Quasi-coherent sheaves on the moduli stack of formal groups

TL;DR

This paper develops decomposition results for quasi-coherent sheaves on the moduli stack of formal groups, utilizing the stack's geometry, especially the height filtration, to achieve algebraic chromatic convergence and fracture square decompositions.

Contribution

It introduces new algebraic chromatic convergence and fracture square theorems for sheaves on the moduli stack of formal groups, overcoming technical challenges due to the stack's non-finiteness.

Findings

Decomposition results for quasi-coherent sheaves based on height filtration.

Algebraic chromatic convergence theorems established.

Fracture square decompositions derived for the stack.

Abstract

The central aim of this monograph is to provide decomposition results for quasi-coherent sheaves on the moduli stack of one-dimensional formal groups. These results will be based on the geometry of the stack itself, particularly the height filtration and an analysis of the formal neighborhoods of the geometric points. The main theorems are algebraic chromatic convergence results and fracture square decompositions. There is a major technical hurdle in this story, as the moduli stack of formal groups does not have the finitness properties required of an algebraic stack as usually defined. This is not a conceptual problem, but in order to be clear on this point and to write down a self-contained narrative, I have included a great deal of discussion of the geometry of the stack itself, giving various equivalent descriptions.