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Arithmetic D-modules on locally noetherian formal schemes

Richard Crew

Abstract

We extend Berthelot's theory of arithmetic D-modules to a class of morphisms that are not necessarily of finite type. As an application we give a new construction of the category of convergent isocrystals on a separated scheme of finite type over a field, and show that the pullback by Frobenius is an auto-equivalence. This extends results of Berthelot that were proven in the smooth case.

Arithmetic D-modules on locally noetherian formal schemes

Abstract

We extend Berthelot's theory of arithmetic D-modules to a class of morphisms that are not necessarily of finite type. As an application we give a new construction of the category of convergent isocrystals on a separated scheme of finite type over a field, and show that the pullback by Frobenius is an auto-equivalence. This extends results of Berthelot that were proven in the smooth case.

Paper Structure

This paper contains 34 sections, 92 theorems, 192 equations.

Table of Contents

  1. Formal Geometry
  2. Flatness and formal smoothness.
  3. Universally noetherian morphisms.
  4. Differentials and Smoothness
  5. Differential invariants.
  6. Deformations.
  7. Globalization.
  8. Smooth and quasi-smooth morphisms.
  9. Ordinary Differential Operators.
  10. Stratifications.
  11. Arithmetic Differential Operators
  12. $m$-PD-structures.
  13. Partial Divided Powers.
  14. The $m$-PD-adic Filtration.
  15. The $m$-PD-envelope of an ideal.
  16. ...and 19 more sections

Key Result

Lemma 1.1.2

Let $A$ be an adic noetherian ring with ideal of definition $J$, $x\in{\operatorname{Spf}({A})}$, ${\mathfrak m}\subset A$ the open prime ideal corresponding to $x$. If $M$ is a coherent $A$-module let ${\cal M}$ be the sheaf on ${\operatorname{Spf}({A})}$ corresponding to $M$ and ${\cal M}_x$ the s

Theorems & Definitions (115)

  • Definition 1.1.1
  • Lemma 1.1.2
  • Lemma 1.1.3
  • Proposition 1.1.4
  • Corollary 1.1.5
  • Definition 1.2.1
  • Lemma 1.2.2
  • Proposition 1.2.3
  • Proposition 1.2.4
  • Proposition 1.2.5
  • ...and 105 more