Moduli stack of oriented formal groups and the chromatic filtration

TL;DR

This paper introduces a new open filtration on the spectral moduli stack of formal oriented groups, linking chromatic filtration in homotopy theory with classical formal group stratification, and expresses key constructions via spectral algebraic geometry.

Contribution

It defines a novel open filtration that unifies chromatic and height stratifications, enabling new geometric interpretations of chromatic homotopy theory concepts.

Findings

Provides a geometric framework for chromatic localization and $K(n)$-localization.

Expresses classical chromatic constructions as sheaf restrictions and completions.

Establishes connections between spectral algebraic geometry and chromatic homotopy theory.

Abstract

We define a filtration by open substacks on the non-connective spectral moduli stack of formal oriented groups, which simultaneously encodes and relates the chromatic filtration of spectra and the height stratification of the classical moduli stack of formal groups. Using this open filtration, we express various classical constructions in chromatic homotopy theory, such as chromatic localization, the monochromatic layer, and $K(n)$-localization, in terms of restriction and completion of sheaves in non-connective spectral algebraic geometry.