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

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.

arXiv comment26 pages. Fixed glaring typo in the abstract

References in corpus (15)

• From algebraic cobordism to motivic cohomology
• Motivic cell structures
• The Picard group of topological modular forms via descent theory
• Local duality in algebra and topology
• The Adams-Novikov spectral sequence and Voevodsky's slice tower
• The geometry of filtrations
• C-motivic modular forms
• Exotic Motivic Periodicities
• The motivic Adams-Novikov spectral sequence at odd primes over $\mathbb{C}$ and $\mathbb{R}$
• A^1-homotopy invariance in spectral algebraic geometry
• Moduli stack of oriented formal groups and periodic complex bordism
• Moduli stack of oriented formal groups and the chromatic filtration
• ${\mathbb{E}}_{\infty }$ automorphisms of motivic Morava $E$-theories
• The Structure of Motivic Homotopy Groups
• Transchromatic extensions in motivic and Real bordism