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Simultaneously nonvanishing higher derived limits

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Pith's one-line read If the dominating number is $\omega_n$, the $n$-th derived limit of the canonical inverse system with coefficients $\mathbb{Z}^{(\omega_n)}$ does not vanish.

desk verdict This paper proves the first ZFC implication from the dominating number alone to nonvanishing of higher derived limits, and the first consistency of simultaneous nonvanishing at all finite levels; it deserves a serious referee. read the letter →

arxiv 2411.15856 v2 pith:FTWMC67L submitted 2024-11-24 math.LO math.ATmath.CT

Matteo Casarosa , Chris Lambie-Hanson This is my paper · ORCID
classification math.LOmath.ATmath.CT MSC 03E3503E0503E1703E7518G10
keywords derivedlimitsdominatingnumberweakdiamondsquareprinciplescoherentfamiliesinversesystemscontinuumnonvanishing
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 shows that a single classical cardinal invariant, the dominating number $\mathfrak{d}$, already controls whether certain higher derived limits vanish. Specifically, if $\mathfrak{d}=\omega_n$, then $\lim^n \mathbf{A}[\mathbb{Z}^{(\omega_n)}]\neq 0$, and with weak diamond assumptions the same holds for the canonical system with integer coefficients. Since vanishing of all such limits is tied to additivity properties in strong homology and condensed mathematics, the paper derives a lower bound on the continuum: if $\lim^n \mathbf{A}[H]=0$ for all $n\geq 1$ and all abelian groups $H$, then $2^{\aleph_0}\geq \aleph_{\omega+1}$. It also establishes the consistency of simultaneous nonvanishing of $\lim^k \mathbf{A}$ for all $k\geq 2$.

What carries the argument

The central object is a coherent $n$-family: an alternating family of functions $\varphi_u:\bigcap_{i<n}u(i)\to H$ indexed by $n$-tuples of sets, satisfying a mod-finite alternating-sum relation. By the background equivalence restated as Fact 2.21, $\lim^n \mathbf{A}_I[H]=0$ exactly when every coherent $H$-valued $n$-family indexed along $I$ is trivial, so the paper's entire strategy is to construct nontrivial coherent families. The new combinatorial mechanism is an ascending sequence of sets, a weakening of a $\subseteq^*$-increasing chain that still supports recursive constructions of nontrivial families; combined with weak diamond and square principles, it yields both the single-cardinal theorem and the simultaneous-nonvanishing consistency results.

What would settle it

The central claim would be refuted by a ZFC model with $\mathfrak{d}=\omega_n$ and $\lim^n\mathbf{A}[\mathbb{Z}^{(\omega_n)}]=0$, or by a model with continuum below $\aleph_{\omega+1}$ in which every $\lim^n\mathbf{A}[H]$ vanishes for all finite $n$ and all abelian groups $H$. A direct witness would be a coherent $\mathbb{Z}^{(\omega_n)}$-valued $n$-family that is nontrivial even though the corresponding derived limit vanishes, contradicting the bridge equivalence.

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Extended reading notes

Core claim

The paper proves Theorem A: whenever $\mathfrak{d}=\omega_n$, the derived limit $\lim^n \mathbf{A}[\mathbb{Z}^{(\omega_n)}]$ is nonzero, and with $\mathrm{w}\diamondsuit(S^{k+1}_k)$ for all $k<n$, even $\lim^n \mathbf{A}\neq 0$. The immediate corollary answers an open question: universal vanishing of all such derived limits forces the continuum to be at least $\aleph_{\omega+1}$. The paper also proves Theorem B: for each fixed $n$ it is consistent that $\mathfrak{b}=\mathfrak{d}=\omega_n$ and $\bigwedge_{2\leq k\leq n}\lim^k\mathbf{A}\neq 0$, and it is consistent that $\bigwedge_{2\leq k<\omega}\lim^k\mathbf{A}\neq 0$ with $\mathfrak{b}=\mathfrak{d}=\aleph_{\omega+2}$, nearly matching the lower bound $\aleph_{\omega+1}$ from Goblot's theorem.

