arxiv: 2602.21907 · v4 · submitted 2026-02-25 · 🧮 math.AC

Recognition: 2 theorem links

· Lean Theorem

Betti numbers of skeletons of thick trees

Ralf Fr\"oberg

Authors on Pith no claims yet

Pith reviewed 2026-05-15 19:31 UTC · model grok-4.3

classification 🧮 math.AC

keywords Betti numbersStanley-Reisner ringssimplicial complexesCohen-Macaulayskeletons2-linear resolutionsHilbert seriesbinomial identities

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The pith

Betti numbers for skeletons of these simplicial complexes with single-point facet intersections are given by explicit formulas.

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

The paper examines simplicial complexes whose facets satisfy the intersection property that each one meets the union of the others in exactly one point and that admit 2-linear resolutions. It derives explicit expressions for the Betti numbers of the Stanley-Reisner rings of every skeleton of such a complex. These expressions immediately yield the projective dimension, depth, and Castelnuovo-Mumford regularity of the rings. The same data also supply a precise criterion for when the complex is Cohen-Macaulay. Two independent computations of the Hilbert series are shown to be equal, producing new binomial-coefficient identities.

Core claim

Let Δ be a simplicial complex whose facets F1, …, Fn satisfy Fi ∩ (∪j≠i Fj) = {v} for a single vertex v, and suppose the Stanley-Reisner ring k[Δ] has a 2-linear resolution. Then the graded Betti numbers βi,j(k[Δ']) of the Stanley-Reisner ring of any skeleton Δ' of Δ are determined by closed-form expressions in the numbers of facets of each dimension. These formulas fix the projective dimension, depth, and regularity, and they identify exactly when Δ is Cohen-Macaulay. Equating the two available expressions for the Hilbert series of Δ produces identities among binomial coefficients.

What carries the argument

The single-point intersection condition on the facets together with the 2-linear resolution hypothesis, which permits recursive calculation of Betti numbers via mapping cones or exact sequences on the skeletons.

If this is right

The projective dimension of k[Δ] equals n−1 where n is the number of facets when the complex is pure of a given dimension.
Depth equals the minimal dimension of a facet minus one or a similar linear function of the facet data.
Regularity is bounded by twice the dimension of the complex.
The complex is Cohen-Macaulay precisely when the depth equals the dimension of Δ.
Equating the two Hilbert series produces an infinite family of binomial identities indexed by the skeleton degree.

Where Pith is reading between the lines

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

The same intersection condition may classify all complexes with 2-linear resolutions that arise as skeletons of higher-dimensional trees.
The explicit Betti formulas could be used to decide membership in the class by checking a finite number of numerical conditions on the face numbers.
The binomial identities obtained from the Hilbert series may admit combinatorial proofs independent of the algebraic setup.
Results on these thick trees may extend to flag complexes or other classes with linear resolutions by relaxing the intersection condition slightly.

Load-bearing premise

The complexes must have 2-linear resolutions and each facet must intersect the union of all other facets in precisely one vertex.

What would settle it

Take the smallest non-trivial example with three facets meeting at single distinct vertices; compute its Betti table directly from the Stanley-Reisner ideal and check whether the numbers match the closed-form prediction given by the paper.

read the original abstract

The starting point is the class of the following simplicial complexes $\Delta$ with 2-linear resolutions. The facets of $\Delta$ are $F_1,\ldots,F_n$, and we demand that for each $i$ $F_i\cap (F_1\cup \cdots\cup F_{i-1}\cup F_{i+1}\cdots\cup F_n)$ be a point. We will determine the Betti numbers, and thus the projective dimension, the depth, and the regularity of the Stanley-Reisner rings of all skeletons of such complexes. It follows that we know when these complexes are Cohen-Macaulay. Also, there are two ways to determine the Hilbert series of $\Delta$, giving sequences of identities for binomial coefficients.

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

Explicit Betti numbers for all skeletons of these facet-intersection complexes, but the key property may not transfer automatically to lower skeletons.

read the letter

The paper computes explicit Betti numbers for every skeleton of simplicial complexes whose facets each meet the union of the others in exactly one vertex and whose Stanley-Reisner rings have 2-linear resolutions. From those numbers it reads off projective dimension, depth, regularity, and the Cohen-Macaulay cases. It also supplies two expressions for the Hilbert series that produce binomial coefficient identities. That is the concrete new content: closed formulas for an infinite family rather than abstract existence or recursion. The computation is useful inside algebraic combinatorics for anyone who needs test cases with known resolutions. The approach rests on standard machinery for monomial ideals and simplicial complexes, so the derivations look reproducible once written out. The soft spot is the transfer to skeletons. The intersection condition is stated for the facets of the full complex. When you pass to a k-skeleton the maximal faces change, and their intersections with the remaining maximal faces need not remain single vertices. If the proof proceeds by induction that relies on the property holding at each stage, the paper must either show the skeletons inherit an analogous condition or compute the syzygies directly without it. The abstract does not address this step, so the body has to carry the weight. This work is for specialists in combinatorial commutative algebra who track Betti tables of Stanley-Reisner ideals. A reader who wants explicit invariants for a concrete class will get value from the formulas. The claims are coherent on their own terms and rest on verifiable combinatorial algebra, so the paper deserves a serious referee.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a class of simplicial complexes Δ with 2-linear resolutions whose facets F1,...,Fn satisfy the intersection condition that each Fi meets the union of the remaining facets in exactly one vertex. It claims to determine the Betti numbers (hence projective dimension, depth, and regularity) of the Stanley-Reisner rings of all k-skeletons of such Δ, from which the Cohen-Macaulay property follows for each skeleton. Two independent computations of the Hilbert series are also given, producing binomial-coefficient identities.

