{"id":"99d6b15e-681c-4ee1-b244-69460f8a721b","arxiv_id":"2412.19605","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper proves several new theorems connecting infinite combinatorics to condensed mathematics and strong homology, including a ZFC counterexample to additivity of strong homology on compact spaces. It shows that one family of abstract limit computations drives failures of fullness, Banach-Smith duality, and compactness of products of condensed objects.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing concern for the central claim is the under-specified ∞-categorical reduction in §2.2, not the reader's flagged higher-degree Lemma 2.9, which is unused by Corollaries 2.11 and 3.4.","rationale":"The reader's weakest assumption, Lemma 2.9 for n > 1, is delegated to [18] and plausibly affects Theorem 2.20 and Theorem 4.11, but it is not used in the proof of the strongest_claim as stated: Theorem 2.10(5) and Corollaries 2.11 and 3.4 require only the n = 1 case of the dictionary, which the paper proves in detail. The central claim's most fragile step is the conversion from a nonvanishing derived limit to a failure of fullness. That conversion is outsourced to a chain of ∞-categorical reductions in Section 2.2, which the authors themselves treat with a 'naive' reading and without complete rigor. This is a load-bearing externality because a gap there would sever the link between the very solid combinatorial computation of Section 2.3 and the headline theorem on fullness. The paper is otherwise careful and transparent, and I found no internal inconsistency in the combinatorial arguments underpinning lim^1 A_{ω,ω1} ≠ 0. The appropriate verdict remains CONDITIONAL, but the condition should be placed on the §2.2 reduction, not on the higher-degree dictionary.","tokens_in":44266,"tokens_out":21782,"duration_ms":190793,"concrete_test":"Re-derive the reduction of Section 2.2 in the ∞-categorical formalism of [53]/[54], verifying each step (dévissage, passage to free resolutions, use of solid projectivity of ∏ Z_{f(i)}) as an equivalence for Pro(D(Ab))^b. In particular, test whether equality (4) for all free groups G_i = ⊕_J Z and H = ⊕_K Z is sufficient for fullness; search for a bounded complex E and system C_i for which the natural map is full yet the corresponding derived-limit expression fails to be concentrated in degree zero. If such a counterexample exists, Theorem A does not follow from Theorem 2.10(5).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The ZFC computation lim^1 A_{ω,ω1} ≠ 0 (Theorem 2.10(5)) is proved directly and appears sound. The step from this computation to non-fullness of Pro(D(Ab))^b → D(Cond(Ab)) (Corollary 2.11) runs through the reduction in Section 2.2. That reduction is not fully established: the paper explicitly adopts a 'relaxed or naive' ∞-categorical reading, and it invokes [53, 4.4.2.7], [52, Prop. 15.4.2], [77, Thm 5.8], and [39] without spelling out boundedness and size conditions. In particular, the claim that fullness reduces to equality (4) for free groups G_i = ⊕_J Z and H = ⊕_K Z, and then to vanishing of Rlim A_{I,J}[H], is an equivalence asserted after 'standard dévissage' but not proved. If the natural map is full on the bounded subcategory while equation (4) fails for some non-free or degree-shifted reason, the nonvanishing lim^1 would not imply non-fullness. This is a genuine external dependency, flagged by the authors themselves, and it is the real load-bearing point for the 'not full' half of the central claim. The reader's cited Lemma 2.9(n>1) does not affect Corollaries 2.11 or 3.4, which use only the n = 1 dictionary proved in the paper.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a package of infinitary combinatorial tools—multidimensional coherent families, the inverse systems A_{κ,λ}, and their higher derived limits lim^n A_{κ,λ}[H]—and applies them to four main theorems. Theorem A asserts that the natural functor Pro(D(Ab))^b → D(Cond(Ab)) is not full. Theorem B gives consistency results, conditional on lim^n A[H] = 0, concerning Ext groups in Cond(Ab), the structure of RHom against products of free abelian groups, and solid Q_p-Banach duality. Theorem C gives a ZFC counterexample to additivity of strong homology, witnessed by a countable sum of compact spaces Y^{n,ℵ_1}. Theorem D states that constant sheaves on extremally disconnected spaces are