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Homological dimensions of Banach spaces

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Pith's one-line read Kadec's separable Banach space has nonzero self-extensions at every length, making it the first known example of a separable space with infinite homological dimension.

desk verdict The first separable Banach space with nonzero self-extensions at every finite length is real and the central proof holds up, but the abstract overclaims the Hilbert-space result and an auxiliary assertion is unproved. read the letter →

arxiv 1908.06526 v2 pith:DVITI3CX submitted 2019-08-18 math.FA

Félix Cabello Sánchez , Jesús M .F. Castillo , Ricardo García This is my paper · ORCID
classification math.FA MSC 46M1546M1846M10
keywords ExtfunctorBanachspacehomologicaldimensionprojectiveinjectiveKadecL1-spaceexactsequence
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 develops the theory of higher self-extensions ($\operatorname{Ext}^n$) for Banach spaces and exhibits the first separable Banach space whose self-extensions are nonzero at every length: Kadec's space $K$. The authors prove that $\operatorname{Ext}^n(K,K) \neq 0$ for every positive integer $n$, so both the projective and the injective dimension of $K$ are infinite. This matters because it provides the first concrete example of a separable Banach space with infinite homological dimension, a possibility long expected but previously unsupported.

What carries the argument

The main workhorse is the iterated projective kernel $\kappa^n(X)$, obtained by repeatedly taking the kernel of a projective ($\ell_1$) presentation of $X$, together with the reduction formula $\operatorname{Ext}^n(X,Y) = \operatorname{Ext}^1(\kappa^{n-1}(X),Y)$. For $L_1$-spaces, a result of Lindenstrauss and Rosenthal ensures each $\kappa^n(L_1)$ is again an $L_1$-space; the non-triviality comes from the fact that these kernels are not isomorphic to $\ell_1$ and are uncomplemented in their bidual, so the corresponding projective presentations cannot split. Kadec's space $K$ then acts as a universal receptacle: because every separable space with the bounded approximation property, including $L_1$ and its kernels, appears as a complemented subspace of $K$, the heredity lemma transfers the nonzero $\operatorname{Ext}^n$ from $L_1$ to $K$ itself.

What would settle it

Find some $n \geq 1$ for which the $n$-th iterated projective kernel $\kappa^n(L_1)$ is isomorphic to $\ell_1$ (or is a projective Banach space). Then $\operatorname{Ext}^n(L_1, \kappa^n L_1) = 0$, and the heredity argument yielding $\operatorname{Ext}^n(K,K) \neq 0$ would fail. More directly, exhibit any $n$ with $\operatorname{Ext}^n(K,K) = 0$ for Kadec's space; that would contradict Proposition 5.1.

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Extended reading notes

Core claim

The paper's central claim is that Kadec's separable Banach space $K$, which contains complemented copies of every separable Banach space with the bounded approximation property, satisfies $\operatorname{Ext}^n(K,K) \neq 0$ for all $n \geq 1$. Consequently $\operatorname{pd}(K) = \operatorname{id}(K) = \infty$. The argument first shows that for $X = L_1$ every iterated projective kernel $\kappa^n(X)$ is an $L_1$-space not isomorphic to $\ell_1$ and uncomplemented in its bidual, so the $n$-exact sequences built from these kernels do not split and $\operatorname{Ext}^n(L_1, \kappa^n L_1) \neq 0$. Since $L_1$ and all $\kappa^n(L_1)$ have the bounded approximation property, they embed as complemented subspaces of $K$, and the heredity lemma for $\operatorname{Ext}$ (if $\operatorname{Ext}^n(X,Y)=0$ then $\operatorname{Ext}^n(A,B)=0$ for complemented subspaces) forces $\operatorname{Ext}^n(K,K) \neq 0$.

Load-bearing premise

The proof requires that every iterated projective kernel of $L_1$ is an $L_1$-space not isomorphic to $\ell_1$ and uncomplemented in its bidual; if any one of these kernels were projective or isomorphic to $\ell_1$, the chain of nonzero self-extensions for Kadec's space would collapse.

Editorial extensions

If this is right

  • This is the first separable Banach space known to have a nonzero self-extension at every finite length, so its projective and injective dimensions are both infinite.
  • The reduction formula $\operatorname{Ext}^n(X,Y) = \operatorname{Ext}^1(\kappa^{n-1}(X),Y)$ gives an iterative way to compute higher Ext: pass to the projective kernel and compute ordinary Ext.
  • The result confirms Wodzicki's expectation that homological dimensions are infinite for most classical spaces, with $L_1$ itself already having infinite projective dimension.
  • Because $K$ contains complemented copies of every separable BAP space, any pair of separable BAP spaces $X,Y$ with $\operatorname{Ext}^n(X,Y) \neq 0$ yields nonzero $\operatorname{Ext}^n(K,Y)$ and $\operatorname{Ext}^n(X,K)$, as the authors explicitly note.

