Algebraic chromatic homotopy theory for $BP_*BP$-comodules

TL;DR

This paper develops an algebraic framework for chromatic homotopy theory related to $BP_*BP$-comodules, establishing key theorems like nilpotence, convergence, and spectral sequences, paralleling topological results.

Contribution

It introduces an algebraic analog of the derived category of formal groups, proving foundational theorems and conjectures in this new setting.

Findings

Proved an algebraic version of the nilpotence theorem.

Established the chromatic convergence theorem algebraically.

Constructed a generalized chromatic spectral sequence.

Abstract

In this paper, we study the global structure of an algebraic avatar of the derived category of ind-coherent sheaves on the moduli stack of formal groups. In analogy with the stable homotopy category, we prove a version of the nilpotence theorem as well as the chromatic convergence theorem, and construct a generalized chromatic spectral sequence. Furthermore, we discuss analogs of the telescope conjecture and chromatic splitting conjecture in this setting, using the local duality techniques established earlier in joint work with Valenzuela.