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A stacky approach to prismatic crystals via $q$-prism charts

Zeyu Liu

TL;DR

This work develops a stacky, $q$-prism-based framework to classify prismatic crystals by realizing them as quasi-coherent complexes on prismatizations of $p$-adic schemes. Central to the approach is the construction of $q$-connections (and $q$-Higgs derivations) on pullbacks to $q$-prisms, yielding a derived-equivalence between $\mathcal{D}(X_n^{\Delta})$ and modules over noncommutative Ore extensions $A/d^n[\partial; \gamma_A, \partial_A]$, subject to completeness and nilpotence conditions. The authors extend this to relative prismatizations and LCIs, obtain Galois-descent results to Cartier–Witt, and use these to compute cohomology on the Cartier–Witt stack and the absolute prismatic site of $\mathbb{Z}_p$. The framework unifies and extends HT, Sen-operator, and Frobenius data in a single stacky, noncommutative algebraic setting, and provides explicit, computable descriptions of prismatic crystals across absolute and relative contexts with broad applications in $p$-adic Hodge theory and related K-theoretic questions.

Abstract

Let $Y$ be a locally complete intersection over $\mathcal{O}_K$ containing a $p$-power root of unity $ζ_p$. We classify the derived category of prismatic crystals on the absolute prismatic site of $Y$ by studying quasi-coherent complexes on the prismatization of $Y$ via $q$-prism charts. We also develop a Galois descent mechanism to remove the assumption on $\mathcal{O}_K$. As an application, we classify quasi-coherent complexes on the Cartier-Witt stack and give a purely algebraic calculation of the cohomology of the structure sheaf on the absolute prismatic site of $\mathbb{Z}_p$. Along the way, for $Y$ a locally complete intersection over $\overline{A}$ with $A$ lying over a $q$-prism, we classify quasi-coherent complexes on the relative prismatization of $Y$.

A stacky approach to prismatic crystals via $q$-prism charts

TL;DR

This work develops a stacky, -prism-based framework to classify prismatic crystals by realizing them as quasi-coherent complexes on prismatizations of -adic schemes. Central to the approach is the construction of -connections (and -Higgs derivations) on pullbacks to -prisms, yielding a derived-equivalence between and modules over noncommutative Ore extensions , subject to completeness and nilpotence conditions. The authors extend this to relative prismatizations and LCIs, obtain Galois-descent results to Cartier–Witt, and use these to compute cohomology on the Cartier–Witt stack and the absolute prismatic site of . The framework unifies and extends HT, Sen-operator, and Frobenius data in a single stacky, noncommutative algebraic setting, and provides explicit, computable descriptions of prismatic crystals across absolute and relative contexts with broad applications in -adic Hodge theory and related K-theoretic questions.

Abstract

Let be a locally complete intersection over containing a -power root of unity . We classify the derived category of prismatic crystals on the absolute prismatic site of by studying quasi-coherent complexes on the prismatization of via -prism charts. We also develop a Galois descent mechanism to remove the assumption on . As an application, we classify quasi-coherent complexes on the Cartier-Witt stack and give a purely algebraic calculation of the cohomology of the structure sheaf on the absolute prismatic site of . Along the way, for a locally complete intersection over with lying over a -prism, we classify quasi-coherent complexes on the relative prismatization of .

Paper Structure

This paper contains 14 sections, 86 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Recollections on the prismatization functor
  3. $q$-connections on the prismatization of the absolute cyclotomic ring
  4. Construction of the $q$-connection after pulling back to the $q$-prism
  5. $q$-connections on the Hodge-Tate locus
  6. $q$-connections beyond the Hodge-Tate locus
  7. Comparison with the $q$-de Rham prism
  8. Applications to quasi-coherent complexes on the Cartier-Witt stack
  9. Complexes on the relative prismatization over a $q$-prism base
  10. Local computation
  11. Locally complete intersection case
  12. On the absolute prismatization
  13. Smooth case
  14. Locally complete intersection case

Key Result

Theorem 1.0.2

Assume that $p>2$ or $\alpha>0$. With the preceding notations, for $n\in \mathbb{N}\cup \{\infty\}$, the pullback along the covering $\rho: \mathrm{Spf}(A/d^n)\to X_n^{{\mathlarger{\mathbbl{\Delta}}}}$By abuse of notation, when $n=\infty$, this just means $\rho: \mathrm{Spf}(A)\to X^{{\mathlarger{\m whose essential image consists of those objects $M\in \mathcal{D}(A/d^n[\partial; \gamma_A, \partia

Theorems & Definitions (209)

  • Theorem 1.0.2: \ref{['thmt.main classification']}. Quasi-coherent complexes on the prismatization of cyclotomic rings
  • Remark 1.0.3
  • Remark 1.0.4
  • Remark 1.0.5
  • Theorem 1.0.6: \ref{['thmt.main classification II']}
  • Remark 1.0.7: The relation between \ref{['into thmt.main classification II']} with \ref{['intro.main thm1']}
  • Theorem 1.0.8: \ref{['thm. descent for the cartier witt stack to q prism']}
  • Corollary 1.0.9: \ref{['cor. main cor 1']}
  • Remark 1.0.10
  • Proposition 1.0.11: \ref{['prop. cohomology of Zp']}
  • ...and 199 more