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Even periodization of spectral stacks

Rok Gregoric

TL;DR

This work introduces and develops the operation of even periodization to geometrize chromatic phenomena in nonconnective spectral geometry. By collapsing to even-periodic affines, it recovers the Hahn–Raksit–Wilson even filtration, and it provides a Nygaard-complete prismatization framework that globalizes prismatic ideas beyond $p$-adic geometry. The authors identify key moduli stacks as evp-images, including the chromatic base stack $ ext{M}$ from $ ext{Spec}(f S)$ and the oriented elliptic curve moduli $ ext{M}^{ ext{or}}_{ ext{Ell}}$ from $ ext{Spec}( ext{TMF})$, and prove that evp is fully faithful on a broad class of even-convergent affines. They further relate evp to MU-shearing, establish complex-periodization as a parallel construction, and prove a central result that $ ext{Spec}( ext{TMF})^{ ext{evp}} o ext{M}^{ ext{or}}_{ ext{Ell}}$ is an equivalence of stacks, yielding a geometric bridge between TMF and elliptic moduli. The paper also develops a robust foundation for formal spectral stacks and their quasi-coherent sheaves, extending the even-periodization program to formal and $I$-adic contexts. Overall, the work provides a unified, geometric realization of chromatic filtrations and TMF-related moduli, with potential implications for prismatic geometry and synthetic approaches in stable homotopy theory.

Abstract

We introduce the operation of even periodization on nonconnective spectral stacks. We show how to recover from it the even filtration of Hahn-Raksit-Wilson, and (a Nygaard-completion of) the filtered prismatization stack of Bhatt-Lurie and Drinfeld. We prove that the moduli stack of oriented elliptic curves of Goerss-Hopkins-Miller and Lurie is the even periodization of the spectrum of topological modular forms. Likewise, the chromatic moduli stack, which we previously studied under the name of the moduli stack of oriented formal groups, arises via even periodization from the sphere spectrum. Over the complex bordism spectrum, we relate even periodization with the symmetric monoidal shearing, in the sense of Devalapurkar, of free gradings.

Even periodization of spectral stacks

TL;DR

This work introduces and develops the operation of even periodization to geometrize chromatic phenomena in nonconnective spectral geometry. By collapsing to even-periodic affines, it recovers the Hahn–Raksit–Wilson even filtration, and it provides a Nygaard-complete prismatization framework that globalizes prismatic ideas beyond -adic geometry. The authors identify key moduli stacks as evp-images, including the chromatic base stack from and the oriented elliptic curve moduli from , and prove that evp is fully faithful on a broad class of even-convergent affines. They further relate evp to MU-shearing, establish complex-periodization as a parallel construction, and prove a central result that is an equivalence of stacks, yielding a geometric bridge between TMF and elliptic moduli. The paper also develops a robust foundation for formal spectral stacks and their quasi-coherent sheaves, extending the even-periodization program to formal and -adic contexts. Overall, the work provides a unified, geometric realization of chromatic filtrations and TMF-related moduli, with potential implications for prismatic geometry and synthetic approaches in stable homotopy theory.

Abstract

We introduce the operation of even periodization on nonconnective spectral stacks. We show how to recover from it the even filtration of Hahn-Raksit-Wilson, and (a Nygaard-completion of) the filtered prismatization stack of Bhatt-Lurie and Drinfeld. We prove that the moduli stack of oriented elliptic curves of Goerss-Hopkins-Miller and Lurie is the even periodization of the spectrum of topological modular forms. Likewise, the chromatic moduli stack, which we previously studied under the name of the moduli stack of oriented formal groups, arises via even periodization from the sphere spectrum. Over the complex bordism spectrum, we relate even periodization with the symmetric monoidal shearing, in the sense of Devalapurkar, of free gradings.

Paper Structure

This paper contains 36 sections, 109 theorems, 391 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Related work
  4. Notation and terminology
  5. Acknowledgments
  6. The even filtration
  7. Even and even periodic filtration
  8. The Adams-Novikov filtration
  9. $I$-adic even filtration
  10. Spectral algebraic geometry via the functor of points
  11. The setup of nonconnective spectral stacks
  12. The connective localization
  13. Underlying classical stacks
  14. Quasi-coherent sheaves on a nonconnective spectral stack
  15. Affine morphisms of nonconnective spectral stacks
  16. ...and 21 more sections

Key Result

Theorem I.2

Even periodization in the following cases recovers other previously known even periodic spectral stacks: where respectively:

Theorems & Definitions (342)

  • Definition I.1
  • Theorem I.2
  • Theorem I.3
  • Theorem I.4
  • Theorem I.5
  • Theorem I.6
  • Theorem I.7
  • Definition 2.1.1
  • Remark 2.1.2
  • Remark 2.1.3
  • ...and 332 more