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arxiv: 2512.22614 · v2 · submitted 2025-12-27 · 🧮 math.NT · math.AG

Lichtenbaum-van Hamel duality for singular varieties over p-adic fields

Felipe Rivera-Mesas This is my paper

Pith reviewed 2026-05-16 19:20 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords Lichtenbaum dualityalgebraic Brauer groupp-adic fieldstruncated homologysingular varietiesperfect pairingprofinite completion
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The pith

There exists a natural continuous perfect pairing between the algebraic Brauer group Br_1(X) and the profinite completion of truncated homology H_0(X,ℤ)_τ for proper geometrically integral varieties over p-adic fields.

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

The paper extends the van Hamel-Lichtenbaum duality theorem from smooth varieties to the singular case. It shows that for any proper geometrically integral variety X over a p-adic field k the algebraic Brauer group pairs perfectly and continuously with the profinite completion of a certain zeroth truncated homology group. The construction uses the structure morphism to define the homology via a Hom in the derived category of sheaves. A reader cares because the result removes the smoothness hypothesis that limited earlier versions of the duality, allowing the same arithmetic invariants to be compared on a wider class of varieties.

Core claim

For a proper geometrically integral variety X over a p-adic field k there exists a natural continuous perfect pairing Br_1(X) × H_0(X,ℤ)_τ^∧ → ℚ/ℤ, where Br_1(X) is the kernel of Br(X) to Br of the geometric base change and H_0(X,ℤ)_τ is the group Hom_{D(k_sm)}(τ_≤1 Rφ_* 𝔾_{m,X}, 𝔾_{m,k}) with φ the structure morphism.

What carries the argument

The truncated homology group H_0(X,ℤ)_τ obtained as the Hom in the derived category from the truncation of the pushforward of the multiplicative group sheaf to the base multiplicative group, which supplies the homological side of the pairing.

If this is right

  • Duality statements now apply to varieties with singularities without extra resolution assumptions.
  • The algebraic Brauer group of such varieties can be recovered from homological data up to profinite completion.
  • The pairing respects the natural topologies and is functorial with respect to morphisms of varieties.

Where Pith is reading between the lines

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

  • The result may simplify calculations of Brauer-Manin obstructions when the variety is singular.
  • One could test whether the same truncation construction yields duality after base change to finite extensions of the p-adic field.
  • Combining the pairing with known vanishing results for higher homology might yield new finiteness statements for Brauer groups.

Load-bearing premise

The variety X must be proper and geometrically integral over the p-adic field so that the structure morphism defines a well-behaved truncated homology group in the derived category.

What would settle it

A specific proper geometrically integral singular variety over a p-adic field where the map induced by the pairing fails to be bijective or continuous after profinite completion would disprove the claim.

read the original abstract

In this article, we extend the van Hamel-Lichtenbaum duality theorem to (not necessarily smooth) proper and geometrically integral varieties defined over a $p$-adic field $k$. More precisely, we prove that for such variety $X$ there exists a natural continuous perfect pairing \[ \mathrm{Br}_1(X)\times H_0(X,\mathbb{Z})_\tau^{\wedge} \to \mathbb{Q}/\mathbb{Z}, \] where $\mathrm{Br}_1(X):=\ker(\mathrm{Br}(X)\to\mathrm{Br}(\overline{X}))$ is the algebraic Brauer group of $X$, $H_0(X,\mathbb{Z})_\tau$ is the zeroth group of truncated homology $\mathrm{Hom}_{D(k_{\mathrm{sm}})}(\tau_{\leq 1}R\phi_*\mathbb{G}_{m,X},\mathbb{G}_{m,k})$, $\phi$ is the structure morphism of $X$, and $(-)^{\wedge}$ is the profinite completion functor.

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

2 major / 2 minor

Summary. The paper extends the Lichtenbaum-van Hamel duality theorem to proper geometrically integral (possibly singular) varieties X over a p-adic field k. It proves the existence of a natural continuous perfect pairing Br_1(X) × H_0(X,ℤ)_τ^∧ → ℚ/ℤ, where Br_1(X) is the kernel of Br(X) → Br(X̄), and H_0(X,ℤ)_τ is defined as Hom_{D(k_sm)}(τ_≤1 Rφ_* ℂ_m,X, ℂ_m,k) with φ the structure morphism and (−)^∧ the profinite completion.