Reading between the lines

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

  • The same 'universal receptacle' strategy could be applied to any space that contains complemented copies of a family of spaces with non-splitting iterated kernels, potentially producing many more examples of infinite homological dimension.
  • The argument relies on the bounded approximation property to place $L_1$ and its kernels inside $K$; a separable space lacking BAP might sidestep the heredity transfer, so the role of BAP here is genuinely load-bearing.
  • The Hilbert-space question (whether $\operatorname{pd}(\ell_2)$ and $\operatorname{id}(\ell_2)$ are infinite) remains open, but the paper's reduction formula suggests that a negative answer would require exhibiting reflexive spaces with non-splitting higher extensions, which the authors note is currently out of reach.
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Editorial analysis

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

Referee Report

0 major / 6 minor

Summary. The paper develops the theory of the functors Ext^n in the category of Banach spaces, giving length-reduction formulas via projective and injective presentations, several vanishing and non-vanishing results, and a comparison of Ext^2 in the Banach and quasi-Banach settings. Its central theorem (Proposition 5.1) states that for Kadec's separable universal space K, Ext^n(K,K) is nonzero for every n, so that both the projective and injective dimensions of K are infinite. The proof combines Wodzicki's result that L1 has infinite projective dimension, the fact that L1-spaces have the bounded approximation property, Kadec's theorem that K contains complemented copies of every separable Banach space with the BAP, and Lemma 4.1 on complemented subspaces. The paper also records several auxiliary examples and remarks, and it explicitly leaves open the computation of pd(ℓ2) and id(ℓ2).

Significance. If correct, the main theorem provides the first example of a separable Banach space with nonzero self-extensions at every finite length, a natural and previously open question. The proof is short and elegant, and the paper lays useful foundations for the study of Ext^n in Banach spaces. The paper is also honest about unresolved issues, such as the homological dimensions of the Hilbert space. The central argument is internally coherent and does not rely on fitted parameters or circular reasoning; it properly credits Wodzicki and Kadec for the key ingredients. The main weakness is the presence of several auxiliary unproved assertions and a mismatch between the arXiv metadata abstract and the content of the manuscript, none of which affects the validity of Proposition 5.1.

minor comments (6)
  1. [Section 5, paragraph before Proposition 5.1] The paper states without proof that Ext^n(C_p) ≠ 0 for all n, saying 'we omit the proof'. Since this is presented as a fact rather than a conjecture, please either supply a proof or clearly label the statement as a conjecture; it is not needed for the main theorem, but as written it is an unsupported claim.
  2. [Abstract (arXiv metadata) vs. Section 5, Problem 1] The arXiv metadata abstract claims that 'the homological dimension/codimension of Hilbert spaces is infinite', but the v2 text does not prove this and indeed Problem 1 asks to compute pd(ℓ2), id(ℓ2), noting that only the lower bound ≥3 is known. Please align the abstract with the actual content of the paper.
  3. [Section 4, remark on Ext^3] The paper relies on the same-team preprint [7] for the nontriviality of Ext^2(ℓp) and Ext^2(ℓ2). Since [7] is a preprint rather than a published reference, please clarify its status or include a proof of the needed facts; the main theorem does not depend on these results.
  4. [Section 4, Lemma 4.1] The proof of Lemma 4.1 uses the notation 'Q F P ı' which is difficult to parse. A clearer description of the pullback/pushout operations (for example, explicitly writing the pullback along P and the pushout along the inclusion B→Y) would improve readability.
  5. [Section 4, Proposition 4.2] The proof of Proposition 4.2 is terse, especially the construction of X = c0 ⊕ (ℓ∞(c)/(ℓ∞/c0)) and the statement that 'X can replace cκ^2(c0)'. Please expand this argument for clarity, as the diagram is difficult to follow.
  6. [General] The manuscript contains numerous typographical and OCR artifacts (e.g., 'Kadec' space', 'develop ed', 'the the', 'i t'). A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 2.0 of 10

No structural circularity in Proposition 5.1; the Kadec-space result rests on independent external results, with only peripheral same-team citations.

