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Prismatization

Vladimir Drinfeld

Abstract

The goal is to construct three related "prismatization" functors from the category of p-adic formal schemes to that of formal stacks. This should provide a good category of coefficients for prismatic cohomology in the spirit of F-gauges. In this article we define and study the three versions of the prismatization of the formal spectrum of the ring of p-adic integers.

Prismatization

Abstract

The goal is to construct three related "prismatization" functors from the category of p-adic formal schemes to that of formal stacks. This should provide a good category of coefficients for prismatic cohomology in the spirit of F-gauges. In this article we define and study the three versions of the prismatization of the formal spectrum of the ring of p-adic integers.

Paper Structure

This paper contains 230 sections, 104 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Format of the Bhatt-Lurie approach to prismatic theory
  3. Non-derived variant
  4. Derived variant
  5. The stack $\Sigma =(\mathop{\mathrm{Spf}}\nolimits{\mathbb{Z}}_p)^{\mathlarger{\mathbbl{\Delta}}}$
  6. Definition of $\Sigma =(\mathop{\mathrm{Spf}}\nolimits{\mathbb{Z}}_p)^{\mathlarger{\mathbbl{\Delta}}}$
  7. $S$-points of $\Sigma$
  8. On the language of $W_S$-modules
  9. Cones
  10. Cone of a morphism of abelian groups
  11. Cone of a morphism of commutative group schemes
  12. Quasi-ideals
  13. Cone of a quasi-ideal
  14. Where to find details
  15. Outline of the construction of the prismatization functor
  16. ...and 215 more sections

Key Result

Lemma 2.4.4

Let $\pi :S'\to S$ be a faithfully flat quasi-compact morphism of schemes. Let $Y'$ be an $S'$-scheme equipped with a descent datum with respect to $\pi$. Suppose that $Y'$ is a (possibly infinite) disjoint union of schemes affine over $S'$. Then $Y'$ descends to a scheme $Y$ over $S$.

Theorems & Definitions (234)

  • Definition 2.3.1
  • Remark 2.3.2
  • Remark 2.3.3
  • Definition 2.3.4
  • Definition 2.3.5
  • Remark 2.3.6
  • Remark 2.3.7
  • Example 2.3.8
  • Definition 2.4.1
  • Remark 2.4.2
  • ...and 224 more