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On Lurie's theorem and applications

Jack Morgan Davies

TL;DR

This work provides a comprehensive formulation and proof of Lurie’s theorem, establishing an étale sheaf $\mathscr{O}^{\mathrm{top}}_n$ of $\mathbf{E}_\infty$-rings on the spectral moduli stack of $p$-divisible groups, with precise even-degree cohomology and orientation data linking to formal groups. The construction rests on a blend of formal geometry, deformation theory, and descent for spectral Deligne--Mumford stacks, culminating in a global refinement of the Landweber exact functor theorem that glues Lubin--Tate, TMF, and related theories. The paper also develops orientation classifiers and a robust deformation-theoretic framework to define the universal spectral deformations, enabling automorphism actions and Adams operations on the resulting spectra. The resulting machinery yields concrete new cohomology theories (e.g., Lubin--Tate spectra, TMF, TAF) and clarifies how higher-categorical moduli enhance classical constructions, with broad implications for stable homotopy theory and topological modular forms. Overall, the work significantly deepens the link between chromatic homotopy theory and spectral algebraic geometry, expanding the toolbox for constructing and organizing structured ring spectra from moduli problems.

Abstract

Lurie's theorem states that there exists a sheaf of ring spectra on the site of formally étale Deligne--Mumford stacks over the moduli stack of $p$-divisible groups of height $n$, which agrees with the classical Landweber exact functor theorem (LEFT) on affines. In other words, this theorem is a global, higher categorical refinement of the LEFT. In recent work, Lurie has introduced many of the ingredients one needs to prove this theorem, and in this article, we gather these ingredients together and prove Lurie's theorem. Applications of this theorem to Lubin--Tate theories, topological modular and automorphism forms, and Adams operations are also discussed.

On Lurie's theorem and applications

TL;DR

This work provides a comprehensive formulation and proof of Lurie’s theorem, establishing an étale sheaf of -rings on the spectral moduli stack of -divisible groups, with precise even-degree cohomology and orientation data linking to formal groups. The construction rests on a blend of formal geometry, deformation theory, and descent for spectral Deligne--Mumford stacks, culminating in a global refinement of the Landweber exact functor theorem that glues Lubin--Tate, TMF, and related theories. The paper also develops orientation classifiers and a robust deformation-theoretic framework to define the universal spectral deformations, enabling automorphism actions and Adams operations on the resulting spectra. The resulting machinery yields concrete new cohomology theories (e.g., Lubin--Tate spectra, TMF, TAF) and clarifies how higher-categorical moduli enhance classical constructions, with broad implications for stable homotopy theory and topological modular forms. Overall, the work significantly deepens the link between chromatic homotopy theory and spectral algebraic geometry, expanding the toolbox for constructing and organizing structured ring spectra from moduli problems.

Abstract

Lurie's theorem states that there exists a sheaf of ring spectra on the site of formally étale Deligne--Mumford stacks over the moduli stack of -divisible groups of height , which agrees with the classical Landweber exact functor theorem (LEFT) on affines. In other words, this theorem is a global, higher categorical refinement of the LEFT. In recent work, Lurie has introduced many of the ingredients one needs to prove this theorem, and in this article, we gather these ingredients together and prove Lurie's theorem. Applications of this theorem to Lubin--Tate theories, topological modular and automorphism forms, and Adams operations are also discussed.

Paper Structure

This paper contains 24 sections, 35 theorems, 100 equations.

Table of Contents

  1. (1)
  2. (2)
  3. (3)
  4. The statement of Lurie's theorem
  5. (Locally Noetherian)
  6. (Qcqs)
  7. (Formal geometry)
  8. (Closed points have perfect residue fields)
  9. (Formally étale over ${\mathcal{M}}_n$)
  10. (Cotangent complex conditions in $\mathcal{C}_{A_0}$)
  11. Formally étale natural transformations
  12. On presheaves of discrete rings
  13. On presheaves of connective $\mathbf{E}_\infty$-rings
  14. Deformation theory on the moduli stack of $p$-divisible groups
  15. Orientations of $p$-divisible groups
  16. ...and 9 more sections

Key Result

Theorem 1.11

Using the notation of definitionofsites, there is an étale hypersheaf of $\mathbf{E}_\infty$-rings $\mathscr{O}^\mathrm{top}_n$ on $\mathcal{C}_{A_0}$ such that for a formal affine $\mathbf{G}_0\colon\mathop{\mathrm{Spf}}\nolimits B_0\to {\mathcal{M}}_n^\heartsuit$ in $\mathcal{C}_{A_0}$ the $\mathb

Theorems & Definitions (133)

  • Definition 1: Sheaves
  • Definition 2: Hypersheaves (sag)
  • Definition 3
  • Definition 4
  • Remark 5: Invariance of $\mathop{\mathrm{Spf}}\nolimits$ under completion
  • Definition 6
  • Definition 11
  • Remark 13
  • Definition 14
  • Definition 15
  • ...and 123 more