory both of interest in its own right and with ample and unexpected applications to classical open problems.