injective for finite fields, that certain products of Čech–Stone compactifications have infinite injective dimension, and that products of compact projective condensed anima need not be compact. The combinatorial core includes the ZFC computation lim^1 A_{ω,ω_1} ≠ 0 (Theorem 2.10(5)), the consistent higher-degree nonvanishing of Theorem 2.20, and the construction of nontrivial n-coherent families in Theorem 4.11.","tokens_in":44492,"tokens_out":25184,"duration_ms":247989,"significance":"If the ∞-categorical transfer in §2.2 is made fully rigorous, Theorems A, B, and D are substantial and will be influential. The strong homology part of the paper, Theorem C together with Theorem 3.3 and Corollary 3.4, is largely independent of that transfer and appears sound; it provides a notably simpler, compact ZFC counterexample to additivity than Prasolov's. The explicit combinatorial constructions, especially the proof of Theorem 2.10(5) and the recursive construction in Theorem 4.11, are a genuine strength: they are detailed, checkable, and likely to be reusable. The paper also performs a useful service by organizing several Clausen–Scholze questions and connecting them to a substantial set-theoretic literature. The main weakness is that the reduction from fullness of the natural functor to equality (4), and hence to vanishing of the derived limits, is asserted through a 'relaxed or naive' ∞-categorical reading and is not proved with the necessary hypotheses.","major_comments":[{"comment":"The reduction from full faithfulness of the natural functor (3) to equality (4), and then to vanishing of lim^n A_{I,J}[H], is the load-bearing step for Theorem A and for Theorem 2.13 (hence Theorem B), but it is not established in the manuscript. The paper explicitly says it works with a 'relaxed or naive' reading of the ∞-categorical manipulations, and the specific claims 'RHom commutes with all finite limits and colimits' and the 'standard dévissage' reduction to free groups concentrated in degree zero are not proved; the cited references [53, 4.4.2.7], [52, Prop. 15.4.2], [77, Thm. 5.8], and [39] are not accompanied by the boundedness, size, and projectivity checks needed here. In particular, the identification of the right-hand side of (4) with Rlim A_{I,J}[H] depends on the projectivity of ∏_i Z^{f(i)} in Solid and on slenderness, and the reader cannot verify that these apply for arbitrary I and J. Since Corollary 2.11 is literally 'immediate from item (5) of Theorem 2.10, together with Proposition 2.5', the non-fullness theorem is only as solid as this reduction. Please either prove Proposition 2.5 directly for the specific objects G_i = ⊕_J Z and H = ⊕_K Z, which would avoid most of the problematic reduction, or supply a complete ∞-categorical proof with all hypotheses verified.","section":"§2.2 (pp. 8–10), Proposition 2.5, Corollary 2.11"},{"comment":"The n = 1 case of Lemma 2.9 is proved in the text, but for n > 1 the proof is delegated to [18, Section 2.1] with a brief assurance that the argument is 'close in spirit' and 'only a bit more tedious'. This lemma is the dictionary used to translate nonvanishing of lim^n into nontrivial n-coherent families in Theorem 2.10, Theorem 2.20, and Theorem 4.11. Because those theorems are among the paper's central new contributions, the higher-degree case should either be proved in full or the precise statement from [18] should be quoted with the hypotheses verified for the systems A_{κ,λ}[H]. In particular, the alternating conventions and the use of equation (7) need to be checked explicitly.","section":"Lemma 2.9 (p. 12)"},{"comment":"The proof establishes, at most, that the natural functor is not fully faithful in the ∞-categorical sense: nonvanishing lim^1 contradicts the implication 'fully faithful ⇒ vanishing' of Proposition 2.5. The theorem, however, states that the functor is 'not full'. For ordinary categories, 'full' and 'fully faithful' are not interchangeable unless faithfulness is known separately, and the paper does not prove faithfulness of the functor (3). Please either change the statement and abstract to 'not fully faithful', or add an argument showing that the induced map on Hom sets is not surjective (or that the functor is faithful), so that 'not full' follows.","section":"Theorem A (p. 2) and Corollary 2.11 (p. 13)"}],"minor_comments":[{"comment":"The sentence 'RHom commutes with