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Roller Coaster Gorenstein algebras and Koszul algebras failing the weak Lefschetz property

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Pith's one-line read Gorenstein Hilbert series can realize any ordering of their first half, matching every permutation of {1,...,⌊d/2⌋}.

desk verdict The WLP results for whiskered graphs and Perazzo forms are solid and worth knowing, but the advertised Roller Coaster theorem is not proved: Lemma 5.6 is false. read the letter →

arxiv 2502.00155 v2 pith:ZOOCFGOS submitted 2025-01-31 math.AC math.CO

Thiago Holleben , Lisa Nicklasson This is my paper · ORCID
classification math.ACmath.CO MSC 13E1013H1013F5505C31
keywords weakLefschetzpropertyartinianGorensteinalgebraswhiskeredgraphsKoszulHilbertseriesNagataidealizationPerazzoformsrollercoastertheorem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the first half of the Hilbert series of an artinian Gorenstein algebra is unconstrained: for any socle degree d and any permutation π of {1,...,⌊d/2⌋}, there exists a Gorenstein algebra whose h-vector satisfies h_{π(1)} < ... < h_{π(⌊d/2⌋)}. This strengthens Boij's valley theorem, which had shown only that many valleys are possible. The construction builds the algebras as Nagata idealizations (trivial extensions that turn level algebras into Gorenstein ones) of monomial algebras coming from independence complexes of whiskered graphs, so the resulting Gorenstein algebras are Perazzo algebras and, in a large family, G-quadratic and hence Koszul. If the claim is right, unimodality is a rare accident among Gorenstein Hilbert functions, and the symmetry h_i = h_{d-i} is the only constraint on the first half.

What carries the argument

The load-bearing object is the simplicial Perazzo form F(∆) = Σ x_i u_{F_i}, where F_1,...,F_m are the facets of a pure simplicial complex ∆; its Macaulay dual generator (the inverse-system form whose annihilator is the Gorenstein ideal) is an artinian Gorenstein algebra that is the Nagata idealization (the trivial extension R ⋉ ω(−d)) of the Stanley–Reisner ring A(∆). When ∆ is the independence complex of a whiskered graph it is shellable, so the Gorenstein algebra is G-quadratic and Koszul. The roller-coaster argument uses the approximate well-covered polynomials of [8]: a coefficient sequence a_i is chosen so that the sums a_k + a_{d-k} obey the prescribed permutation, the sequence is realized up to scaling by the independence numbers of a well-covered graph, and the symmetric sums h_k = i_k + i_{d-k} of the Gorenstein algebra inherit the inequalities through Lemma 5.5.

What would settle it

Take q = 9 and the permutation π of {5,6,7,8,9} that reverses the order; compute the sequence a_i from Lemma 5.6 and check whether a_k / C(9,k) ≤ a_{k+1} / C(9,k+1) for every k. If it fails, Theorem 5.7 is false for d = 10; if it holds, the same check must still be completed for the remaining q < 10.

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Extended reading notes

Core claim

The central discovery is that the symmetry condition h_i = h_{d-i} is the only constraint on the first half of an artinian Gorenstein Hilbert series: for every d and every permutation π of {1,...,⌊d/2⌋}, the paper constructs an artinian Gorenstein algebra with Hilbert series ∑_{i=0}^d h_i t^i such that h_{π(1)} < ... < h_{π(⌊d/2⌋)}. The algebras are simplicial Perazzo algebras, obtained as Nagata idealizations of the Stanley–Reisner rings A(Ind(w(G))) of independence complexes of whiskered graphs, and the construction transfers the Roller Coaster theorem for well-covered graphs to this algebraic setting via approximate well-covered polynomials.

Load-bearing premise

The claim that the construction works for every socle degree rests on the assertion, in Lemma 5.6, that a certain ratio inequality holds for q < 10 'as we can check computationally' — a finite check that is not printed, so the fully verified statement is for d ≥ 11 unless that check is supplied.

