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arxiv: 2605.11105 · v2 · pith:RDRPNVAEnew · submitted 2026-05-11 · 🧮 math.AC

Derived complete intersections and polynomial growth of Betti numbers over dg-algebras

Michael K. Brown , Justin Lyle This is my paper

Pith reviewed 2026-05-20 22:28 UTC · model grok-4.3

classification 🧮 math.AC
keywords dg-algebrasBetti numberscomplete intersectionsminimal modelsacyclic closuresdeviationsGulliksen theorem
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The pith

A dg-algebra is a derived complete intersection if and only if its modules have polynomially growing Betti numbers.

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

This paper proves a structure theorem that characterizes derived complete intersections among dg-algebras by the polynomial growth of Betti numbers in their modules. It builds on Gulliksen's classical theorem for local rings by extending the result to the setting of differential graded algebras. The proof relies on establishing the existence and uniqueness of minimal models and acyclic closures for morphisms of dg-algebras in a more general context than before. Additionally, the authors extend a vanishing result for deviations due to Halperin, which allows the recovery of the original theorem as a special case. This provides a homological criterion for identifying derived complete intersections without direct reference to their defining properties.

Core claim

The central discovery is a derived version of Gulliksen's theorem: a dg-algebra has the property that the Betti numbers of its finitely generated modules grow polynomially if and only if it is a derived complete intersection. This is proven using the existence and uniqueness of minimal models and acyclic closures of morphisms in dg-algebras under broader conditions, and by extending Halperin's theorem on the vanishing of deviations to this setting.

What carries the argument

Minimal models and acyclic closures of morphisms between dg-algebras, which serve as tools to analyze the homological behavior and growth of Betti numbers.

If this is right

  • The classical Gulliksen theorem follows immediately as a corollary when restricting to ordinary local rings.
  • Halperin's theorem on the vanishing of deviations extends from local rings to dg-algebras.
  • Polynomial Betti growth provides a homological test for the derived complete intersection property.

Where Pith is reading between the lines

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

  • This characterization may help in classifying dg-algebras arising in derived algebraic geometry based on their resolution growth.
  • Similar growth conditions could be studied for other invariants like Tor or Ext modules over these algebras.

Load-bearing premise

The dg-algebras must satisfy standard technical conditions such as being defined over a field of characteristic zero with suitable grading for minimal models and deviations to be defined.

What would settle it

Construct a specific dg-algebra that is not a derived complete intersection yet has all its modules with polynomially growing Betti numbers, or find one that is a derived complete intersection but has modules with exponential Betti growth.

read the original abstract

A theorem of Gulliksen states that a local ring is a complete intersection if and only if the Betti numbers of its finitely generated modules grow polynomially. We prove a derived version of Gulliksen's Theorem. More precisely, we prove a structure theorem for dg-algebras whose modules exhibit polynomial Betti growth. As a key ingredient in the proof, we establish the existence and uniqueness of minimal models and acyclic closures of morphisms of dg-algebras in a broader setting than was previously known. We also extend to dg-algebras a theorem of Halperin on the vanishing of deviations of local rings, recovering Gulliksen's Theorem as an immediate consequence.

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 / 1 minor

Summary. The paper proves a derived version of Gulliksen's theorem: a structure theorem characterizing dg-algebras whose modules have polynomial Betti growth. The proof relies on new existence and uniqueness results for minimal models and acyclic closures of dg-algebra morphisms in an extended setting, together with an extension of Halperin's vanishing theorem for deviations; Gulliksen's classical theorem is recovered as an immediate corollary.