all finite limits and colimits' should be made precise. In the relevant stable ∞-categorical setting, mapping spectra preserve finite limits in each variable, but they do not preserve arbitrary colimits; the colimits over I([J]<ω) used later are not finite, so the intended statement needs qualification.","section":"§2.2 (p. 9)"},{"comment":"The final nontriviality argument in the induction step is very compressed. In particular, the step from the failure of equation (20) to the claim that the associated family Ψ is trivial, and the later notation 'supp(φ_α)' for the fixed tuple α obtained from Fodor's lemma, should be expanded for the reader to check the contradiction.","section":"§4.1, proof of Theorem 4.11 (pp. 38–40)"},{"comment":"Several key references are unpublished or online lecture notes ([76], [77], [27]) and personal communication ([78]). This is common in the field, but for the journal version the authors should give the most stable available references or precise pointers (theorem numbers, dates, or published versions) for the claims cited from them.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely to be a strong contribution after revision. The set-theoretic and combinatorial core—especially Theorem 2.10(5), Theorem 2.20, and Theorem 4.11—is valuable and appears sound. My main concern is the under-specified ∞-categorical reduction in §2.2, which is explicitly flagged by the authors as 'naive'; I would not be comfortable accepting Theorem A and the relevant parts of Theorem B until that reduction is either replaced by a direct proof of Proposition 2.5 for the needed special case or made fully rigorous with all size and boundedness hypotheses checked. The terminology issue around 'full' versus 'fully faithful' also affects the precise statement of the main theorem. The strong homology result (Theorem C) is, in my reading, independent of the problematic reduction and can be cited as a solid contribution now."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper delivers on its promises: it gives a ZFC proof that the natural functor Pro(D(Ab))^b → D(Cond(Ab)) is not full, a new compact ZFC counterexample to additivity of strong homology, and a negative answer to the compact-projectives-product conjecture. The computation of lim^1 A_{ℵ0,ℵ1} ≠ 0 is detailed and looks sound; that's the engine of Theorem C and the non-fullness claim.\n\nWhat's genuinely new: Theorem C's counterexample via Y^{n,ℵ1} is simpler and compact, improving on Prasolov's non-metrizable example. The paper is also admirably transparent about what is due to Clausen–Scholze; the authors' contribution is the derived limit analysis, and they say so clearly.\n\nThe soft spots are real but concentrated. The reduction in Section 2.2 from the full faithfulness question to equation (4) and then to vanishing of Rlim A_{I,J}[H] is asserted after 'standard dévissage' and a 'relaxed or naive' ∞-categorical reading. The citations are there, but the boundedness and size conditions are not checked. This is the load-bearing step for Theorem A, and a referee should push for a more careful treatment. The reader's concern about Lemma 2.9 for n>1 is less important for the main ZFC results, since Corollaries 2.11 and 3.4 only use the n=1 case proved in the paper; but the higher-degree dictionary does support Theorem 2.20 and Theorem 4.11, so that delegation to [18] is worth verifying. Theorem 4.4 relies on a personal communication from Scholze for the final step; that's acceptable but should be flagged.\n\nNone of this looks fatal. The concrete computation of the RHS for free groups seems to stand on its own, so I expect the non-fullness conclusion survives even if the general reduction needs tightening. The paper is clearly written, well-organized, and the combinatorial arguments are worked out in detail. It deserves a serious referee. I'd send it to peer review with a request to expand Section 2.2 and to clarify the size conditions. I'd also bring it to my reading group.","headline":"Solid paper with a real result; the main soft spot is the under-specified ∞-categorical reduction in §2.2, not the higher-degree dictionary the reader flagged.","tokens_in":45104,"tokens_out":7844,"would_cite":true,"duration_ms":71425,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18F10","18G80","03E05","03E35","03E75","13D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that