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arxiv: 2604.24986 · v1 · submitted 2026-04-27 · 🧮 math.AT · math.AC· math.GR

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Koszul modules, holonomy Lie algebras, and resonance of groups and CDGAs

Alexander I. Suciu

Pith reviewed 2026-05-07 17:11 UTC · model grok-4.3

classification 🧮 math.AT math.ACmath.GR
keywords Koszul modulesholonomy Lie algebrasresonance varietiesAlexander invariantsChen ranksCDGAsformalitytangent cone theorem
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The pith

The first Koszul module of a connected CDGA is isomorphic to the infinitesimal Alexander invariant of its holonomy Lie algebra.

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

The paper builds a framework that uses Koszul modules to relate classical Alexander-type invariants of groups to infinitesimal invariants extracted from commutative differential graded algebra models. It proves that for any connected CDGA with finite-dimensional first cohomology, the first Koszul module equals the infinitesimal Alexander invariant of the associated holonomy Lie algebra and supplies explicit formulas for the Chen ranks of that Lie algebra. A tangent cone theorem is shown for resonance varieties, establishing that cohomology determines their first-order local behavior at the origin. For finitely generated groups that possess 1-finite 1-models, the classical Alexander invariants coincide with the Koszul invariants after completion and passage to associated graded objects, so that Chen ranks are determined by the model.

Core claim

To a connected CDGA (A,d) with finite-dimensional H^1(A) we associate Koszul modules B_i(A) over the symmetric algebra on H_1(A). The first Koszul module B_1(A) is isomorphic to the infinitesimal Alexander invariant of the holonomy Lie algebra h(A), yielding explicit formulas for holonomy Chen ranks. A tangent cone theorem is established for resonance varieties. For finitely generated groups admitting 1-finite 1-models, classical Alexander invariants agree with Koszul invariants after completion and passage to associated graded objects, and Chen ranks are determined by the model.

What carries the argument

Koszul linearization, which replaces nonlinear equivariant constructions with functorial algebraic objects (the Koszul modules B_i(A)) defined directly from a CDGA.

If this is right

  • Explicit formulas for the Chen ranks of holonomy Lie algebras follow from the isomorphism with the first Koszul module.
  • The tangent cone theorem implies that the first-order behavior of resonance varieties at the origin is controlled by cohomology.
  • For groups admitting 1-finite 1-models, Chen ranks are completely determined by the CDGA model after completion and grading.
  • The framework yields Chen rank computations for 2-step nilpotent Lie algebras and pure elliptic braid groups.
  • Partial formality results for Sasakian manifolds and a general method for detecting non-formality of spaces, groups, and maps become available.

Where Pith is reading between the lines

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

  • The tangent cone result suggests that resonance varieties may admit higher-order approximations controlled by higher cohomology operations.
  • The agreement after associated graded objects raises the question of whether similar relations hold for other invariants such as Massey products or higher nilpotent quotients.
  • The non-formality detection method could be tested on additional classes of manifolds or maps that admit algebraic models but are suspected to be non-formal.
  • Extending the Koszul module construction beyond the first degree might produce formulas for higher Chen ranks or other infinitesimal invariants.

Load-bearing premise

The comparisons between classical and Koszul invariants require that the groups admit 1-finite 1-models and that the CDGA is connected with finite-dimensional H^1.

What would settle it

A counterexample would be a connected CDGA with finite-dimensional H^1 where B_1(A) fails to be isomorphic to the infinitesimal Alexander invariant of h(A), or a finitely generated group with a 1-finite 1-model whose completed Alexander invariants differ from the associated graded Koszul invariants.

read the original abstract

We develop a Koszul-theoretic framework for comparing classical Alexander-type invariants with infinitesimal invariants arising from finite-type commutative differential graded algebra models. The central mechanism is Koszul linearization, which replaces nonlinear equivariant constructions with functorial algebraic objects defined from a CDGA. To a connected CDGA $(A,d)$ with finite-dimensional $H^1(A)$ we associate Koszul modules $\mathcal{B}_i(A)$ over the symmetric algebra on $H_1(A)$. We prove that the first Koszul module $\mathcal{B}_1(A)$ is isomorphic to the infinitesimal Alexander invariant of the holonomy Lie algebra $\mathfrak{h}(A)$, yielding explicit formulas for holonomy Chen ranks. We establish a tangent cone theorem for resonance varieties, showing that cohomology controls their first-order behavior at the origin. For finitely generated groups admitting 1-finite 1-models, we prove that classical Alexander invariants agree with Koszul invariants after completion and passage to associated graded objects, and that Chen ranks are determined by the model. Applications include Chen rank computations for 2-step nilpotent Lie algebras and pure elliptic braid groups, partial formality results for Sasakian manifolds, and a general framework for detecting non-formality of spaces, groups, and maps.

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 manuscript develops a Koszul-theoretic framework for comparing classical Alexander-type invariants with infinitesimal invariants from finite-type CDGA models. To a connected CDGA (A, d) with finite-dimensional H^1(A), it associates Koszul modules B_i(A) over the symmetric algebra on H_1(A). The central result is that the first Koszul module B_1(A) is isomorphic to the infinitesimal Alexander invariant of the holonomy Lie algebra h(A), which yields explicit formulas for holonomy Chen ranks. It also establishes a tangent cone theorem for resonance varieties and, for finitely generated groups admitting 1-finite 1-models, proves agreement between classical and Koszul invariants after completion and passage to associated graded objects, with applications to Chen rank computations, partial formality results, and detecting non-formality.

Significance. If the main isomorphism and theorems hold, this framework provides a functorial algebraic method to compute and relate invariants that are otherwise difficult to access, particularly Chen ranks via holonomy Lie algebras. The tangent cone theorem links resonance varieties to cohomology in a first-order sense, and the agreement results for groups with 1-finite 1-models bridge classical and model-based approaches. This could enable new computations for examples like 2-step nilpotent Lie algebras and pure elliptic braid groups, and offer a general tool for non-formality detection in spaces, groups, and maps. The approach appears to avoid ad-hoc parameters and relies on standard CDGA data.

minor comments (2)
  1. [Abstract] The abstract mentions 'Koszul linearization' as the central mechanism; a brief definition or reference to its introduction in the text would aid readers.
  2. [Applications] The applications to Sasakian manifolds and non-formality detection are mentioned; including at least one concrete computation or example would strengthen the presentation of the results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, for the accurate summary of our Koszul-theoretic framework relating Alexander-type invariants to infinitesimal invariants from CDGA models, and for the positive assessment of its significance and potential applications. We are pleased with the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines Koszul modules B_i(A) functorially from the CDGA data (A,d) with finite-dimensional H^1(A), then proves an isomorphism B_1(A) ≅ infinitesimal Alexander invariant of h(A) as a theorem, not by construction or renaming. The tangent cone theorem for resonance varieties and the agreement of classical/Koszul invariants after completion for groups with 1-finite 1-models are derived under explicitly stated hypotheses, without fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations that reduce the central claims to inputs. The framework introduces new algebraic objects and derives independent relations from them.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard homological algebra for CDGAs and Lie algebras together with the new definition of Koszul modules; no free parameters or externally postulated entities are introduced.

axioms (1)
  • domain assumption A connected CDGA with finite-dimensional H^1(A) admits a well-defined holonomy Lie algebra and Koszul modules over Sym(H_1(A)).
    This is the explicit setup stated for associating the modules B_i(A).

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