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arxiv: 2310.12770 · v2 · submitted 2023-10-19 · 🧮 math.AG · math.KT

Prismatic cohomology relative to δ-rings

Benjamin Antieau , Achim Krause , Thomas Nikolaus This is my paper

Pith reviewed 2026-05-24 06:42 UTC · model grok-4.3

classification 🧮 math.AG math.KT
keywords prismatic cohomologyδ-ringssyntomic cohomologyprismatic crystalsfiltered ringsalgebraic geometrysite-theoretic cohomology
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The pith

Relative prismatic cohomology depends only on the underlying δ-ring and not on any chosen prism structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs prismatic and syntomic cohomology relative to an arbitrary δ-ring. This construction simultaneously extends the absolute prismatic cohomology and the relative version previously defined only over prisms. It establishes that the relative theory is independent of the extra prism data and is determined solely by the δ-ring. Three candidate definitions are supplied—one via a site, one via prismatic crystals, and one via stacks—and shown to agree under mild syntomicity conditions on the base. The new setup makes the prismatic cohomology of filtered rings arise directly as a special case.

Core claim

Prismatic cohomology relative to a δ-ring generalizes both the absolute and relative theories of Bhatt-Scholze; the relative theory, originally defined with respect to a prism, depends only on the underlying δ-ring. Site-theoretic, crystal-theoretic, and stack-theoretic definitions are introduced and proved equivalent when the base satisfies mild syntomicity hypotheses. As a direct consequence, the prismatic cohomology of filtered rings appears naturally inside the relative theory.

What carries the argument

prismatic cohomology relative to a δ-ring, defined equivalently by a site, by prismatic crystals, or by a stack

If this is right

  • The relative theory now applies to any δ-ring without requiring the existence of a prism lift.
  • Syntomic cohomology acquires a relative version defined over an arbitrary δ-ring.
  • Prismatic cohomology of filtered rings is recovered as the special case of the relative theory over a filtered δ-ring.
  • Absolute prismatic cohomology appears as the case relative to the initial δ-ring.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Computations that previously required choosing a prism can now be performed directly on the δ-ring.
  • The independence result may simplify descent arguments or base-change statements that mix different prisms.
  • The stack-theoretic definition suggests a natural extension to derived or higher-categorical settings over δ-rings.

Load-bearing premise

The three proposed definitions of relative prismatic cohomology agree whenever the base satisfies mild syntomicity hypotheses.

What would settle it

An explicit δ-ring equipped with two distinct prism structures whose associated relative prismatic cohomologies differ on some test object would show that the theory is not independent of the prism.

read the original abstract

We develop prismatic and syntomic cohomology relative to a $\delta$-ring. This simultaneously generalizes Bhatt and Scholze's absolute and relative prismatic cohomology and shows that the latter, which was defined relative to a prism, is in fact independent of the prism structure and only depends on the underlying $\delta$-ring. We give several possible definitions of our new version of prismatic cohomology: a site theoretic definition, one using prismatic crystals, and a stack theoretic definition. These are equivalent under mild syntomicity hypotheses. As an application, we note how the theory of prismatic cohomology of filtered rings arises naturally in this context.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper develops prismatic and syntomic cohomology relative to an arbitrary δ-ring. It generalizes both the absolute prismatic cohomology and the relative version of Bhatt-Scholze (originally defined relative to a prism), and proves that the latter depends only on the underlying δ-ring rather than the choice of prism. Three definitions are supplied—a site-theoretic one, one via prismatic crystals, and a stack-theoretic one—and shown to be equivalent under mild syntomicity hypotheses on the base. An application to the prismatic cohomology of filtered rings is indicated.

Significance. If the stated equivalences and independence hold, the work supplies a more intrinsic and flexible foundation for prismatic cohomology that removes an auxiliary choice of prism. This has the potential to simplify many constructions in p-adic cohomology and arithmetic geometry while preserving compatibility with existing Bhatt-Scholze theory. The provision of three independent but equivalent definitions adds robustness; the explicit reduction showing that the prism-relative theory factors through the δ-ring is a concrete technical contribution.

minor comments (3)
  1. The abstract refers to 'mild syntomicity hypotheses' without listing them; the main theorem statements should record the precise conditions (e.g., which morphisms are required to be syntomic or quasi-syntomic) so that readers can immediately check applicability.
  2. The application to filtered rings is described only as 'noted'; if this is intended as a substantive illustration, a short dedicated subsection with at least one concrete computation or compatibility statement would strengthen the manuscript.
  3. Notation for the three definitions (site-theoretic, crystal, stack) should be introduced with consistent symbols or labels early in the text so that later comparisons are easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines prismatic cohomology relative to an arbitrary δ-ring via three independent constructions (site-theoretic, prismatic crystals, stack-theoretic) and proves their equivalence under syntomicity hypotheses as theorems. It then shows that the Bhatt-Scholze prism-relative theory factors through the underlying δ-ring, again as an explicit reduction theorem rather than by redefinition or fitting. No step reduces a claimed prediction or uniqueness result to a self-citation chain, ansatz smuggled via prior work, or input parameter; the central independence claim is established externally to the definitions themselves. Self-citations, if present, are not load-bearing for the main results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard axioms of commutative algebra, algebraic geometry, and the existing theory of prisms and δ-rings from prior literature; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard axioms of commutative rings and δ-structures as in prior δ-ring literature
    Invoked throughout the definitions of relative prismatic cohomology
  • domain assumption Mild syntomicity hypotheses suffice for equivalence of the three definitions
    Stated explicitly as the condition under which site, crystal, and stack versions agree

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Syntomification and crystalline local systems

    math.NT 2025-10 unverdicted novelty 5.0

    Equivalence of reflexive sheaves on syntomic stack X^Syn with Z_p-lattices in crystalline local systems on generic fiber X_η, plus results on etale realization and filtered F-isocrystals for proper smooth X.

Reference graph

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30 extracted references · 30 canonical work pages · cited by 1 Pith paper

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