Significance. If the central claim holds, the result is significant for arithmetic geometry: it removes the smoothness hypothesis from a key duality between algebraic Brauer groups and truncated homology over p-adics, thereby extending the range of varieties to which Lichtenbaum-type pairings apply. The construction is presented as natural and built from standard derived-category operations on the multiplicative sheaf.

major comments (2)
  1. [§4] §4 (proof of the perfect pairing): the argument that τ_≤1 Rφ_* ℂ_m,X yields a group whose profinite completion is Pontryagin dual to Br_1(X) must explicitly control the higher direct images R^i φ_* ℂ_m,X for i ≥ 2 that are supported on the singular locus; without a separate vanishing or exact-sequence argument showing these do not affect the Hom in degree 0, the perfectness claim for singular X remains unverified.
  2. [Definition of H_0] Definition of H_0(X,ℤ)_τ (immediately after the statement of the main theorem): the truncation τ_≤1 is applied to Rφ_* ℂ_m,X, but the manuscript does not supply a computation or spectral-sequence argument confirming that the resulting Hom group coincides with the expected H_0 when X is singular; this step is load-bearing for the duality statement.
minor comments (2)
  1. [Notation] The notation ℂ_m,X versus G_m,X is used interchangeably; adopt a single convention throughout.
  2. [Introduction] Add a short comparison paragraph in the introduction recalling the precise statement of the original Lichtenbaum-van Hamel theorem for smooth X.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. The comments highlight places where the exposition on higher direct images and the truncation for singular varieties can be strengthened. We will revise the manuscript to include the requested explicit arguments while preserving the core claims.

read point-by-point responses

  1. Referee: [§4] §4 (proof of the perfect pairing): the argument that τ_≤1 Rφ_* ℂ_m,X yields a group whose profinite completion is Pontryagin dual to Br_1(X) must explicitly control the higher direct images R^i φ_* ℂ_m,X for i ≥ 2 that are supported on the singular locus; without a separate vanishing or exact-sequence argument showing these do not affect the Hom in degree 0, the perfectness claim for singular X remains unverified.

    Authors: We agree that an explicit control of the higher direct images R^i φ_* ℂ_m,X (i ≥ 2) is necessary to confirm they do not affect the degree-0 Hom. The original argument implicitly uses that these sheaves are supported on the singular locus and vanish under the relevant Hom functor after truncation, but this was not stated as a separate lemma. In the revised manuscript we will insert a new lemma in §4 that provides a short exact sequence (or spectral-sequence fragment) isolating the contribution of the smooth locus and showing that the higher images contribute nothing to Hom_{D(k_sm)}(τ_≤1 Rφ_* ℂ_m,X, ℂ_m,k). This will make the perfectness claim fully rigorous for singular X. revision: yes

  2. Referee: [Definition of H_0] Definition of H_0(X,ℤ)_τ (immediately after the statement of the main theorem): the truncation τ_≤1 is applied to Rφ_* ℂ_m,X, but the manuscript does not supply a computation or spectral-sequence argument confirming that the resulting Hom group coincides with the expected H_0 when X is singular; this step is load-bearing for the duality statement.

    Authors: We accept that a direct verification of the identification Hom_{D(k_sm)}(τ_≤1 Rφ_* ℂ_m,X, ℂ_m,k) ≃ H_0(X,ℤ) for singular X is required. The manuscript relies on the standard properties of the truncation functor and the fact that higher cohomology sheaves are supported on the singular locus, but no explicit spectral-sequence computation was included. We will add a short paragraph (or appendix computation) immediately after the definition that uses the Leray spectral sequence for the structure morphism and shows that the truncation indeed recovers the expected zeroth homology group even when X is singular. This will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained via explicit definitions and prior duality

full rationale

The paper defines H_0(X,ℤ)_τ explicitly as Hom_{D(k_sm)}(τ_≤1 Rφ_* ℂ_m,X, ℂ_m,k) and states the perfect pairing as a theorem extending the known Lichtenbaum-van Hamel result to singular proper geometrically integral X. No equation reduces a claimed prediction to a fitted input by construction, no load-bearing uniqueness theorem is imported solely via self-citation, and the central pairing is not renamed from a known empirical pattern. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard facts from étale cohomology and Brauer groups over p-adic fields together with the definition of truncated homology; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of the Brauer group Br(X) and its kernel Br_1(X) = ker(Br(X) → Br(X̄)) over p-adic fields
    Invoked directly in the definition of the left side of the pairing.
  • domain assumption Existence and properties of the truncated homology H_0(X,ℤ)_τ defined via Hom in D(k_sm) of τ_≤1 Rφ_* G_m,X with G_m,k
    This is the right-hand object in the pairing and is part of the setup for the duality.

pith-pipeline@v0.9.0 · 5469 in / 1456 out tokens · 68223 ms · 2026-05-16T19:20:16.582015+00:00 · methodology

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Reference graph

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