one nonvanishing derived limit, $\\lim^1 A_{\\omega,\\omega_1}$, simultaneously produces failures of fullness in condensed derived categories, of additivity in strong homology, and, in higher degrees, of product…","keywords":["condensed mathematics","anima","compact projectives","derived limits","n-coherence","strong homology","Banach-Smith duality","infinitary combinatorics"],"falsifier":"Take the coherent family $\\Phi=\\langle\\varphi_f:X(f)\\to\\mathbb{Z}\\rangle$ constructed in the proof of Theorem 2.10(5) from a ladder of finite-to-one functions $e_\\alpha:\\alpha\\to\\omega$ such that consecutive restrictions agree modulo finite sets, and check whether a global $\\psi:\\omega\\times\\omega_1\\to\\mathbb{Z}$ exists whose restriction to each $X(f)$ agrees with $\\varphi_f$ modulo finite sets. Claim 2.12 asserts that no such $\\psi$ exists; producing one would force $\\lim^1 A_{\\omega,\\omega_1}=0$ and would collapse Theorems A and C. The higher-degree analogue is to search for an $(n-1)$-trivialization of an $n$-coherent family built in Theorem 4.11; finding one would make the corresponding $\\mathrm{H}^n(U;\\bigoplus_\\kappa K)$ vanish.","tokens_in":43980,"feed_emoji":"🧮","tokens_out":12225,"duration_ms":109287,"temperature":0.7,"pith_summary":"This paper establishes that a single family of inverse systems — the groups $A_{\\kappa,\\lambda}[H]$ indexed by finite-support functions from $\\kappa$ to $[\\lambda]^{<\\omega}$ — carries a cluster of structural failures in condensed mathematics and in strong homology. Its central ZFC computation is $\\lim^1 A_{\\omega,\\omega_1}\\neq 0$, a nonvanishing higher derived limit produced from a nontrivially coherent family of finite-to-one functions. From that computation the authors derive that the natural functor from pro-derived abelian groups to derived condensed abelian groups is not full, that strong homology fails to be additive on a countable sum of compact spaces, and that the derived limit functors $\\lim^n$ are not additive for $n=1,2$. In higher degrees, the same machinery shows that products of compact projective condensed anima need not be compact, because constant sheaves on products of large extremally disconnected spaces can have infinite injective dimension. The paper also shows that consistently vanishing higher limits repair these failures and restore a derived Banach–Smith duality.","feed_headline":"Strong homology fails on a countable sum of compact spaces","feed_subtitle":"One infinitary computation also blocks fullness of the condensed derived category and product compactness.","key_machinery":"The central objects are the inverse systems $A_{\\kappa,\\lambda}[H]$ with terms $\\bigoplus_{X(f)}H$ and projection maps, indexed by functions $f:\\kappa\\to[\\lambda]^{<\\omega}$ ordered by inclusion of their graphs $X(f)$. Their higher derived limits are governed by coherence: by Lemma 2.9 and its generalization Lemma 2.18, $\\lim^n A_{\\kappa,\\lambda}[H]=0$ exactly when every $n$-coherent family of functions $X(\\vec f)\\to H$ is trivial. The proof of the ZFC nonvanishing in degree one builds a nontrivially coherent family from a classical ladder of finite-to-one functions $e_\\alpha:\\alpha\\to\\omega$; the key claim is that no single function on $\\omega\\times\\omega_1$ can trivialize all the induced restrictions. In higher degrees the mechanism is a classical vanishing theorem for derived limits above the cofinality of the indexing order, together with transfinite recursion and a pressing-down argument to produce nontrivial $n$-coherent families and to rule out their trivializations.","core_discovery":"On the paper's own terms, the discovery is that the higher derived limits of the systems $A_{\\kappa,\\lambda}[H]=\\bigoplus_{X(f)}H$, with $f:\\kappa\\to[\\lambda]^{<\\omega}$ and $X(f)=\\{(i,j)\\mid j\\in f(i)\\}$, are the common combinatorial heart of several apparently unrelated questions. Theorem 2.10(5) proves $\\lim^1 A_{\\omega,\\omega_1}\\neq 0$ in ZFC; equivalently, there is a nontrivially coherent family of functions indexed by $\\omega([\\omega_1]^{<\\omega})$. The immediate corollaries are that the natural functor $\\mathrm{Pro}(D(\\mathrm{Ab}))^b\\to D(\\mathrm{Cond}(\\mathrm{Ab}))$ is not full, that $\\mathrm{H}^{n-1}(\\coprod_\\omega Y^{n,\\omega_1})\\not\\cong\\bigoplus_\\omega \\mathrm{H}^{n-1}(Y^{n,\\omega_1})$ where $Y^{n,\\omega_1}$ is the compact one-point compactification of a coproduct of $\\omega_1$ open $n$-balls, and that $\\lim^n$ is not additive for $n=1,2$. Under the consistent hypothesis that $\\lim^n A_{\\aleph_0,\\aleph_0}[H]=0$ for all $n>0$ and all $H$, the paper derives degree-zero concentration of certain $\\mathrm{RHom}$ expressions, commutation of $\\mathrm{Ext}$ with countable limits, and a derived Banach–Smith duality for separable solid $\\mathbb{Q}_p$-Banach spaces. With additional set-theoretic hypotheses such as the axiom of constructibility, the nonvanishing extends to $\\lim^{n+1} A_{\\aleph_n,\\aleph_{n+1}}\\neq 0$. Finally, Theorem 4.11 constructs open subsets $U\\subseteq\\beta X$ with $\\mathrm{H}^n(U;\\bigoplus_\\kappa K)\\neq 0$ whenever $|X|,\\kappa\\geq\\aleph_\\omega$, yielding Theorem D: the constant sheaf on $S\\times T$ has infinite injective dimension for large extremally disconnected $S,T$, and products of compact projective condensed anima are not in general compact.","pith_inferences":["If the degree-by-degree pattern of the paper's higher nonvanishing theorem persists, the question of whether $\\lim^3$ is consistently additive is likely independent of ZFC and tied to the behaviour of $\\mathrm{H}^2(\\omega_2;\\mathbb{Z})$; the paper itself leaves this as an open question.","The $\\aleph_\\omega$ threshold in Theorem D suggests a testable dichotomy: either products of extremally disconnected spaces below $\\aleph_\\omega$ are compact in every model of ZFC, or the threshold itself fluctuates with set-theoretic hypotheses such as the strong limit status of $\\aleph_\\omega$.","Since the authors note the pyknotic translation is straightforward, translating Theorems A–D to pyknotic categories would confirm that these failures are not an artifact of the condensed-site conventions.","The same $n$-coherent families may have functional-analytic shadows: nonvanishing derived limits could appear as nonzero $\\mathrm{Ext}$ classes in Banach–Smith duality outside the separable case, giving a concrete place to look for a failure of derived duality."],"forward_implications":["The functor $\\mathrm{Pro}(D(\\mathrm{Ab}))^b\\to D(\\mathrm{Cond}(\\mathrm{Ab}))$ is not full, so the condensed derived category contains homomorphisms that no pro-object morphism can see (Corollary 2.11).","Strong homology is not additive in ZFC on a countable sum of compact spaces: $\\mathrm{H}^{n-1}(\\coprod_\\omega Y^{n,\\omega_1})$ is not isomorphic to $\\bigoplus_\\omega \\mathrm{H}^{n-1}(Y^{n,\\omega_1})$ (Theorem C, Corollary 3.4).","The derived limit functors $\\lim^1$ and $\\lim^2$ on pro-abelian groups are not additive in ZFC (Corollary 3.5).","If all higher limits $\\lim^n A_{\\aleph_0,\\aleph_0}[H]$ vanish, then the $\\mathrm{RHom}$ computations in $D(\\mathrm{Cond}(\\mathrm{Ab}))$ concentrate in degree zero and the classical Banach–Smith duality extends to a derived duality for separable solid $\\mathbb{Q}_p$-Banach spaces (Theorems 2.13 and 2.14).","For extremally disconnected $S,T$ of cardinality at least $\\aleph_\\omega$, the constant sheaf on $S\\times T$ has infinite injective dimension, and products of compact projective condensed anima need not be compact (Theorems 4.3 and 4.4)."],"supporting_citations":[{"why":"Supplies the higher-degree dictionary, cited by Lemma 2.9, that converts vanishing higher derived limits into triviality of n-coherent families.","marker":"[18]"},{"why":"Source of the classical strong homology additivity failure and of the inverse system later generalized to A_{kappa,lambda}; Theorem 3.3 extends its computations to arbitrary cardinalities.","marker":"[57]"},{"why":"Earlier ZFC counterexample to additivity of strong homology; Theorem C gives a compact comparison example.","marker":"[70]"},{"why":"Supplies the walks functions on ordinals used to build the ladder of finite-to-one functions in the proof of Theorem 2.10(5).","marker":"[83]"},{"why":"Establishes the consistency of simultaneously vanishing higher derived limits in Cohen extensions, cited for item (2) of Theorem 2.10 and for Theorem B.","marker":"[16]"},{"why":"Provides the nonvanishing higher derived limit result for A_{omega,omega} cited as item (3) of Theorem 2.10.","marker":"[85]"},{"why":"Foundational account of condensed abelian groups, including the solid-projectivity and embedding facts used to reduce the fullness question to the systems