Load-bearing premise

The whole method rests on the established equivalence that $\lim^n\mathbf{A}_I[H]=0$ exactly when every coherent $H$-valued $n$-family indexed along $I$ is trivial; if that bridge broke, the combinatorial constructions would no longer say anything about derived limits.

Editorial extensions

If this is right

  • If $\mathfrak{d}=\omega_n$, then $\lim^n \mathbf{A}[\mathbb{Z}^{(\omega_n)}]\neq 0$ for the canonical inverse system $\mathbf{A}$.
  • Adding weak diamond assumptions for all $k<n$ upgrades the conclusion to $\lim^n\mathbf{A}\neq 0$ with integer coefficients.
  • Universal vanishing of $\lim^n\mathbf{A}[H]$ for all $n$ and all abelian groups $H$ would force $2^{\aleph_0}\geq\aleph_{\omega+1}$, resolving the open question about the minimum continuum compatible with additivity of derived limits.
  • For each $n$, it is consistent that $\mathfrak{b}=\mathfrak{d}=\omega_n$ and $\lim^k\mathbf{A}\neq 0$ simultaneously for all $2\leq k\leq n$.
  • It is consistent that $\lim^k\mathbf{A}\neq 0$ for every finite $k\geq 2$, with $\mathfrak{d}=\aleph_{\omega+2}$; Goblot's theorem shows $\mathfrak{d}\geq\aleph_{\omega+1}$ is necessary for such a conclusion.

Reading between the lines

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

  • Extension: the paper's ideal-theoretic formulation suggests the dichotomy is not special to $\mathbf{A}$ but holds for any ideal $I$ with $\mathrm{cof}^*(I)=\mathrm{non}^*(I)=\omega_n$, so the same forcing-free core may transfer to other inverse systems of topological or algebraic origin.
  • Extension: the distance between the achievable $\mathfrak{d}=\aleph_{\omega+2}$ and the necessary $\aleph_{\omega+1}$ for simultaneous nonvanishing of all finite $k$ is plausibly closable by a more delicate square construction; nothing in the method forces the extra `$+2$`.
  • Extension: the bridge equivalence (Fact 2.21) is the only place where combinatorial nontriviality is converted into a statement about derived limits; if that equivalence ever failed for some system, the paper's constructions would need a separate homological argument.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies nonvanishing of higher derived limits lim^n A_I[H] of inverse systems of free abelian groups associated with directed families of sets and abelian groups H. The main result, Theorem A, states that if the dominating number d equals ω_n, then lim^n A[Z^{(ω_n)}] ≠ 0, and that under additional weak diamond principles w♢(S^{k+1}_k) for k<n, even lim^n A ≠ 0. This yields the corollary that if lim^n A[H]=0 for all n≥1 and all abelian groups H, then 2^{ℵ0} ≥ ℵ_{ω+1}, answering a question of Bannister. The second main theorem, Theorem B, establishes consistency of simultaneous nonvanishing: for each 2≤n<ω there is a model with b=d=ω_n and ⋀_{2≤k≤n} lim^k A ≠ 0, and there is a model with b=d=ω_{ω+2} and ⋀_{2≤k<ω} lim^k A ≠ 0. The proofs reduce nonvanishing to constructions of nontrivial coherent n-families: Sections 3–4 introduce the notion of an ascending sequence and derive such sequences from cof^*(I)=non^*(I), while Section 5 proves a stepping-up lemma using square principles and weak diamonds.

Significance. If correct, the paper makes substantial advances. It shows that a single natural cardinal invariant, the dominating number, can force nonvanishing of a concrete higher derived limit without bounding-number or scale assumptions, and it gives the first consistency result for simultaneous nonvanishing of infinitely many finite-dimensional derived limits. The main combinatorial machinery—ascending sequences (Lemma 4.2) and the stepping-up lemma (Lemma 5.3)—is explicit and appears sound. The proof strategy is genuine derivation: nontrivial coherent families are constructed from combinatorial hypotheses, and the derived-limit conclusions follow by the standard bridge Fact 2.21. The paper is also honest about limitations, such as the remaining gap between d=ω_{ω+1} and d=ω_{ω+2} in the infinite simultaneous-nonvanishing result.