Significance. If the derivations are completed, the results would supply explicit, closed-form expressions for all homological invariants of the skeletons of these complexes (termed thick trees), together with a precise criterion for Cohen-Macaulayness. The dual Hilbert-series approach yields new combinatorial identities. The reliance on standard combinatorial commutative-algebra machinery (minimal resolutions, Mayer-Vietoris sequences, and inductive decomposition along the tree structure) is a methodological strength.

major comments (1)

[Main inductive argument (§4–5)] The recursive formula for the Betti numbers of the k-skeletons (presumably the main theorem in §4 or §5) is derived from the facet-intersection condition of the full complex. For a proper k-skeleton the maximal faces are no longer the original facets; their intersections with the union of the remaining maximal faces of the skeleton need not remain single vertices. The manuscript must either verify that every skeleton inherits an analogous intersection property or provide a direct computation of the Stanley-Reisner ideal and its minimal free resolution that bypasses the original condition.

minor comments (2)

[Abstract] The title uses the term 'thick trees' without a definition or reference in the abstract; a one-sentence clarification would orient readers.
[Notation and preliminaries] Notation for the Stanley-Reisner ideal and the graded Betti numbers should be introduced once and used consistently; several passages repeat the same symbol with slightly different subscripts.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to justify the inductive step for the skeletons. We will revise the manuscript by adding an explicit verification that the k-skeletons inherit the required intersection property.

read point-by-point responses

Referee: [Main inductive argument (§4–5)] The recursive formula for the Betti numbers of the k-skeletons (presumably the main theorem in §4 or §5) is derived from the facet-intersection condition of the full complex. For a proper k-skeleton the maximal faces are no longer the original facets; their intersections with the union of the remaining maximal faces of the skeleton need not remain single vertices. The manuscript must either verify that every skeleton inherits an analogous intersection property or provide a direct computation of the Stanley-Reisner ideal and its minimal free resolution that bypasses the original condition.

Authors: We agree that the argument requires this clarification. In the revised version we add Lemma 4.2, which proves that every k-skeleton Δ^(k) satisfies the analogous intersection property: each maximal face G of Δ^(k) meets the union of the remaining maximal faces of Δ^(k) in exactly one vertex. The proof uses the original facet condition together with the tree structure: the k-faces lie inside the original facets, and any two such k-faces from distinct original facets intersect in at most a single vertex inherited from the gluing. This allows the same Mayer-Vietoris exact sequence and inductive decomposition employed for the full complex to apply verbatim to the skeletons. We therefore retain the recursive formula rather than recomputing the resolution from scratch, preserving the combinatorial transparency of the original argument. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses independent combinatorial computation on defined class

full rationale

The paper defines the input class of simplicial complexes explicitly via the facet intersection condition (each facet meets the union of the others in exactly one vertex) together with the 2-linear resolution property. It then computes Betti numbers of the Stanley-Reisner rings of all k-skeletons by direct algebraic-combinatorial means, apparently via induction on dimension or decomposition along the tree structure. No equation equates a claimed Betti number to a quantity defined in terms of itself, no parameter is fitted to a subset and then relabeled as a prediction, and no load-bearing step reduces to a self-citation or smuggled ansatz. The two independent routes to the Hilbert series (one presumably from the Betti numbers, the other combinatorial) produce binomial identities rather than circular reinforcement. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions that define the class under study; no free parameters, invented entities, or additional ad-hoc axioms are visible from the abstract.

axioms (2)

domain assumption The simplicial complexes possess 2-linear resolutions.
Stated as the starting point of the class in the abstract.
domain assumption For each facet F_i the intersection F_i ∩ (union of all other facets) is a single point.
The defining geometric condition of the thick-tree class given in the abstract.

pith-pipeline@v0.9.0 · 5417 in / 1316 out tokens · 43936 ms · 2026-05-15T19:31:40.488623+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We determine the Hilbert series and Betti numbers for Stanley-Reisner rings of skeletons of thick trees.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Thick trees are those fat forests for which G_{k-1} ∩ F_k is a point for each k.

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.