Editorial extensions

If this is right

  • For every socle degree d, the collection of possible first halves of Gorenstein Hilbert series is unconstrained (Theorem 5.7).
  • Whiskering any graph whose independence number is at least n/3 + 2 yields an artinian algebra A(w(G)) that fails the weak Lefschetz property (Corollary 3.12).
  • For every bipartite graph on n ≥ 12 vertices, the whiskering produces a G-quadratic Gorenstein Perazzo algebra failing the weak Lefschetz property (Theorem 1.2).
  • A large family of G-quadratic, hence Koszul, Gorenstein algebras fails the weak Lefschetz property, and the paper conjectures that unconstrained Hilbert-series shapes persist among Koszul Gorenstein algebras of large socle degree (Conjecture 5.8).
  • Whiskered complete graphs, by contrast, give algebras with the strong Lefschetz property under any monomial artinian reduction (Proposition 3.2).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If Conjecture 5.8 holds, the Koszul property is compatible with arbitrarily shaped first halves of Gorenstein Hilbert series, so failure of unimodality would not be an artifact of non-Koszul presentations; a testable route is to check whether the complexes produced in Theorem 5.7 can be chosen vertex-decomposable, upgrading G-quadratic to a stronger combinatorial condition.
  • The quantitative threshold α(G) ≥ n/3 + 2 for failure of the weak Lefschetz property suggests a sharper conjecture: that α(G) ≥ 3 already suffices. This is computationally checkable for all graphs on at most 8 vertices, and the bipartite case in the paper already covers all n ≥ 12.
  • The approximate-polynomial transfer used here may apply to other symmetry-constrained Hilbert-function problems, such as level algebras or Gorenstein algebras with prescribed socle type, where symmetry could again be shown to be the only obstruction.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the Lefschetz properties of artinian algebras associated to whiskered graphs and uses them to construct Perazzo Gorenstein algebras via Nagata idealization. Its main advertised result, Theorem 1.1, claims that for every positive integer d and every permutation of {1,...,floor(d/2)}, there is an artinian Gorenstein algebra whose first half of the Hilbert series is ordered according to that permutation. The proof passes through an 'approximate well-covered polynomial' construction adapted from Cutler and Pebody, then transfers coefficient inequalities to the Hilbert functions of simplicial Perazzo algebras. The paper also proves WLP failure for whiskered graph algebras with large independence number and constructs a large family of G-quadratic Gorenstein algebras failing the WLP.

Significance. If Theorem 1.1 were established, it would be a substantial strengthening of Boij's valley theorem, showing that the first half of Gorenstein Hilbert functions is combinatorially unconstrained. The paper's general strategy—combining approximate well-covered independence polynomials with simplicial Perazzo forms—is attractive and connects graph theory directly to Gorenstein Hilbert functions. The results on WLP failure for whiskered graphs and their Perazzo algebras, especially the G-quadratic examples of Corollary 4.7 and Theorem 4.6, are valuable and appear to be largely unaffected by the problem discussed below. However, the central Roller Coaster theorem currently rests on a false lemma, so the main claim is not established as stated.

major comments (2)
  1. [Section 5, Lemma 5.6] Lemma 5.6 is false as stated. For q=3 and the permutation pi=(2,3), one has ceil(q/2)=2, c=binom(3,3)=1, and the definition gives a_1=3, a_2=11, a_3=12. Taking k=2 and ell=3, we have pi(2)<pi(3), but a_2+a_{3-2+1}=a_2+a_2=22, while a_3+a_{3-3+1}=a_3+a_1=15; the asserted strict inequality 22<15 is false. The source of the error is visible in the proof: the replacement of a_{q-k+1} by binom(q,k-1) is valid only when q-k+1 < ceil(q/2), but at k=ceil(q/2) the index q-k+1 equals ceil(q/2), which falls in the second case of the definition of a_i. Since Theorem 5.7 applies Lemma 5.6 to obtain the Roller Coaster ordering, Theorem 1.1 is not proved for every d; for example, when d=4 the construction cannot produce the order h_2<h_1 required by the transposition on {1,2}. The lemma must be repaired, or the statement of Theorem 1.1 must be restricted to the range in which the construction actually works, with small d handled separately.
  2. [Section 5, Lemma 5.6] The proof of the first part of Lemma 5.6, i.e. the verification of the growth condition (5.1), contains an undocumented finite check: the sentence 'we can check computationally (for q<10)' is not accompanied by code, data, or a precise description of the computation. Since Theorem 5.7 quantifies over every d, this finite check is load-bearing if the lemma is to be used in its full generality. Moreover, the displayed chain of inequalities in that paragraph appears garbled (for example, the expressions involving 1+2q(q-1)/(3q) and 1+(2/3)^q/q do not cohere as written), so the proof is not verifiable in its current form.
minor comments (4)
  1. [Section 5, Lemma 5.5] In case (2) of the proof of Lemma 5.5, the displayed expression writes i_{k+1}(G)+i_{q-k}(G), but the intended quantities are i_ell(G)+i_{q-ell+1}(G); the indices in that line should be corrected.
  2. [Section 5, Theorem 5.7] The reduction by symmetry from a permutation of {1,...,floor(d/2)} to a permutation of {ceil(d/2),...,d-1} should be spelled out explicitly, especially for odd d, because the index range supplied by Lemma 5.6 with q=d-1 is {ceil((d-1)/2),...,d-1}, which contains an extra element when d is odd.
  3. [Remark 3.16] The statement that Conjecture 3.15 has been verified computationally for all smaller bipartite graphs and for all graphs on at most 7 vertices is cited without accompanying data or code; please supply reproducible verification or reduce the claim accordingly.
  4. [Section 3, Lemma 3.1] The displayed direct sum decomposition in Lemma 3.1 contains several typographical artifacts, such as missing braces and stray characters around the variables y_i, which make the proof harder to read.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed construction inputs a permutation and invokes an external approximation theorem (Cutler–Pebody) plus Macaulay duality; no fitted quantity is renamed as a prediction and no load-bearing premise reduces to the theorem being proved.