A_{kappa,lambda}.","marker":"[77]"},{"why":"Supplies the vanishing theorem for derived limits above the cofinality of the index, used in the higher-degree constructions of Theorems 2.20 and 4.11.","marker":"[41]"},{"why":"Supplies the descent and adjunction lemmas used to turn infinite injective dimension of constant sheaves into non-compactness of products of condensed anima.","marker":"[46]"}],"fun_headline_variants":["Condensed math's derived category isn't full—new ZFC proof","Strong homology fails: new combinatorial ZFC result","One cardinal computation topples strong homology","Higher derived limits break condensed math and strong homology","Infinitary combinatorics: new ZFC failure in condensed math"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bridge from vanishing higher derived limits to triviality of all $n$-coherent families is stated for every degree $n$, but for $n>1$ its proof is delegated to an earlier source rather than given here; the higher-dimensional claims of the paper stand on that dictionary.","fun_headline_variants_meta":{"raw":{"variants":["Condensed math's derived category isn't full—new ZFC proof","Strong homology fails: new combinatorial ZFC result","One cardinal computation topples strong homology","Higher derived limits break condensed math and strong homology","Infinitary combinatorics: new ZFC failure in condensed math"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000965,"raw_usage":{"total_tokens":4194,"prompt_tokens":1117,"completion_tokens":3077,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":733,"completion_tokens_details":{"reasoning_tokens":2999}},"tokens_in":733,"tokens_out":3077,"duration_ms":20670,"temperature":1.0,"reasoning_tokens":2999,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:10:18.814594+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the coherent family $\\Phi=\\langle\\varphi_f:X(f)\\to\\mathbb{Z}\\rangle$ constructed in the proof of Theorem 2.10(5) from a ladder of finite-to-one functions $e_\\alpha:\\alpha\\to\\omega$ such that consecutive restrictions agree modulo finite sets, and check whether a global $\\psi:\\omega\\times\\omega_1\\to\\mathbb{Z}$ exists whose restriction to each $X(f)$ agrees with $\\varphi_f$ modulo finite sets. Claim 2.12 asserts that no such $\\psi$ exists; producing one would force $\\lim^1 A_{\\omega,\\omega_1}=0$ and would collapse Theorems A and C. The higher-degree analogue is to search for an $(n-1)$-trivialization of an $n$-coherent family built in Theorem 4.11; finding one would make the corresponding $\\mathrm{H}^n(U;\\bigoplus_\\kappa K)$ vanish.","supporting_citations":[{"cited_title":"Simultaneously vanishing higher derived limits","cited_arxiv_id":null,"evidence_quote":"Supplies the higher-degree dictionary, cited by Lemma 2.9, that converts vanishing higher derived limits into triviality of n-coherent families."},{"cited_title":"Mardeˇ si´ c and A","cited_arxiv_id":null,"evidence_quote":"Source of the classical strong homology additivity failure and of the inverse system later generalized to A_{kappa,lambda}; Theorem 3.3 extends its computations to arbitrary cardinalities."},{"cited_title":"Prasolov","cited_arxiv_id":null,"evidence_quote":"Earlier ZFC counterexample to additivity of strong homology; Theorem C gives a compact comparison example."},{"cited_title":"Simultaneously vanishing higher derived limits without large cardinals","cited_arxiv_id":null,"evidence_quote":"Establishes the consistency of simultaneously vanishing higher derived limits in Cohen extensions, cited for item (2) of Theorem 2.10 and for Theorem B."},{"cited_title":"Non-vanishing higher derived limits","cited_arxiv_id":null,"evidence_quote":"Provides the nonvanishing higher derived limit result for A_{omega,omega} cited as item (3) of Theorem 2.10."},{"cited_title":"Lectures on condensed mathematics (all results joint with Dustin Clausen)","cited_arxiv_id":null,"evidence_quote":"Foundational account of condensed abelian groups, including the solid-projectivity and embedding facts used to reduce the fullness question to the systems A_{kappa,lambda}."},{"cited_title":"Sur les d´ eriv´ es de certaines limites projectives","cited_arxiv_id":null,"evidence_quote":"Supplies the vanishing theorem for derived limits above the cofinality of the index, used in the higher-degree constructions of Theorems 2.20 and 4.11."}],"review_version":1}