{"id":"22bf56d0-fb7f-445c-9966-2ac02e659892","arxiv_id":"1908.06526","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"There is a separable Banach space, the Kadec space, for which Ext^n(K,K) is nonzero for every n, so its projective and injective dimensions are infinite.","lead":"The paper develops higher extension functors Ext^n for Banach spaces and gives the first example of a separable Banach space K with a nonzero self-extension Ext^n(K,K) in every degree n. This makes K the first separable space known to have infinite projective and injective homological dimension.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the iterated-kernel premise of Proposition 5.1 is adequately supported and the central claim does not depend on the paper's auxiliary unproved assertions.","rationale":"The reader's weakest_assumption correctly identifies the iterated-kernel premise as the place to scrutinize, but my examination finds that the premise is adequately proved within the paper using standard results. The central claim of Proposition 5.1 does not hinge on the auxiliary unproved assertions that motivate the CONDITIONAL verdict; those issues remain real but do not change the assessment. I therefore see no reason to adjust the reader's verdict.","tokens_in":17794,"tokens_out":23550,"duration_ms":214652,"concrete_test":"Verify the critical external input by consulting [37, Proposition 5.2] and re-deriving the iterative step: assume kappa^n(L1) is complemented in its bidual, apply Theorem 4.1(4) with X = kappa^{n-1}(L1) and Y = kappa^n(L1), and check that the vanishing of Ext^1(kappa^{n-1}(L1), kappa^n(L1)) forces the projective presentation of kappa^{n-1}(L1) to split, contradicting the induction hypothesis and establishing that each kappa^n(L1) is a separable L1-space not isomorphic to l1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I reviewed the proof of Proposition 5.1 and its supporting claim that for X = L1 every iterated projective kernel kappa^n(X) is a separable L1-space not isomorphic to l1 and uncomplemented in its bidual. The argument is sound: [37, Proposition 5.2] yields that the kernel of a quotient map between L1-spaces is an L1-space; separability is preserved by choosing the projective presentation with P = l1; and if kappa^n(X) were complemented in its bidual, Theorem 4.1(4) applied with first variable kappa^{n-1}(X) and second variable kappa^n(X) would force Ext^1(kappa^{n-1}(X), kappa^n(X)) = 0, which would split the projective presentation of kappa^{n-1}(X), making it projective and eventually forcing L1 to be isomorphic to l1. Thus the nonzero of Ext^n(L1, kappa^n L1) follows from non-projectivity, and the use of Lemma 4.1 with Kadec's universal space is valid because L1 and kappa^n(L1) are separable L1-spaces with the BAP. I found no internal inconsistency in this central argument. The paper's conditional status is due to auxiliary issues: the explicitly omitted proof of Ext^n(C_p) != 0 (Section 5), the import of Ext^2(l_p) from the same-team preprint [7], and the arXiv metadata abstract overclaiming a Hilbert-space result that v2 itself leaves open. None of these affects Proposition 5.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":18049,"tokens_out":16044,"duration_ms":141178,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Section 5, paragraph before Proposition 5.1"},{"comment":"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.","section":"Abstract (arXiv metadata) vs. Section 5, Problem 1"},{"comment":"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.","section":"Section 4, remark on Ext^3"},{"comment":"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.","section":"Section 4, Lemma 4.1"},{"comment":"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.","section":"Section 4, Proposition 4.2"},{"comment":"The manuscript contains numerous typographical and OCR artifacts (e.g., 'Kadec' space', 'develop ed', 'the the', 'i t'). A careful proofreading pass is recommended.","section":"General"}],"recommendation":"minor_revision","confidential_remarks":"The main mathematical result — Proposition 5.1 — is sound, and the proof is convincing after checking the auxiliary facts about κ^n(L1). The manuscript would benefit from correcting the arXiv abstract, proving or rephrasing the Ext^n(C_p) assertion, and addressing the reliance on the preprint [7]. The editor should consider whether the use of [7] is acceptable or whether the authors should be asked to include the needed proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the main result is real. The paper gives the first separable Banach space K with Ext^n(K,K) ≠ 0 for every n, hence pd(K) = id(K) = ∞. The proof is a clean application of Wodzicki's theorem that L1 has infinite projective dimension, the Kadec–Pelczyński–Wojtaszczyk universal embedding, and the elementary Lemma 4.1. I read the key step about the iterated kernels κ^n(L1) carefully: the argument that each κ^n(L1) is a separable L1-space not isomorphic to ℓ1, and that Ext^n(L1, κ^n(L1)) ≠ 0, is supported by Lindenstrauss–Rosenthal and Theorem 4.1(4); the stress-test note is right that the contrapositive works. I do not see a load-bearing gap. The paper also does useful organizational work: Theorem 4.1 collects n-th order Sobczyk, Lindenstrauss lifting, and Johnson–Zippin, and Proposition 6.1, distinguishing Ext^2 in Banach versus quasi-Banach spaces, is a genuinely new and cute example.