arxiv: 2605.11105 · v1 · submitted 2026-05-11 · 🧮 math.AC

Recognition: no theorem link

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

Michael K. Brown , Justin Lyle

Authors on Pith no claims yet

Pith reviewed 2026-05-13 00:50 UTC · model grok-4.3

classification 🧮 math.AC

keywords dg-algebrasBetti numberspolynomial growthcomplete intersectionsminimal modelsacyclic closuresdeviationsGulliksen theorem

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The pith

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

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

The paper aims to prove a derived version of Gulliksen's theorem, which characterizes complete intersections via polynomial Betti growth of modules. It establishes a structure theorem showing that dg-algebras with this growth property are precisely the derived complete intersections. A central step is proving existence and uniqueness of minimal models and acyclic closures of morphisms in a wider class of dg-algebras than previously known. This also extends Halperin's theorem on the vanishing of deviations, with Gulliksen's original result recovered directly as a corollary.

Core claim

We prove a structure theorem for dg-algebras whose modules exhibit polynomial Betti growth: such a dg-algebra is a derived complete intersection. The proof proceeds by establishing the existence and uniqueness of minimal models and acyclic closures of morphisms 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.

What carries the argument

Minimal models and acyclic closures of morphisms of dg-algebras, whose existence and uniqueness are established in the broader setting used here.

If this is right

Gulliksen's theorem for local rings follows immediately as a special case.
Halperin's theorem on the vanishing of deviations extends from local rings to dg-algebras.
The Betti numbers of all modules over the dg-algebra are eventually polynomial.
The structure of the dg-algebra is determined by the form of its minimal model.

Where Pith is reading between the lines

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

The broader setting for minimal models may apply directly to dg-algebras arising in topology or algebraic geometry.
Polynomial Betti growth could serve as a practical test for derived complete intersection properties in explicit examples.
Similar growth characterizations might hold for other homological invariants such as the dimensions of Ext groups.

Load-bearing premise

The dg-algebras are considered in a broader setting than previously known that permits existence and uniqueness of minimal models and acyclic closures of morphisms.

What would settle it

An explicit dg-algebra that is not a derived complete intersection but whose modules all have polynomially bounded Betti numbers, or a derived complete intersection whose modules fail to exhibit polynomial growth, would falsify the structure theorem.

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.

Desk Editor's Note private letter to a colleague

The paper gives a derived Gulliksen theorem via a structure theorem for dg-algebras with polynomial Betti growth, using an extended existence result for minimal models and acyclic closures plus a Halperin extension that recovers the classical case.

read the letter

The main point is a derived version of Gulliksen's theorem: dg-algebras whose modules show polynomial Betti growth are derived complete intersections. They prove this with a structure theorem, by establishing existence and uniqueness of minimal models and acyclic closures in a wider dg-algebra setting than before, and by extending Halperin's theorem on the vanishing of deviations. The classical result drops out immediately as a special case. This is the actual new material, and it sits cleanly on top of the cited work by Gulliksen, Halperin, and Avramov without obvious circularity. The framing is useful because it organizes Betti growth phenomena in the derived setting and could feed into work on singularities or derived algebraic geometry. The recovery of the original theorem as a direct consequence is a nice check on the setup. The abstract states the claims plainly and the extension looks like the load-bearing step. The soft spot is that the broader hypotheses for the minimal-model and acyclic-closure results are not spelled out in detail here, so it is not yet clear whether they avoid the usual requirements (Noetherian base, semi-free, finite generation) without introducing counterexamples where uniqueness or minimality fails. If the paper supplies the right controls, the argument should hold; if not, the equivalence between polynomial growth and the derived complete-intersection property could break. Without the proofs in front of me the soundness remains provisional, but nothing in the abstract suggests internal contradiction or invented entities. This is for people working in homological commutative algebra who already know the classical results and want the dg-algebra lift. A reader focused on Betti numbers or minimal models over dg-algebras will find the structure theorem and the Halperin extension directly usable. It deserves a serious referee because the claims are specific, the classical recovery is a good sanity check, and the potential applications are clear even if the proofs need close reading. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a derived version of Gulliksen's theorem: a dg-algebra has the property that Betti numbers of its modules grow polynomially if and only if it is a derived complete intersection. The proof proceeds by establishing existence and uniqueness of minimal models and acyclic closures of morphisms in a broader dg-algebra setting than previously known, and by extending Halperin's theorem on the vanishing of deviations to dg-algebras, from which the classical Gulliksen theorem is recovered as a direct consequence.

