Moduli stack of oriented formal groups and periodic complex bordism

Abstract

We study the non-connective spectral stack M or FG , the moduli stack of oriented formal groups in the sense of Lurie. We show that its descent spectral sequence recovers the Adams-Novikov spectral sequence. For two E∞-forms of periodic complex bordism MP, the Thom spectrum and Snaith construction model, we describe the universal property of the cover Spec(MP) → M or FG . We show that Quillen's celebrated theorem on complex bordism is equivalent to the assertion that the underlying ordinary stack of M or FG is the classical stack of ordinary formal groups M ♡ FG . In order to carry out all of the above, we develop foundations of a functor of points approach to non-connective spectral algebraic geometry.

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

“…It is a non-connective spectral stack, with one of its characteristic properties being a Bott-isomorphism-like equivalence of quasi-coherent sheaves β ∶ Σ 2 (ω M or FG ) ≃ O M or FG , where O M or FG is the structure sheaf and ω M or FG is the dualizing line of the universal oriented formal group. We also showed in [Gre21a,Theorem 2.4.4] that the global sections functor F ↦ Γ(M or FG ; F ) induces an equivalence of ∞-categories IndCoh(M or FG ) ≃ Sp (1) between the stable ∞-category of spectra and ind-coherent sheaves (in the non-standard sense of [Gre21a, Definition 2.4.2], equivalent to Definition 1.5.1) on the non-connective spectral stack M or FG . In this paper, the main object of interest will be its connective cover τ ≥0 (M or FG ), studied in Section 1.4.…”

“…In summation, the results of this paper demonstrate how (at least over the complex numbers and in the p-complete cellular context) stable motivic homotopy theory manifests in spectral algebraic geometry, using the spectral stack τ ≥0 (M or FG ). Our prior work [Gre21a], [Gre21b] shows that the same is true of chromatic homotopy theory, using the non-connective spectral stack M or FG . In this sense, present results may not seem so surprising; Voevodsky's conjecture [Voe02, Conjecture 9] (proved in [Lev14] by relying on work of Hopkins-Morel, see [Hoy15]), and especially the related [Lev14, Theorem 4] and [Lev15], already showcase a deep connection between the non-motivic Adams-Novikov spectral sequence, the progenitor of chromatic homotopy theory, and (the décalage of) the slice filtration, one of the fundamental constructions in motivic stable homotopy theory.…”

“…The connective cover of the moduli stack of oriented formal groups. Recall the stack of oriented formal groups M or FG from [Gre21a]. We showed in [Gre21a, Corollary 2.3.7] that it admits a simplicial presentation…”

“…In this paper, we follow a suggestion of Morava and connect the τ -deformation picture with spectral algebraic geometry. More specifically, we relate it to our prior work [Gre21a], where we examined how chromatic homotopy theory manifests in non-connective spectral algebraic geometry. The main object of interest there, building on the work of Lurie from [Ell2], was the moduli stack of oriented formal groups M or FG .…”

“…The relationship between preorientations of formal groups in the sense of Lurie and motivic homotopy theory has been touched on in the height 1 case by [Hor14], but we hope that the connection with spectral algebraic geometry and the stack M or FG that we demonstrate in this paper may shed some light on the fascinating interactions of chromatic and motivic ideas, as considered in various forms and from various perspectives in [BHSZ21], [Ghe17], [Hor06], [Joa15], [Kra18], [MG19], [NØS09], [Sta21b] etc. This paper was born out of a suggestion by Jack Morava that there should be a relationship between [Gre21a] and the τ -deformation picture of [GWX21] et al That came at much surprise to the author, who is therefore immensely grateful to Morava, both for his generosity in sharing this idea, as well as for his insistence that we give it serious thought.…”