minor comments (6)
  1. [Section 2.2, Fact 2.21] Fact 2.21 is the bridge through which all later combinatorial conclusions are converted into statements about derived limits, but the cited proof in [5, Section 2.2] is explicitly for H=Z and the paper only notes that the generalization to arbitrary abelian groups and directed I is straightforward. Since this fact is load-bearing for every main theorem, a short proof sketch of the general case, or a precise reference to a source that states it in this generality, would make the paper more self-contained.
  2. [Theorem 3.6, coding function G] The construction of the coding function G and the club D is terse. In particular, the claim that G↾γ2 can be chosen surjective onto Pγ for every γ∈D should be justified by the choice of D with gaps of size ω_{n-1} and the bound |Pγ|≤2^{ω_{n-1}}; a few sentences making this explicit would remove any doubt about the cardinal arithmetic.
  3. [Theorem 3.9, nontriviality argument] In the final nontriviality argument for n>1, the text says 'find γ∈Lim(C)∩S^n_{n-1} such that, for all a∈[γ]^n, ψ_a maps into H(γ)'. This should be [γ]^{n-1}, and the existence of such a γ should be justified by a short closure argument on the finitely supported supports of the ψ_a.
  4. [Throughout, notation typos] There are several small typographical issues: in Theorem 3.9 'maps into 2 (β)' and 'maps into 2 (γ)' should presumably be 'maps into H^{(β)}' and 'H^{(γ)}'; in Lemma 5.10 '(ωω, leq∗)' should be '(ω^ω, ≤^*)'; and in Theorem 3.9 there is an extra parenthesis in 'H(γ+ω_{n-1}))'.
  5. [Corollary 4.5 and abstract] The abstract states the Bannister-answer corollary as 2^{ℵ0} ≥ ℵ_{ω+1}, while Corollary 4.5 states it as 2^{ℵ0} > ℵ_ω. These are equivalent, but stating them in the same form would avoid confusion.
  6. [Lemma 5.7 proof] The sentence 'Using [19, Lemma 3.6], one can prove by induction...' is terse. A one-sentence explanation of how Lemma 3.6 of [19] applies for arbitrary countable abelian groups H would improve readability, since this induction is used to provide the base chains for Lemma 5.3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are genuine combinatorial constructions from stated hypotheses.

full rationale

The paper's central claim is a statement about derived limits, and its proof reduces to constructing nontrivial coherent n-families via the equivalence in Fact 2.21. This equivalence is imported from [16, Section III] and [5, Section 2.2], which is a standard algebraic fact independent of the new combinatorial content, so using it is not circular. Theorems 3.5, 3.6, and 3.9 give fully explicit recursive constructions of nontrivial coherent families from ascending sequences; nontriviality is argued directly (e.g., by weak-diamond coding or by placing fresh group elements on infinite sets), not assumed. Section 4 proves Lemma 4.2 and Proposition 4.3 from the definitions of cof* and non*, establishing that d = omega_n supplies the required ascending sequence; no fitted parameter is renamed as a prediction. Section 5's Theorem B uses Lemma 5.3, whose proof is an explicit stepping-up construction using square and weak-diamond hypotheses, with results from [19] used as lemmas under stated hypotheses. Corollary 4.6 explicitly acknowledges prior provenance of a special case, so it is not a renamed known result presented as new. The only support asserted rather than demonstrated is the footnote to Fact 2.21, which says that the arguments in [5, Section 2.2] generalize to arbitrary abelian groups and directed I; this is a proof obligation, not a circular reduction. No identified step reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters: the results are proof-theoretic, with no data fitting or hand-tuned constants. The central claims rest on ZFC, the coherence/triviality equivalence (Fact 2.21), Goblot's theorem, standard facts about the dominating number, and cited results from [19] and [5].

