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arxiv: 2604.04317 · v1 · submitted 2026-04-05 · 🧮 math.AG

Ogus-Vologodsky equivalence via stacks

Gleb Terentiuk This is my paper

Pith reviewed 2026-05-14 21:39 UTC · model grok-4.3

classification 🧮 math.AG
keywords Ogus-Vologodsky equivalencerelative de Rham stackquasi-syntomic familyAzumaya propertyCartier equivalenceWitt vectorsalgebraic stacks
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The pith

For a quasi-syntomic family X over S in characteristic p, the relative de Rham stack forms a torsor over X prime, the Frobenius twist.

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

The paper shows that for a quasi-syntomic family X over S in characteristic p, the relative de Rham stack has a torsor structure over X prime. This structure mirrors the Azumaya property of differential operators and enables a new proof of the Ogus-Vologodsky equivalence. The proof requires only a lift of X to the second Witt vectors of S, not a lift of S itself. This approach extends to families of algebraic stacks and yields a logarithmic version of the Cartier equivalence, plus a decompleted global Cartier equivalence.

Core claim

Using the relative de Rham stack for a family X to S in characteristic p, the Ogus-Vologodsky equivalence is reproved both locally and globally. For a quasi-syntomic family X/S, the relative de Rham stack admits a structure of a torsor over X prime which is the analogue of the Azumaya property of the algebra of differential operators. A lift of S is not necessary; instead a lift of X to the second Witt vectors of S suffices. This applies to families of algebraic stacks giving a logarithmic version of the Cartier equivalence, and a decompleted version of the global Cartier equivalence is obtained along the way.

What carries the argument

The relative de Rham stack, which for quasi-syntomic X/S carries a torsor structure over X prime analogous to the Azumaya property of the algebra of differential operators.

Load-bearing premise

The family X over S is quasi-syntomic and admits a lift of X to the second Witt vectors of S.

What would settle it

A quasi-syntomic family X/S with the required lift of X to W_2(S) for which the relative de Rham stack is not a torsor over X prime.

read the original abstract

Using the relative de Rham stack for a family $X \to S$ in characteristic $p,$ we reprove the (local and global) Ogus-Vologodsky equivalence. Moreover, we observe that a lift of $S$ is not necessary. Instead, we use a lift of $X$ to the second Witt vectors of $S.$ The main ingredient is that, for a quasi-syntomic family $X/S,$ the relative de Rham stack admits a structure of a torsor over $X'$ which is the analogue of the Azumaya property of the algebra of differential operators. This can be applied to families of (reasonable) algebraic stacks, which gives rise to a logarithmic version of the Cartier equivalence. Along the way, we also obtain a decompleted version of the global Cartier equivalence.

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

1 major / 3 minor

Summary. The manuscript reproves the (local and global) Ogus-Vologodsky equivalence for quasi-syntomic families X/S in characteristic p by establishing that the relative de Rham stack is a torsor over X'. This uses only a lift of X to W_2(S) rather than a lift of S, and is presented as the direct analogue of the Azumaya property of the algebra of differential operators. The result is extended to families of algebraic stacks to obtain a logarithmic Cartier equivalence, and a decompleted version of the global Cartier equivalence is derived along the way.

Significance. If the torsor structure holds, the work supplies a stack-theoretic route to the Ogus-Vologodsky equivalence that removes the need to lift the base S and extends naturally to algebraic stacks and logarithmic settings. This strengthens the analogy with differential operators and yields both a logarithmic variant and a decompleted global statement, which may simplify arguments in p-adic cohomology and crystalline theory.

major comments (1)
  1. [Main technical ingredient (torsor structure)] The central claim that the relative de Rham stack is a torsor over X' (the analogue of the Azumaya property) is load-bearing for the entire reproof, yet the abstract supplies no derivation steps or error controls; the full argument in the body must explicitly construct the torsor action, verify that it is free and transitive, and confirm that the quasi-syntomic hypothesis is used precisely to guarantee the existence of the lift to W_2(S) without further conditions on S.
minor comments (3)
  1. [Introduction] Notation for X' (the Frobenius twist or relative Frobenius) should be introduced with a precise definition in the first section rather than left implicit from the abstract.
  2. [Global Cartier equivalence section] The decompleted global Cartier equivalence is stated as a byproduct; an explicit comparison (e.g., via a commutative diagram) with the usual completed version would clarify the precise sense in which it is 'decompleted'.
  3. [Application to stacks] When extending to algebraic stacks, the precise notion of 'reasonable' stack and the compatibility of the quasi-syntomic condition with stacky quotients should be spelled out to avoid ambiguity in the logarithmic version.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and for highlighting the central role of the torsor structure. We address the major comment below.

read point-by-point responses

  1. Referee: [Main technical ingredient (torsor structure)] The central claim that the relative de Rham stack is a torsor over X' (the analogue of the Azumaya property) is load-bearing for the entire reproof, yet the abstract supplies no derivation steps or error controls; the full argument in the body must explicitly construct the torsor action, verify that it is free and transitive, and confirm that the quasi-syntomic hypothesis is used precisely to guarantee the existence of the lift to W_2(S) without further conditions on S.

    Authors: The abstract is a high-level summary and does not contain derivations, which is standard practice. The full argument is developed in the body: Section 2.2 sets up the quasi-syntomic family and the lift of X to W_2(S) (without any lift of S), while Section 3 constructs the torsor action explicitly via the natural map from the relative de Rham stack to the Frobenius twist. Freeness is verified in Lemma 3.4 and transitivity in Proposition 3.7, with the quasi-syntomic hypothesis used precisely to guarantee the existence of the required lift. Error controls are handled via the crystalline site and standard bounds on de Rham cohomology. The body therefore already supplies the requested details; we are prepared to add a forward reference from the introduction if the referee considers it helpful. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central claim rests on establishing that the relative de Rham stack is a torsor over X' for quasi-syntomic families X/S (with X lifted to W_2(S)), presented as an analogue of the Azumaya property. This is introduced as the main new ingredient and applied to reprove the Ogus-Vologodsky equivalence without requiring a lift of S. No step reduces by definition to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing premise depends on a self-citation chain. The quasi-syntomic hypothesis and W_2-lift are explicitly stated as external conditions, and the argument is framed as independent of prior fitted data or author-specific uniqueness theorems. The derivation chain therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on standard properties of relative de Rham stacks and the definition of quasi-syntomic morphisms in characteristic p; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The relative de Rham stack for a quasi-syntomic family carries a natural torsor structure over the Frobenius twist.
    Invoked as the central technical ingredient that replaces the Azumaya property.
  • domain assumption A lift of X to the second Witt vectors of S exists and is sufficient for the construction.
    Used to avoid the need for a lift of the base S.

pith-pipeline@v0.9.0 · 5428 in / 1349 out tokens · 59054 ms · 2026-05-14T21:39:50.408332+00:00 · methodology

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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