(PRE-)Sheaves of Ring Spectra Over the Moduli Stack of Formal Group Laws

Abstract

In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.While some of what I say is quite general, the ring spectra I have in mind will arise from the chromatic point of view, which uses the geometry of formal groups to organize stable homotopy theory. Thus, a subsidiary aim here is to reemphasize this connection between algebraic geometry and homotopy theory.

Diagrams indexed by Grothendieck constructions

“…Recently stacks have come to play an important role in algebraic topology. Complex oriented cohomology theories give rise to stacks over the moduli stack of formal groups and in certain situations, conversely, stacks over the moduli stack of formal groups give rise to spectra [G,R2,GHMR,B]. One fundamental example is the spectrum of topological modular forms [Hp] which is associated to the moduli stack of elliptic curves.…”

On affine morphisms of Hopf algebroids

“…Foundational work has been carried out on a category of algebraic stacks suitable for the application to stable homotopy theory by Hopkins and Miller [8], Goerss [3], Pribble in his thesis [12], Naumann [11] and by others. There is related foundational work on stacks from the homotopy theoretic or derived viewpoint by Hollander in [6,5,4].…”

Descent for quasi-coherent sheaves on stacks

“…More recently stacks have also played an important role in algebraic topology. Complex oriented cohomology theories give rise to stacks over the moduli stack of formal groups and, in certain situations, stacks over the moduli stack of formal groups give rise to spectra (see Goerss [6], Goerss-Hopkins [8] and Rezk [21]) which play an important role in understanding the homotopy groups of spheres (see Goerss-Henn-Mahowald-Rezk [7] and Behrens [2]). One fundamental example is the spectrum of topological modular forms (see Hopkins [12]) which is associated to the moduli stack of elliptic curves.…”

“…The moduli stack of formal groups M F G is of special importance in stable homotopy theory (see Goerss [6], Pribble [18] and Naumann [17]). The Lazard ring provides an atlas Spec L !…”