{"id":"8039c3f3-5592-4263-a04e-ff8bf5709fcf","arxiv_id":"2508.20331","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors build n-cotorsion pairs in the middle of a recollement from the outer terms, and conversely, under exactness conditions, for extriangulated categories.","lead":"This paper proves gluing and ungluing theorems that build n-cotorsion pairs in the middle category of a recollement from the outer categories, and vice versa, inside the framework of extriangulated categories. A general reader might care because this extends a standard homological algebra tool to a very inclusive categorical setting, but the core result is largely already present in a cited preprint by He and He.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Omitted proof of Lemma 3.9 is load-bearing: Theorem 3.10 applies it to i* and j!, which the lemma as stated does not cover, so contravariant finiteness of X in B is not rigorously established.","rationale":"The central claim is Theorem 3.10, a gluing construction for n-cotorsion pairs. Its proof has two ingredients: (a) the orthogonality identities, which follow from Proposition 3.8 and are sound; (b) the finiteness of the glued subcategories X and Y, which rests on Lemma 3.9. The reader correctly identifies Lemma 3.9 as the weakest point. In my reading, the lemma is true in the stated form and in the more general form needed for i* and j!, because the preservation of contravariant/covariant finiteness is a purely categorical adjunction fact: a right adjoint sends right approximations in the source to right approximations in the target via the counit, and a left adjoint sends left approximations via the unit. No exactness of the functor is required. However, the paper does not prove this, and the citation to [20] and [22] is not a direct reference for the extriangulated case. Moreover, the proof of Theorem 3.10 applies Lemma 3.9 to functors that are not covered by its statement (i* and j! rather than j*), so even a reader who grants Lemma 3.9 would need to run the argument again. This is a genuine rigor gap, but it is fixable, so the conditional verdict is appropriate. I found no internal inconsistency in the orthogonality arguments or in the converse Theorem 3.15. The overlap with He-He [10] is a novelty concern, not a correctness one.","tokens_in":13041,"tokens_out":21422,"duration_ms":171342,"concrete_test":"Give a self-contained proof of the preservation statement needed in Theorem 3.10: for an additive functor F between extriangulated categories with right adjoint G, show that if X is contravariantly finite in the source then F(X) is contravariantly finite in the target, by using the counit FG→Id to push right approximations forward. Then apply this to F = i*: A→B (G = i!) and F = j!: C→B (G = j*), and similarly for covariantly finite subcategories under left adjoints. If the argument uses exactness or extriangulated structure beyond the adjunction, identify the missing hypothesis. Alternatively, provide a counterexample in a small extriangulated category if the statement fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.10 constructs (X,Y) and needs to show X is contravariantly finite in B. The proof invokes Lemma 3.9 to conclude that i*X' and j!X'' are contravariantly finite. But Lemma 3.9, as stated, covers only a functor j*: B→C, not the functors i*: A→B or j!: C→B that are actually used. The standard adjunction argument (a functor with a right adjoint preserves contravariantly finite subcategories) would supply the needed statements, but that argument is not written down or precisely referenced, and the lemma's own proof is omitted entirely, with a pointer to [20] and [22] that does not cover the extriangulated case verbatim. The statement of Lemma 3.9(1) also contains a typo (it concludes about j*X from a hypothesis about Y). Because the finiteness conditions are part of Definition 3.5, a failure of this step would invalidate the n-cotorsion pair claim in Theorem 3.10.