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Affineness and reconstruction in complex-periodic geometry

William Balderrama, Jack Morgan Davies, Sil Linskens

TL;DR

The paper builds a comprehensive framework for affineness and descent in derived algebro-geometric geometry by constructing the spectral-stack category $ ext{Stk}$ over the fpqc topology on $ ext{CAlg}$ and developing notions like $0$-affineness, descent stacks, and reconstruction. It then specializes to nonconnective spectral algebraic geometry and to complex-periodic geometry, tying these notions to chromatic homotopy theory via the moduli stack of oriented formal groups $ ext{M}_{ ext{FG}}^{ ext{or}}$ and the descent stack $ ext{D}_{ ext{MP}}$. The authors prove a sharp bounded-height $0$-affineness result, give a new proof emphasizing categorical structure, and establish reconstruction theorems showing that many complex-periodic stacks are determined by their global sections. They further develop a robust theory of descent, locally descendable morphisms, and weak Tannaka duality in this generalized setting, yielding powerful tools for understanding spectral stacks such as those arising from TMF, KO, and related objects. The work provides foundational methods for identifying when a stack is reconstructible or $0$-affine, with wide implications for chromatic phenomena and the interplay between derived geometry and stable homotopy theory.

Abstract

Working in a generic derived algebro-geometric context, we lay the foundations for the general study of affineness and local descendability. When applied to $\mathbf{E}_\infty$ rings equipped with the fpqc topology, these foundations give an $\infty$-category of spectral stacks, a viable functor-of-points alternative to Lurie's approach to nonconnective spectral algebraic geometry. Specializing further to spectral stacks over the moduli stack of oriented formal groups, we use chromatic homotopy theory to obtain a large class of $0$-affine stacks, generalizing Mathew--Meier's famous $0$-affineness result. We introduce a spectral refinement of Hopkins' stack construction of an $\mathbf{E}_\infty$ ring, and study when it provides an inverse to the global sections of a spectral stack. We use this to show that a large class of stacks, which we call reconstructible, are naturally determined by their global sections, including moduli stacks of oriented formal groups of bounded height and the moduli stack of oriented elliptic curves.

Affineness and reconstruction in complex-periodic geometry

TL;DR

The paper builds a comprehensive framework for affineness and descent in derived algebro-geometric geometry by constructing the spectral-stack category over the fpqc topology on and developing notions like -affineness, descent stacks, and reconstruction. It then specializes to nonconnective spectral algebraic geometry and to complex-periodic geometry, tying these notions to chromatic homotopy theory via the moduli stack of oriented formal groups and the descent stack . The authors prove a sharp bounded-height -affineness result, give a new proof emphasizing categorical structure, and establish reconstruction theorems showing that many complex-periodic stacks are determined by their global sections. They further develop a robust theory of descent, locally descendable morphisms, and weak Tannaka duality in this generalized setting, yielding powerful tools for understanding spectral stacks such as those arising from TMF, KO, and related objects. The work provides foundational methods for identifying when a stack is reconstructible or -affine, with wide implications for chromatic phenomena and the interplay between derived geometry and stable homotopy theory.

Abstract

Working in a generic derived algebro-geometric context, we lay the foundations for the general study of affineness and local descendability. When applied to rings equipped with the fpqc topology, these foundations give an -category of spectral stacks, a viable functor-of-points alternative to Lurie's approach to nonconnective spectral algebraic geometry. Specializing further to spectral stacks over the moduli stack of oriented formal groups, we use chromatic homotopy theory to obtain a large class of -affine stacks, generalizing Mathew--Meier's famous -affineness result. We introduce a spectral refinement of Hopkins' stack construction of an ring, and study when it provides an inverse to the global sections of a spectral stack. We use this to show that a large class of stacks, which we call reconstructible, are naturally determined by their global sections, including moduli stacks of oriented formal groups of bounded height and the moduli stack of oriented elliptic curves.

Paper Structure

This paper contains 61 sections, 103 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. A proof of \ref{['maintheorem:boundedaffineness']}
  4. Conservativity of $f_\ast$:
  5. Cocontinuity of $f_\ast$:
  6. The functor $\pi_\ast$:
  7. Outline
  8. (\ref{['sec:sheaves']})
  9. (\ref{['sec:nonconsag']})
  10. (\ref{['sec:complexperiodicgeometry']})
  11. General derived geometries
  12. Generalized stacks and quasi-coherent sheaves
  13. Stacks on large categories
  14. Basic properties of stacks
  15. Quasi-coherent sheaf contexts and global sections
  16. ...and 46 more sections

Key Result

Theorem 1

Let $\mathsf{X}$ be a complex-periodic stack of bounded height. If the unique morphism $\mathsf{X} \to \mathsf{M}_\mathrm{FG}^\mathrm{or}$ is quasi-affine, then $\mathsf{X}$ is $0$-affine.

Theorems & Definitions (286)

  • Theorem 1: $0$-affineness
  • Theorem 2: $0$-affineness of complex-periodifications
  • Theorem 3: Reconstruction
  • Theorem 4: Bounded reconstruction
  • proof : Proof of \ref{['maintheorem:boundedaffineness']}
  • Definition 1.2.0.1: \ref{['def:univ0affine']}
  • Corollary 1.2.0.2: \ref{['thm:affinenessbounded']}
  • Definition 1.2.0.3: \ref{['def:descentstacks']}
  • Definition 1.2.0.4: \ref{['def:ldgeneral', 'def:ld']}
  • Definition 2.1.1.1
  • ...and 276 more