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arxiv: 2606.10573 · v1 · pith:EABGH4XPnew · submitted 2026-06-09 · 🧮 math.AG · math.NT

The convergent stack

Marco D'Addezio This is my paper

Pith reviewed 2026-06-27 11:54 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords convergent stackconvergent isocrystalspositive characteristicquasi-coherent modulesp-adic formal schemesf-semiperfect schemescohomology
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The pith

A convergent stack attached to any scheme over F_p equates its finitely generated quasi-coherent O[1/p]-modules with convergent isocrystals when the scheme is finite type over a perfect field.

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

The paper constructs a convergent stack X_conv for every scheme X over the finite field F_p. This stack is designed so that its quasi-coherent sheaves capture convergent isocrystals and their cohomology in positive characteristic. The central result states that, when X is finite type over a perfect field, the finitely generated quasi-coherent O[1/p]-modules on X_conv are equivalent to the category of convergent isocrystals on X, and the equivalence preserves cohomology groups. Additional explicit presentations are given when X embeds into a smooth p-adic formal scheme or when X is f-semiperfect. The construction is motivated by earlier stacky models for de Rham and crystalline cohomology.

Core claim

To any scheme X over F_p the paper attaches a convergent stack X_conv. When X is of finite type over a perfect field, its finitely generated quasi-coherent O[1/p]-modules are equivalent to convergent isocrystals over X, compatibly with cohomology. When X embeds into a smooth p-adic formal scheme, X_conv is described explicitly as the quotient of an open tube by a p-adic formal groupoid. For f-semiperfect schemes the convergent stack is representable by a preperfectoid adic space over Q_p.

What carries the argument

The convergent stack X_conv, whose finitely generated quasi-coherent O[1/p]-modules are defined to be the convergent isocrystals on X.

If this is right

  • The equivalence is compatible with cohomology, so convergent cohomology groups can be read off from the stack.
  • For schemes embedding into smooth p-adic formal schemes the stack is realized concretely as a quotient by a p-adic formal groupoid.
  • For f-semiperfect schemes the stack becomes representable by a preperfectoid adic space over Q_p.
  • The construction applies to arbitrary schemes over F_p, not only those satisfying the finite-type or embedding hypotheses.

Where Pith is reading between the lines

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

  • The stacky language may allow convergent isocrystals to be manipulated with the same formal tools used for quasi-coherent sheaves on ordinary schemes.
  • Because the construction is functorial in X, it could be used to define convergent cohomology for morphisms or families without choosing embeddings.

Load-bearing premise

A convergent stack X_conv can be attached to every scheme over F_p and satisfies the module equivalence and explicit descriptions under the listed conditions on X.

What would settle it

Take a specific smooth curve X of finite type over a perfect field of characteristic p, compute the category of convergent isocrystals on X by direct means, and compare it with the category of finitely generated quasi-coherent O[1/p]-modules on the constructed stack X_conv.

read the original abstract

Inspired by Simpson's de Rham stack and Drinfeld's crystalline stack, we develop a stacky approach to convergent cohomology and convergent isocrystals in positive characteristic. To any scheme $X$ over $\mathbb{F}_p$ we attach a convergent stack $X_{\mathrm{conv}}$. When $X$ is of finite type over a perfect field, its finitely generated quasi-coherent $\mathcal{O}[\tfrac1p]$-modules are equivalent to convergent isocrystals over $X$, compatibly with cohomology. When $X$ embeds into a smooth $p$-adic formal scheme, we describe $X_{\mathrm{conv}}$ explicitly as the quotient of an open tube by a $p$-adic formal groupoid. For $f$-semiperfect schemes, by contrast, the convergent stack is representable by a preperfectoid adic space over $\mathbb{Q}_p$.

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 / 2 minor

Summary. The paper constructs a convergent stack X_conv associated to any scheme X over F_p. For X of finite type over a perfect field, it claims an equivalence between the category of finitely generated quasi-coherent O[1/p]-modules on X and the category of convergent isocrystals over X, compatible with cohomology. Explicit descriptions are given in two further cases: when X embeds into a smooth p-adic formal scheme, X_conv is the quotient of an open tube by a p-adic formal groupoid; when X is f-semiperfect, X_conv is representable by a preperfectoid adic space over Q_p. The construction is inspired by Simpson's de Rham stack and Drinfeld's crystalline stack.

Significance. If the stated equivalences and explicit descriptions hold with the claimed compatibilities, the work supplies a new stack-theoretic language for convergent isocrystals and convergent cohomology in positive characteristic. This could streamline comparisons between different p-adic cohomology theories and provide a uniform framework for schemes over F_p. The explicit geometric presentations in the embedding and f-semiperfect cases are potentially useful for computations.

minor comments (2)
  1. The abstract states the main equivalences but does not indicate where the proofs appear; adding a sentence locating the central theorem (e.g., 'Theorem 4.2') would improve readability.
  2. Notation for the convergent stack is introduced as X_conv; ensure consistent use of mathrm throughout the text and in all diagrams.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript, for highlighting its potential significance, and for recommending minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs the convergent stack X_conv for any scheme X over F_p, building explicitly on external prior notions (Simpson's de Rham stack and Drinfeld's crystalline stack) and providing separate explicit descriptions for the three listed cases (finite type over perfect field, embedding into smooth p-adic formal scheme, f-semiperfect). The central equivalence between finitely generated quasi-coherent O[1/p]-modules and convergent isocrystals is presented as a consequence of this construction rather than a self-referential definition or fitted input renamed as prediction. No load-bearing self-citations, uniqueness theorems imported from the same authors, or ansatzes smuggled via prior work are indicated in the abstract or case descriptions, so the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Based solely on the abstract, the paper introduces the convergent stack as the central new object; no free parameters, background axioms, or additional invented entities are mentioned.

invented entities (1)
  • convergent stack X_conv no independent evidence
    purpose: To provide a stacky model for convergent cohomology and isocrystals on schemes over F_p
    The main new mathematical object defined in the paper and used to state the equivalences.

pith-pipeline@v0.9.1-grok · 5667 in / 1155 out tokens · 24153 ms · 2026-06-27T11:54:14.105254+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 1 linked inside Pith

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