full rationale

The central derivation is not circular. Proposition 5.1 is proved by embedding L1 and κ^n L1 as complemented subspaces of Kadec's universal space K and applying Lemma 4.1. The required non-vanishing Ext^n(L1,κ^n L1) is obtained in Section 4 from a splitting argument: if κ^n L1 were complemented in its bidual, Theorem 4.1(4) (Lindenstrauss lifting, based on [35,37]) would force the projective presentation of κ^{n-1}L1 to split, making κ^{n-1}L1 projective and eventually forcing L1 to be isomorphic to ℓ1, a contradiction. The L1-kernel property is cited from [37, Proposition 5.2], and pd(L1)=∞ is credited to Wodzicki [56]; both are external, parameter-free results that do not include the target space K. Kadec's universal space is attributed to [26,46,47]. No fitted parameter is renamed as a prediction, and no displayed identity reduces to its own input. There are peripheral self-citations, notably [7] (same-team preprint) for Ext^2(ℓ_p)≠0 and Ext^2(ℓ_2)≠0 in illustrative remarks, and [2,6,13] for characterizations in side statements; none is load-bearing for Proposition 5.1. The manuscript itself flags missing support: the assertion 'Ext^n(C_p)≠0 for all n' is said to be easy but its proof is omitted, and the abstract's claim that homological dimension/codimension of Hilbert spaces is infinite is not established in v2—Section 5 states only pd(ℓ2), id(ℓ2)≥3 and poses Problem 1. These are completeness/correctness concerns, not circularity. Overall the derivation chain for the main theorem is self-contained modulo independent external results.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted to data. The paper uses standard Banach-space constants such as Bourgain's constant as existential inputs, not fitted values. It introduces no new entities; Kadec's space and Bourgain's B are prior constructions. The central claim rests mainly on Wodzicki's pd(L1) = infinity and Kadec's universal space, with auxiliary reliance on same-team preprints.

assumptions (7)
  • standard math Yoneda Ext and the long homology sequences are exact in the category of Banach spaces
    Invoked throughout Sections 2-3 and Appendix 7; cited to Frerick-Sieg [20] and Mitchell [42].
  • standard math Every Banach space admits projective presentations by l1(I) and injective presentations into l-infinity
    Used in Section 3 to reduce Ext^n to Ext^1 of iterated kernels; cites Kothe [34] and Lindenstrauss-Rosenthal [37].
  • standard math Order-1 lifting and extension theorems such as Sobczyk, Lindenstrauss, and Johnson-Zippin extend to all orders
    Theorem 4.1 is proved from these via Proposition 3.1; the n-order transfer is the paper's own argument.
  • domain assumption A separable L1-space not isomorphic to l1 has infinite projective dimension
    The proof in Sections 4-5 of pd(L1) = infinity is a sketch relying on the uncomplemented-in-bidual property of iterated kernels; this is load-bearing for Proposition 5.1.
  • standard math Kadec's space exists and contains complemented copies of all separable Banach spaces with the bounded approximation property
    Used in Proposition 5.1; prior results by Kadec, Pelczynski, and Wojtaszczyk [26,46,47].
  • domain assumption Ext^2(lp) is nonzero for 1 < p < infinity, including l2
    Cited to the same-team preprint [7]; used for the Ext^3 remark at the end of Section 4, not for the Kadec theorem.
  • standard math Bourgain's finite-dimensional complementation construction and Amir's non-injectivity of l-infinity/c0
    Used in Propositions 4.1 and 4.2 to build B and L-infinity spaces with nonzero Ext^2.

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Pith. "Pith review of Homological dimensions of Banach spaces." pith.science (2026). https://pith.science/paper/DVITI3CX

@misc{pith2026190806526,
  author       = {Pith},
  title        = {Pith review of: Homological dimensions of Banach spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DVITI3CX}},
  note         = {Machine review of arXiv:1908.06526}
}
abstract

The purpose of this paper is to lay the foundations for the study of the problem of when $\Ext^n(X, Y)=0$ in Banach/quasi-Banach spaces. We provide a number of examples of couples $X,Y$ so that $\Ext^n(X,Y)$ is (or is not ) $0$, including the first example of a separable Banach space $\mathscr K$ so that $\Ext^n(\mathscr K, \mathscr K)\neq 0$ for all $n\in \N$. Such space moreover provides the first example of Banach spaces with infinite homological dimension/codimension. We also show that the homological dimension/codimension of Hilbert spaces is infinite. The final section is devoted to compare $\Ext^2(\cdot, \cdot)$ in Banach and Quasi-Banach spaces.

Discussion (0). Continue with ORCID to comment.

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