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies n-cotorsion pairs in a recollement (A,B,C) of extriangulated categories. The main forward direction, Theorem 3.10, defines subcategories X={B in B | j*B in X'', i*B in X'} and Y={B in B | j*B in Y'', i!B in Y'} from n-cotorsion pairs (X',Y') in A and (X'',Y'') in C, and asserts that (X,Y) is an n-cotorsion pair in B that restricts back to the given pairs, assuming i* and i! are exact. The converse direction, Theorem 3.15, shows that an n-cotorsion pair (X,Y) in B induces n-cotorsion pairs (i*X,i!Y) in A and (j*X,j*Y) in C under closure conditions of the form i*i*X subset X, i*i*Y subset Y and j*j*Y subset Y, and characterizes X and Y by these restrictions. The paper also derives corollaries for triangulated categories and (n+1)-cluster tilting subcategories.","tokens_in":13364,"tokens_out":13591,"duration_ms":113380,"significance":"The results are plausible and sit naturally in the existing program of gluing homological structures in recollements: they generalize the n=1 cotorsion-pair results of Ma-Zhou and He-He to n-cotorsion pairs, and the triangulated corollaries recover and extend results of Chen and of Long-Zhang-Zhou. The proofs use standard dimension-shifting and approximation arguments, and the paper is transparent about relying on prior work. However, the main theorem currently depends on a preservation lemma that is neither proved nor stated in the required generality, so the paper's contribution cannot be fully verified as written. If the missing arguments are supplied, the paper would be a useful reference; as it stands the novelty is incremental and overlaps with the authors' earlier [24] and with [10].","major_comments":[{"comment":"Lemma 3.9 is load-bearing but is not proved and is not applicable as stated. The lemma concerns additive functors j*:B->C, whereas Theorem 3.10(2) invokes it for the functors i*:A->B and j!:C->B to conclude that i*X' and j!X'' are contravariantly finite in B. The needed statement is the standard fact that a functor with a right adjoint preserves contravariantly finite subcategories; Lemma 3.9(2) is the special case of this fact for j*:B->C, and the general version is neither stated nor proved. In addition, Lemma 3.9(1) contains a typo: from a covariantly finite subcategory Y of B it concludes that j*X is covariantly finite in C, and the conclusion should presumably concern j*Y. Since contravariant and covariant finiteness is part of Definition 3.5, the proof of Theorem 3.10(2) is incomplete until the preservation facts are stated in the needed generality and proved.","section":"Lemma 3.9 and Theorem 3.10(2)"},{"comment":"The proof asserts 'One can see that X is closed under E-extensions' without giving the argument, and this closure is then used to conclude that the object X, constructed as an extension of X1 in X and X2 in X, lies in X. The closure does follow from the definition of X together with exactness of i* and j* and closure of X' and X'' under E-extensions (Remark 3.6(1)), but the one-line verification should be included; as written the proof has a gap at the point where X in X is concluded.","section":"Theorem 3.10(2), proof"}],"minor_comments":[{"comment":"The heading 'From A andB toC' is misleading; the section constructs n-cotorsion pairs in B from data in A and C, so it should read 'From A and C to B'.","section":"Section 3.1 heading"},{"comment":"There are typos in the proof: 'n-cotosion' should be 'n-cotorsion', and 'Proposiion' should be 'Proposition'.","section":"Theorem 3.10 proof"},{"comment":"In the proof, 'suc that' should be 'such that', and the index in the displayed intersection 'nT i=k' should read 'nT k=1'.","section":"Theorem 3.15(1) proof"},{"comment":"In both corollaries, 'otorsion pair' should be 'cotorsion pair'.","section":"Corollaries 3.17 and 3.18"},{"comment":"The word 'Specially' should be 'Specifically' in the sentences introducing the triangulated consequences.","section":"Corollaries 3.12 and 3.19"},{"comment":"The dual argument proving that Y is covariantly finite is not written out; a sentence indicating the corresponding left-approximation argument would make the proof easier to check.","section":"Theorem 3.10(2), dual part"}],"recommendation":"major_revision","confidential_remarks":"The editor may wish to check the novelty relative to [10] and [24]. Corollary 3.17 is explicitly a restatement of [24, Theorem 4.6] and [27, Theorem 4.4], and reference [10] is described as having already extended n-cotorsion-pair recollement results to extriangulated categories. The authors should clarify in the introduction what Theorem 3.10 and Theorem 3.15 add beyond those existing results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper's Theorem 3.10 and Theorem 3.15 are the same gluing and ungluing statements that He and He [10] already proved for extriangulated categories, and the authors know this—they say so in the introduction. So the abstract's claim that \"we first show\" is misleading. What is actually new: the corollaries on (n+1)-cluster tilting subcategories and the restatements for triangulated categories. Those are straightforward consequences of the main theorems, but they are stated cleanly and collected in one place.