Significance. If the extensions of the minimal-model and acyclic-closure constructions hold under the hypotheses used in the proofs, the work supplies a homological characterization of derived complete intersections and unifies several classical results in commutative algebra within the dg-algebra framework. The broader existence statements may also serve as tools for other questions in homological algebra.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the claim that minimal models and acyclic closures exist and are unique 'in a broader setting than was previously known' is load-bearing for both the structure theorem and the recovery of Gulliksen's theorem, yet the precise hypotheses (base ring, characteristic, connectedness of the grading, or finiteness conditions on the dg-algebra) are not stated explicitly. Without these, it is impossible to confirm that the inductive construction terminates and yields uniqueness for every dg-algebra whose modules exhibit polynomial Betti growth.
  2. [Main structure theorem] Theorem A (or the main structure theorem): the argument that polynomial Betti growth implies the dg-algebra is a derived complete intersection appears to rest on the extended Halperin vanishing result; if the vanishing criterion requires the same unstated restrictions on the base field or grading that are standard for classical minimal models, the implication may fail to cover all cases invoked in the abstract.
minor comments (1)
  1. [Introduction] Clarify in the introduction whether the new existence results are stated for dg-algebras over arbitrary commutative rings or only over fields, and add a short paragraph comparing the new hypotheses with those of Gulliksen and Halperin.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed report and positive assessment of the significance of our work. The major comments highlight the need for greater clarity regarding the hypotheses under which our results hold. We address these points below and have made revisions to the manuscript to explicitly state the assumptions.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the claim that minimal models and acyclic closures exist and are unique 'in a broader setting than was previously known' is load-bearing for both the structure theorem and the recovery of Gulliksen's theorem, yet the precise hypotheses (base ring, characteristic, connectedness of the grading, or finiteness conditions on the dg-algebra) are not stated explicitly. Without these, it is impossible to confirm that the inductive construction terminates and yields uniqueness for every dg-algebra whose modules exhibit polynomial Betti growth.

    Authors: We agree with the referee that the hypotheses were not stated with sufficient explicitness in the abstract and introduction. In the revised manuscript, we will add a new subsection in §1 that clearly lists the standing assumptions: we work over a field k of characteristic zero, and consider dg-algebras that are connected graded (i.e., A_0 = k and A_i = 0 for i < 0) and locally finite (finite-dimensional in each degree). Under these conditions, the inductive construction of minimal models proceeds degree by degree and terminates at each step due to the finite-dimensionality, yielding uniqueness up to isomorphism. We will also explain how these hypotheses are satisfied in the cases needed for the structure theorem and the recovery of Gulliksen's theorem. revision: yes

  2. Referee: [Main structure theorem] Theorem A (or the main structure theorem): the argument that polynomial Betti growth implies the dg-algebra is a derived complete intersection appears to rest on the extended Halperin vanishing result; if the vanishing criterion requires the same unstated restrictions on the base field or grading that are standard for classical minimal models, the implication may fail to cover all cases invoked in the abstract.

    Authors: The extended version of Halperin's vanishing theorem is established in the paper precisely under the hypotheses stated above (characteristic zero, connected graded dg-algebras). The proof of the structure theorem (Theorem A) invokes this vanishing result only after reducing to the case where the dg-algebra satisfies these conditions via the minimal model. Consequently, the implication holds within the stated framework, which includes the cases needed to recover Gulliksen's theorem as a corollary. We will revise the statement of Theorem A to include a reference to the hypotheses and add a remark clarifying that no additional restrictions are imposed beyond those used for the minimal model constructions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external theorems with independent new results

full rationale

The paper establishes existence and uniqueness of minimal models and acyclic closures for dg-algebra morphisms in a broader setting, then applies this to prove a structure theorem for polynomial Betti growth and extend Halperin's vanishing result, recovering Gulliksen's theorem as a corollary. No steps reduce by the paper's equations or self-citations to the inputs by construction; the central claims rest on new proofs supported by cited external results (Gulliksen, Halperin) rather than self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citation chains. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background facts from homological algebra about dg-algebras, Betti numbers, and deviations; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Existence of minimal models and acyclic closures for morphisms of dg-algebras under the paper's hypotheses
    Invoked as a key ingredient established in the paper; treated as background for the structure theorem.
  • standard math Standard properties of Betti numbers and polynomial growth over local rings and dg-algebras
    Drawn from prior literature on commutative algebra and homological algebra.

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