Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf C$

TL;DR

This paper establishes a deep connection between motivic homotopy theory and spectral algebraic geometry by relating cellular motivic spectra over complex numbers to ind-coherent sheaves on a spectral stack linked to oriented formal groups.

Contribution

It identifies cellular motivic spectra over C with ind-coherent sheaves on a spectral stack related to the moduli of oriented formal groups, introducing a geometric perspective on the motivic tau-deformation.

Findings

Cellular motivic spectra over C correspond to ind-coherent sheaves on a spectral stack.

The spectral stack is the connective cover of the moduli stack of oriented formal groups.

A geometric origin for tau-deformation behavior is provided via spectral algebraic geometry.

Abstract

We exhibit a relationship between motivic homotopy theory and spectral algebraic geometry, based on the motivic $\tau$-deformation picture of Gheorghe, Isaksen, Wang, Xu. More precisely, we identify cellular motivic spectra over $\mathbf C$ with ind-coherent sheaves (in a slightly non-standard sense) on a certain spectral stack $\tau_{\ge 0}(\mathcal M_\mathrm{FG}^\mathrm{or})$. The latter is the connective cover of the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups, which we have introduced previously and studied in connection with chromatic homotopy theory. We also provide a geometric origin on the level of stacks for the observed $\tau$-deformation behavior on the level of sheaves, based on a notion of extended effective Cartier divisors in spectral algebraic geometry.