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The prismatization of $p$-adic formal schemes

Bhargav Bhatt, Jacob Lurie

TL;DR

The paper develops a comprehensive stack-theoretic framework for p-adic prisms by introducing the Cartier-Witt prismatization WCart_X for bounded p-adic formal and derived schemes. It extends prisms to animated delta-rings, defines derived relative and absolute prismatizations, and analyzes the derived Hodge-Tate stack as a gerbe with deformation-theoretic control, linking these constructions to derived prismatic cohomology and crystals. It also provides explicit descriptions in regular settings, demonstrates comparisons with site-based prismatic cohomology, and raises conjectures about regularity criteria through the HT stack. Overall, the approach unifies prismatic and crystalline perspectives via a geometric, functorial prismatization, with significant implications for cohomology theories in mixed characteristic and deformation theory.

Abstract

In this note, we introduce and study the Cartier--Witt stack $\mathrm{WCart}_X$ attached to a $p$-adic formal scheme $X$ as well as some variants. In particular, we reinterpret the notion of prismatic crystals on $X$ and their cohomology in terms of quasicoherent sheaf theory on $\mathrm{WCart}_X$ in favorable situations.

The prismatization of $p$-adic formal schemes

TL;DR

The paper develops a comprehensive stack-theoretic framework for p-adic prisms by introducing the Cartier-Witt prismatization WCart_X for bounded p-adic formal and derived schemes. It extends prisms to animated delta-rings, defines derived relative and absolute prismatizations, and analyzes the derived Hodge-Tate stack as a gerbe with deformation-theoretic control, linking these constructions to derived prismatic cohomology and crystals. It also provides explicit descriptions in regular settings, demonstrates comparisons with site-based prismatic cohomology, and raises conjectures about regularity criteria through the HT stack. Overall, the approach unifies prismatic and crystalline perspectives via a geometric, functorial prismatization, with significant implications for cohomology theories in mixed characteristic and deformation theory.

Abstract

In this note, we introduce and study the Cartier--Witt stack attached to a -adic formal scheme as well as some variants. In particular, we reinterpret the notion of prismatic crystals on and their cohomology in terms of quasicoherent sheaf theory on in favorable situations.
Paper Structure (15 sections, 36 theorems, 124 equations)

This paper contains 15 sections, 36 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Animated prisms
  3. Absolute prismatization
  4. Revisiting the prismatic logarithm
  5. The relative prismatization
  6. Comparison with prismatic cohomology
  7. Derived relative prismatization
  8. Definitions
  9. The derived Hodge-Tate stack
  10. Affineness of $\mathop{\mathrm{WCart}}\nolimits_{X/A}$
  11. Comparison with derived prismatic cohomology
  12. Derived absolute prismatization
  13. Examples: The Hodge-Tate stack of some regular schemes
  14. Some questions on the Hodge-Tate stack and regularity
  15. Animated delta-rings

Key Result

Corollary 2.10

The $\infty$-category of animated prisms over a fixed animated prism $(A \to A/I)$ identifies with the $\infty$-category of $(p,I)$-complete animated $\delta$-$A$-algebras via the forgetful functor.

Theorems & Definitions (142)

  • Remark 1.1
  • Remark 1.2
  • Definition 2.4: Animated prisms
  • Example 2.6
  • Remark 2.7: Detecting the prismatic condition on $\pi_0(-)$
  • Remark 2.8: Base changing animated prism structures along $\delta$-maps
  • Remark 2.9: Rigidity of maps between animated prisms
  • Corollary 2.10
  • proof
  • Example 2.11: Cartier-Witt divisors and animated prism structures on $W(R)$
  • ...and 132 more