Moduli stack of oriented formal groups and periodic complex bordism

Abstract

We introduce and study the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups. We realize some results of chromatic homotopy theory in terms of the geometry of this stack. For instance, we show that its descent spectral sequence recovers the Adams-Novikov spectral sequence. For two $\mathbb E_\infty$-forms of periodic complex bordism $\mathrm{MP}$, the Thom spectrum and Snaith construction model, we describe the universal property of the cover $\mathrm{Spec}(\mathrm{MP})\to\mathcal M_\mathrm{FG}^\mathrm{or}$. We show that Quillen's celebrated theorem on complex bordism is equivalent to the assertion that the underlying ordinary stack of $\mathcal M_\mathrm{FG}^\mathrm{or}$ is the classical stack of ordinary formal groups $\mathcal M^\heartsuit_\mathrm{FG}$. In order to carry out all of the above, we develop foundations of a functor of points approach to non-connective spectral algebraic geometry.