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.

In this note, we show how a continuous action of the Morava stabilizer group Gn on the Lubin-Tate spectrum En, satisfying the conclusion E hGn n ≃ L K(n) S of the Devinatz-Hopkins Theorem, may be obtained by monodromy on the stack of oriented deformations of formal groups in the context of formal spectral algebraic geometry.

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.

We exhibit a relationship between motivic homotopy theory and spectral algebraic geometry, based on the motivic τ -deformation picture of Gheorghe, Isaksen, Wang, Xu. More precisely, we identify cellular motivic spectra over C with ind-coherent sheaves (in a slighly non-standard sense) on a certain spectral stack τ≥0(M or FG ). The latter is the connective cover of the non-connective spectral stack M or FG , 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 τ -deformation behavior on the level of sheaves, based on a notion of extended effective Cartier divisors in spectral algebraic geometry.