assumptions (6)
  • standard math ZFC
    All theorems are proved in ZFC; consistency results are forcing extensions of L, hence relative to consistency of ZFC.
  • domain assumption Equivalence between vanishing of lim^n A_I[H] and triviality of all coherent H-valued n-families (Fact 2.21)
    Imported from [16] and [5, Section 2.2]; this is the bridge connecting homological algebra to the combinatorial constructions.
  • standard math Goblot's vanishing theorem (Prop 2.7)
    Used repeatedly to obtain trivializations at ordinals below omega_n; the paper provides a proof, but it is a known theorem from [10].
  • domain assumption Results from Velickovic-Vignati [19] (Lemma 3.6, Theorem 4.5, Lemma 4.3)
    The consistency proofs in Section 5 rely on these external results; they are cited, not proved.
  • standard math cf(d) > omega (dominating number has uncountable cofinality)
    Used implicitly in Corollary 4.5 to conclude from c <= aleph_omega that d = aleph_n for some finite n; standard ZFC fact.
  • standard math Godel's constructibility V=L in the ground model
    Used to obtain stationary S subset S^lambda_kappa with square( lambda, S) from Jensen [13]; L is used as the ground model for forcing.

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Cite this review

Pith. "Pith review of Simultaneously nonvanishing higher derived limits." pith.science (2026). https://pith.science/paper/FTWMC67L

@misc{pith2026241115856,
  author       = {Pith},
  title        = {Pith review of: Simultaneously nonvanishing higher derived limits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FTWMC67L}},
  note         = {Machine review of arXiv:2411.15856}
}
abstract

The derived functors $\lim^n$ of the inverse limit find many applications in algebra and topology. In particular, the vanishing of certain derived limits $\lim^n \mathbf{A}[H]$, parametrized by an abelian group $H$, has implications for strong homology and condensed mathematics. In this paper, we prove that if $\mathfrak{d}=\omega_n$, then $\lim^n \mathbf{A}[H] \neq 0$ holds for $H=\mathbb{Z}^{(\omega_n)}$ (i.e. the direct sum of $\omega_n$-many copies of $\mathbb{Z}$). The same holds for $H=\mathbb{Z}$ under the assumption that $\mathrm{w}\diamondsuit(S^{k+1}_k)$ holds for all $k < n$. In particular, this shows that if $\lim^n \mathbf{A}[H] = 0$ holds for all $n \geq 1$ and all abelian groups $H$, then $2^{\aleph_0} \geq \aleph_{\omega+1}$, thus answering a question of Bannister. Finally, we prove some consistency results regarding simultaneous nonvanishing of derived limits, again in the case of $H = \mathbb{Z}$. In particular, we show the consistency, relative to $\mathsf{ZFC}$, of $\bigwedge_{2 \leq k < \omega} \lim^k \mathbf{A} \neq 0$.

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Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Higher limits of wider systems

    math.LO 2025-07 unverdicted novelty 8.0 of 10

    Under GCH plus diamond principles, and in Gödel's constructible universe, the higher derived limits lim^n A_λ are nonzero for every cardinal λ where Goblot's vanishing theorem does not force them to zero.

  2. Nonvanishing Higher Derived Limits without $w\diamondsuit_{\omega_1}$

    math.LO 2025-06 conditional novelty 8.0 of 10

    Under hypotheses d=ℵ_n plus weak diamonds, the nth derived limit of a natural inverse system of abelian groups is nonzero, and the second derived limit is nonzero in the Miller and Mitchell models.

  3. Merging $\lim^1 \mathbf{A} \ne 0$ with other nonvanishing constructions

    math.LO 2026-07 accept novelty 6.5 of 10

    It is consistent that b=d=ω_n and lim^k A ≠ 0 for all 1≤k≤n, and that b=d=ω_{ω+2} with lim^k A ≠ 0 for every k≥1, by new forcings for lim^1 A ≠ 0 compatible with prior nonvanishing methods.

  4. Infinitary combinatorics in condensed math and strong homology

    math.AT 2024-12 conditional novelty 6.0 of 10

    Higher derived limits of the systems A_kappa_lambda are shown to control non-fullness, non-additivity of strong homology, and non-compactness of products of compact projective condensed anima.

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