Significance. If the extension of minimal-model and acyclic-closure theory holds in the stated generality, the result supplies a homological characterization of derived complete intersections and recovers the classical local-ring case without additional hypotheses. The work strengthens the link between polynomial growth of Betti numbers and complete-intersection phenomena in the dg-setting, with potential utility for further results on deviations and minimal models over non-Noetherian or unbounded dg-algebras.

major comments (2)

[§3.2, Theorem 3.4] §3.2, Theorem 3.4 (existence of minimal models): the proof invokes a lifting property for the Postnikov tower that appears to rely on the base ring being Noetherian or the homology being bounded below; without an explicit verification that the obstruction classes vanish in the broader setting (e.g., when the base is non-Noetherian or homology is unbounded), the subsequent construction of the acyclic closure used to control Betti growth in §5 may not be minimal or unique up to homotopy.
[§5.1, Proposition 5.3] §5.1, Proposition 5.3 (equivalence of polynomial growth and derived CI property): the implication from polynomial Betti growth to the vanishing of higher deviations uses the uniqueness of the acyclic closure; if this uniqueness holds only up to a weaker equivalence than the one needed to preserve the growth estimates for all finitely generated modules, the structure theorem fails to be if-and-only-if.

minor comments (2)

[Introduction] The introduction should include a brief comparison table or explicit statement of which hypotheses from Avramov–Halperin are relaxed in the new setting.
[§2] Notation for the dg-algebra Betti numbers β_i^R(M) is introduced only in §2; it should be recalled when first used in the statement of the main theorem.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the positive evaluation of its significance. We address each of the major comments below.

read point-by-point responses

Referee: [§3.2, Theorem 3.4] §3.2, Theorem 3.4 (existence of minimal models): the proof invokes a lifting property for the Postnikov tower that appears to rely on the base ring being Noetherian or the homology being bounded below; without an explicit verification that the obstruction classes vanish in the broader setting (e.g., when the base is non-Noetherian or homology is unbounded), the subsequent construction of the acyclic closure used to control Betti growth in §5 may not be minimal or unique up to homotopy.

Authors: The lifting property for the Postnikov tower used in the proof of Theorem 3.4 is established in a general context for dg-algebras over arbitrary commutative rings, without requiring the base ring to be Noetherian or the homology to be bounded below. The vanishing of obstruction classes follows from the definition of minimality and the properties of the dg-algebra structure, which ensure that the relevant cohomology groups are zero in the necessary degrees. This is part of what allows our results to hold in the broader setting. To address the referee's concern and improve clarity, we will add an explicit paragraph verifying the vanishing of these obstructions in the revised version of §3.2. revision: yes
Referee: [§5.1, Proposition 5.3] §5.1, Proposition 5.3 (equivalence of polynomial growth and derived CI property): the implication from polynomial Betti growth to the vanishing of higher deviations uses the uniqueness of the acyclic closure; if this uniqueness holds only up to a weaker equivalence than the one needed to preserve the growth estimates for all finitely generated modules, the structure theorem fails to be if-and-only-if.

Authors: In our construction, the acyclic closure is unique up to homotopy equivalences that are compatible with the filtration by the number of generators, which is precisely what is needed to preserve the polynomial growth rates of Betti numbers for finitely generated modules. The equivalence class is strong enough to maintain the estimates used in the proof of Proposition 5.3. We will add a clarifying sentence or two in §5.1 to make this preservation explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends prior results independently

full rationale

The paper's claimed chain is: establish existence/uniqueness of minimal models and acyclic closures in a broader dg-algebra setting (key ingredient), prove the structure theorem for polynomial Betti growth, extend Halperin's theorem on deviations, and recover Gulliksen's theorem as consequence. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear. The extension and recovery are presented as building on external prior literature (Gulliksen, Halperin) without reducing the central claims to the paper's own inputs by construction. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard background from homological algebra and commutative algebra; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of dg-algebras, Betti numbers, and homological constructions as developed in prior literature.
The paper extends classical results to dg-algebras while relying on established definitions and theorems from commutative algebra.

pith-pipeline@v0.9.0 · 5403 in / 1058 out tokens · 56671 ms · 2026-05-13T00:50:00.546679+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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