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We introduce a derived stack of Laurent F-crystals, demonstrating its equivalence to the Emerton-Gee stack. This derived stack is proven to be classical when analyzed with truncated animated rings.

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Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Arithmetic Geometry

Background:

  • The study builds upon the theory of F-crystals and p-adic geometry.
  • It relates to the moduli stack of étale (φ,Γ)-modules, known as the Emerton-Gee stack.

Purpose of the Study:

  • To construct and analyze a derived stack of Laurent F-crystals.
  • To establish the relationship between this derived stack and the Emerton-Gee stack.
  • To prove the classical nature of the derived stack in a specific context.

Main Methods:

  • Construction of a derived stack of Laurent F-crystals over the ring of integers of a finite extension K of Qp.
  • Comparison of the underlying classical stack with the Emerton-Gee stack.
  • Investigation of the derived stack's behavior on truncated animated rings using sheafification and Kan extensions.

Main Results:

  • The underlying classical stack of the constructed derived stack is shown to coincide with the Emerton-Gee stack.
  • The derived stack is proven to be classical, equivalent to the sheafification of the Emerton-Gee stack's Kan extension.

Conclusions:

  • The derived stack provides a new perspective on the Emerton-Gee stack and (φ,Γ)-modules.
  • The classical nature of the derived stack has implications for understanding structures in arithmetic geometry.