\n\nThe soft spots are real but mostly peripheral. The arXiv abstract claims the homological dimension of Hilbert space is infinite, but the body says only pd(ℓ2), id(ℓ2) ≥ 3 and leaves it as Problem 1. That is a direct contradiction and should be fixed before this circulates. The assertion that Ext^n(C_p) ≠ 0 for all n is explicitly stated without proof (\"we omit the proof\"). It is not used in Proposition 5.1, but an unproved claim in a foundational paper stands out. The paper also leans on the same-team preprint [7] for Ext^2(ℓ_p) ≠ 0; that is only used in the closing remark of Section 4, not the main theorem, so it is not a fatal issue, but a public version or a clear pointer would help. The authors acknowledge the text was error-prone; the current version still has typos, but I did not find a mathematical error in the central argument.\n\nWho it is for: anyone working on Ext functors in Banach spaces or three-space problems. It deserves a serious referee: the main theorem is significant and short enough to check. The referee should push for an honest abstract, a proof or deletion of the C_p claim, and either a citation to a public version of [7] or a clear statement that the Hilbert-space problem remains open.","headline":"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.","tokens_in":18680,"tokens_out":3257,"would_cite":true,"duration_ms":31404,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46M15","46M18","46M10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Ext functor","Banach space","homological dimension","projective dimension","injective dimension","Kadec space","L1-space","exact sequence"],"falsifier":"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.","tokens_in":17539,"feed_emoji":"♾️","tokens_out":15548,"duration_ms":116509,"temperature":0.7,"pith_summary":"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.","feed_headline":"Kadec's space has nonzero self-extensions at every length","feed_subtitle":"The result makes K the first separable Banach space with infinite projective and injective dimension.","key_machinery":"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.","core_discovery":"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 $\\operatorname{Ext}^n(K,K) \\neq 0$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Proposition 5.2 used to show the projective kernel of an L1-space is again an L1-space, the key iteration step.","marker":"[37]"},{"why":"Introduces projective, injective, and flat dimensions for Banach spaces and the expectation that they are infinite for classical spaces; also supplies parts of Proposition 4.1.","marker":"[56]"},{"why":"Provides the construction of Kadec's space K as a complementably universal separable BAP space.","marker":"[26]"},{"why":"One of the independent constructions of the same universal space with a basis.","marker":"[46]"},{"why":"The other independent construction of the universal space K cited for its properties.","marker":"[47]"}],"fun_headline_variants":["Kadec space has nonzero self-Ext at every degree","A separable Banach space with infinite homological dimension","Self-extensions never vanish in Kadec's space","Ext^n(K,K) ≠ 0: Kadec's space, all n","Infinite homological dimension for Kadec's space"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Kadec space has nonzero self-Ext at every degree","A separable Banach space with infinite homological dimension","Self-extensions never vanish in Kadec's space","Ext^n(K,K) ≠ 0: Kadec's space, all n","Infinite homological dimension for Kadec's space"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000375,"raw_usage":{"total_tokens":1988,"prompt_tokens":924,"completion_tokens":1064,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":977}},"tokens_in":540,"tokens_out":1064,"duration_ms":10359,"temperature":1.0,"reasoning_tokens":977,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:42:30.770097+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Lindenstrauss and H","cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 5.2 used to show the projective kernel of an L1-space is again an L1-space, the key iteration step."},{"cited_title":"Wodzicki, Homological dimensions of Banach spaces , in Linear and Complex Analysis Problem Book 3, Part I , V.P","cited_arxiv_id":null,"evidence_quote":"Introduces projective, injective, and flat dimensions for Banach spaces and the expectation that they are infinite for classical spaces; also supplies parts of Proposition 4.1."},{"cited_title":"I: Kadets, On complementably universal Banach spaces , Studia Math","cited_arxiv_id":null,"evidence_quote":"Provides the construction of Kadec's space K as a complementably universal separable BAP space."},{"cited_title":"Pe/suppress lczy´ nski,Universal bases, Studia Math","cited_arxiv_id":null,"evidence_quote":"One of the independent constructions of the same universal space with a basis."},{"cited_title":"Pe/suppress lczy´ nski and P","cited_arxiv_id":null,"evidence_quote":"The other independent construction of the universal space K cited for its properties."}],"review_version":1}