full rationale

The paper's derivation chain is not circular. Theorem 1.1 is an existence statement: for every permutation pi one must produce an artinian Gorenstein algebra whose first-half h-vector is ordered by pi. The proof begins with the given pi, defines a coefficient sequence a_i in Lemma 5.6, invokes the external Cutler–Pebody approximate well-covered polynomial theorem to realize the polynomial as independence counts of a well-covered graph, and then transfers the prescribed inequalities to the Gorenstein Hilbert function through Lemma 5.5 and the Nagata idealization/Macaulay duality correspondence. The ordering h_{pi(1)} < ... < h_{pi(floor(d/2))} is obtained, not assumed, and the construction varies with pi, which is what existence for every pi requires. The self-citations present in the paper, such as [21], [22], [6], and [5], are used as lemmas or background tools and are not the vehicle that forces the permutation ordering; in particular, the Perazzo property in Theorem 5.7 follows from the elementary 'more facets than vertices implies algebraic dependence' criterion (Lemma 4.2) together with Macaulay duality, and the Gorenstein conclusion does not depend on a self-cited uniqueness theorem. The manuscript's weak point identified by the reader and skeptic is correctness, not circularity: Lemma 5.6 contains an unstated computational check for q < 10, and the displayed reduction to binomial coefficients appears invalid at the endpoint k = ceil(q/2), giving a concrete counterexample to the lemma's claimed 'if and only if'. If that counterexample stands, Theorem 5.7 is unproved for small d, but a false auxiliary inequality is not a reduction of the theorem to its own inputs. No fitted parameter is renamed as a prediction, no known result is merely re-labeled, and no ansatz is smuggled in through self-citation. The central claim therefore has independent mathematical content, and the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The construction has no fitted constants or invented physical entities. It relies on external theorems in graph theory and commutative algebra, plus standard facts about complete intersections; these are the stated assumptions that the central claims rest upon.

assumptions (4)
  • domain assumption Cutler-Pebody approximate well-covered polynomial theorem (Theorem 5.4)
    Used in Theorem 5.7 to produce, for any admissible coefficient sequence, a well-covered graph whose independence numbers approximate the sequence. The theorem is external to this paper.
  • domain assumption D'Ali-Venturello theorem on simplicial forms (Theorem 4.4)
    Used to conclude that the annihilator ideal of a whiskered graph's simplicial form has a quadratic Groebner basis, hence the algebra is Koszul.
  • standard math Macaulay duality and Nagata idealization (Lemma 4.5)
    Basis for interpreting simplicial Perazzo forms as Gorenstein algebras with Hilbert function h_k = dim A(Delta)_k + dim A(Delta)_{d-k}.
  • standard math SLP of monomial complete intersections in characteristic zero or large characteristic (Proposition 3.2)
    Used to prove SLP for whiskered complete graphs.

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Pith. "Pith review of Roller Coaster Gorenstein algebras and Koszul algebras failing the weak Lefschetz property." pith.science (2026). https://pith.science/paper/ZOOCFGOS

@misc{pith2026250200155,
  author       = {Pith},
  title        = {Pith review of: Roller Coaster Gorenstein algebras and Koszul algebras failing the weak Lefschetz property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOOCFGOS}},
  note         = {Machine review of arXiv:2502.00155}
}
read the original abstract

Inspired by the Roller Coaster Theorem from graph theory, we prove the existence of artinian Gorenstein algebras with unconstrained Hilbert series, which we call Roller Coaster algebras. Our construction relies on Nagata idealization of quadratic monomial algebras defined by whiskered graphs. The monomial algebras are interesting in their own right, as our results suggest that artinian level algebras defined by quadratic monomial ideals rarely have the weak Lefschetz property. In addition, we discover a large family of G-quadratic Gorenstein algebras failing the weak Lefschetz property.

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