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arxiv: 2101.10262 · v3 · pith:WYTXJRQXnew · submitted 2021-01-25 · 🧮 math.AG · math.AT· math.KT

Filtered formal groups, Cartier duality, and derived algebraic geometry

Tasos Moulinos This is my paper

Pith reviewed 2026-05-24 13:50 UTC · model grok-4.3

classification 🧮 math.AG math.ATmath.KT
keywords filtered formal groupsCartier dualitydeformation to the normal conederived algebraic geometryHochschild homologyformal groupsadic filtration
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The pith

A deformation to the normal cone in derived algebraic geometry produces a G_m-equivariant degeneration of a formal group to its tangent Lie algebra, identified with the adic filtration by a unicity theorem on complete filtrations.

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

The paper introduces formal groups equipped with filtrations and a duality relating them to filtered Hopf algebras. It constructs a degeneration of a formal group to its tangent Lie algebra using the deformation to the normal cone applied at the unit section. A unicity result is established for complete filtrations, showing that the filtration induced on the coordinate algebra of the degeneration coincides with the adic filtration of the original formal group. This identification is applied to recover the filtration on the filtered circle and to examine lifts of Hochschild homology invariants into spectral algebraic geometry.

Core claim

The deformation to the normal cone construction, when applied to the unit section of a formal groupwidehat{G}, yields a G_m-equivariant degeneration ofwidehat{G} to its tangent Lie algebra. There is a unicity result on complete filtrations that identifies the resulting filtration on the coordinate algebra of this deformation with the adic filtration on the coordinate algebra ofwidehat{G}. In a special case this recovers the filtration on the filtered circle, and the construction extends to studywidehat{G}-Hochschild homology and its lifts to spectral algebraic geometry.

What carries the argument

The deformation to the normal cone construction applied to the unit section of a formal group, which produces the G_m-equivariant degeneration to the tangent Lie algebra and carries the filtration data.

If this is right

  • The filtration on the coordinate algebra of the degeneration is the adic filtration of the formal group.
  • The filtration on the filtered circle is recovered as a special case of the construction.
  • Properties ofwidehat{G}-Hochschild homology can be investigated using the filtered setting.
  • These invariants admit lifts from derived to spectral algebraic geometry.

Where Pith is reading between the lines

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

  • The unicity of filtrations may allow similar degenerations to be defined for other geometric objects equipped with group structures.
  • The duality between filtered formal groups and filtered Hopf algebras could extend to produce new invariants in non-commutative or higher categorical settings.
  • The G_m-equivariance of the degeneration suggests compatibility with circle actions in related contexts such as equivariant homotopy theory.

Load-bearing premise

The deformation to the normal cone construction in derived algebraic geometry, when applied to the unit section of a formal group, provides a G_m-equivariant degeneration of the group to its tangent Lie algebra.

What would settle it

An explicit formal group where the degeneration to the normal cone fails to be G_m-equivariant, or where two distinct complete filtrations on the coordinate algebra agree after degeneration but differ on the adic filtration of the original group.

read the original abstract

We develop a notion of formal groups in the filtered setting and describe a duality relating these to a specified class of filtered Hopf algebras. We then study a deformation to the normal cone construction in the setting of derived algebraic geometry. Applied to the unit section of a formal group $\widehat{\mathbb{G}}$, this provides a $\mathbb{G}_m$-equivariant degeneration of $\widehat{\mathbb{G}}$ to its tangent Lie algebra. We prove a unicity result on complete filtrations, which, in particular, identifies the resulting filtration on the coordinate algebra of this deformation with the adic filtration on the coordinate algebra of $\widehat{\mathbb{G}}$. We use this in a special case, together with the aforementioned notion of Cartier duality, to recover the filtration on the filtered circle of [MRT19]. Finally, we investigate some properties of $\widehat{\mathbb{G}}$-Hochschild homology set out in loc. cit., and describe "lifts" of these invariants to the setting of spectral algebraic geometry.

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 a notion of filtered formal groups together with a Cartier duality relating them to a class of filtered Hopf algebras. It studies a deformation-to-the-normal-cone construction in derived algebraic geometry applied to the unit section of a formal group Ĝ, producing a Gm-equivariant degeneration of Ĝ to its tangent Lie algebra. A unicity theorem for complete filtrations is proved that identifies the induced filtration on the coordinate algebra of the deformation with the adic filtration on the coordinate algebra of Ĝ. In a special case this is combined with the duality to recover the filtration on the filtered circle of MRT19. The paper also examines properties of Ĝ-Hochschild homology from loc. cit. and describes lifts of these invariants to spectral algebraic geometry.

Significance. If the central claims hold, the work supplies an independent, self-contained framework for filtered formal groups in derived algebraic geometry and a parameter-free unicity result that recovers a known filtration without additional data. The deformation construction and its application to Hochschild homology invariants provide concrete tools that connect filtered derived geometry with Cartier duality and spectral methods. These features are genuine strengths of the manuscript.

minor comments (3)
  1. [Abstract] The phrase 'in a special case' for the recovery of the MRT19 filtration appears in the abstract and introduction; stating the precise hypotheses of that case at the first mention would improve readability.
  2. [§2] Notation for the filtered Hopf algebras and their duality is introduced in §2; a short table summarizing the correspondence between filtered formal groups and the dual objects would aid navigation.
  3. [§3] The statement of the unicity result (Theorem 3.12) refers to 'complete filtrations' without an explicit cross-reference to the definition of completeness given earlier in the section; adding the reference would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper develops independent notions of filtered formal groups and filtered Hopf algebras, proves a unicity result for complete filtrations on the deformation to the normal cone applied to the unit section of a formal group, and uses this to recover a special case from prior work [MRT19]. No equations or definitions reduce the central claims (unicity theorem, duality, Hochschild homology lifts) to self-referential inputs, fitted parameters renamed as predictions, or load-bearing self-citations whose validity depends on the present paper. The derivation chain is self-contained against external benchmarks in derived algebraic geometry and Cartier duality.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The constructions rest on standard background results in algebraic geometry and derived categories; the new objects are defined rather than postulated without evidence.

axioms (1)
  • standard math Standard properties of formal groups, Hopf algebras, and the deformation to the normal cone in derived algebraic geometry
    The paper invokes established frameworks from the literature to define and apply the new filtered notions.
invented entities (1)
  • Filtered formal group no independent evidence
    purpose: To extend classical formal groups to a filtered setting with an accompanying duality
    New mathematical object introduced and studied in the paper

pith-pipeline@v0.9.0 · 5698 in / 1208 out tokens · 31099 ms · 2026-05-24T13:50:59.529357+00:00 · methodology

discussion (0)

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Reference graph

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