\n\nThe paper does several things well. It uses standard approximation and dimension-shifting arguments, the hypotheses (i* and i! exact) are the right ones, and the writing is clear enough to follow. The citations to [10] and [24] are honest, and the paper does not hide its debt. If you work on recollements of extriangulated categories and want one place with the n-cotorsion gluing theorem plus cluster-tilting consequences, this is usable.\n\nThe soft spots are real but not fatal. Lemma 3.9 is the biggest one: it is stated only for a functor j*: B→C, but Theorem 3.10 invokes it for i*: A→B and j!: C→B, which the statement does not cover. The proof of the lemma is omitted, with a pointer to [20] and [22]. The stress-test note is right that this gap is load-bearing. It is also routine: any functor with a right adjoint preserves contravariantly finite subcategories, and any functor with a left adjoint preserves covariantly finite subcategories, so the theorem can be fixed by writing down the standard adjunction argument. The lemma's statement also contains a typo in part (1), where the conclusion mentions X while the hypothesis mentions Y. Minor and easily corrected.\n\nTwo smaller proof gaps: the claim that X is closed under E-extensions in Theorem 3.10 is asserted without demonstration (it follows from exactness of i* and j*, but the proof does not say that), and the dual arguments for Y are sketched rather than written. These are standard, not structural.\n\nBottom line: if the journal's novelty bar requires new theorems, this paper should probably not be accepted as is, because the central results already exist. If the editor is willing to consider a consolidation with new corollaries, it deserves a serious referee—not a desk reject—because the mathematics is coherent and the cluster-tilting applications are useful. I would ask the authors to remove \"we first show\" from the abstract and to fix Lemma 3.9 before sending it out.","headline":"Main gluing theorems duplicate He–He [10]; the cluster-tilting corollaries are the real addition, and the omitted Lemma 3.9 is fixable but makes the proof incomplete as written.","tokens_in":13816,"tokens_out":5582,"would_cite":false,"duration_ms":52401,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18E40","18G80","18E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"n-cotorsion pairs glue across extriangulated recollements","keywords":["extriangulated categories","recollement","n-cotorsion pairs","(n+1)-cluster tilting subcategories","cotorsion pairs","higher extension groups","approximations","triangulated categories"],"falsifier":"Compute the two subcategories $X$ and $Y$ defined in Theorem 3.10 for a concrete recollement with $i^*$ and $i^!$ exact, using known $n$-cotorsion pairs in the outer categories, and check the three defining properties: the mutual orthogonality equalities and the existence of right $X$-approximations and left $Y$-approximations for every object. A failure of any one of these in such an example would disprove the theorem; a natural first place to look is a recollement induced by a torsion pair in a module category, where the functors and extension groups can be computed by hand.","tokens_in":12853,"feed_emoji":"🧩","tokens_out":17403,"duration_ms":143991,"temperature":0.7,"pith_summary":"The paper claims that, in a recollement of extriangulated categories, $n$-cotorsion pairs in the two outer categories glue to an $n$-cotorsion pair in the middle category whenever the two adjoint functors from the middle to the left category are exact, and that the glue is reversible: restricting the new pair back to the outer categories recovers the original pairs. The same framework gives a converse: an $n$-cotorsion pair in the middle category induces $n$-cotorsion pairs in the outer categories under closure conditions, and the middle pair can be reconstructed from them. This matters because extriangulated categories are a common home for exact and triangulated categories, so the construction is a single mechanism that specializes to both classical settings. A sympathetic reader should take the main messages to be the explicit formulas for the glued subcategories and the two-way restriction procedure.","feed_headline":"n-cotorsion pairs glue across extriangulated recollements","feed_subtitle":"An explicit recipe builds one n-cotorsion pair in the middle category and recovers the original pairs on both sides.","key_machinery":"The load-bearing object is the $n$-cotorsion pair in an extriangulated category: a pair $(X,Y)$ of subcategories closed under direct summands that are mutual higher orthogonals and satisfy approximation finiteness ($X$ contravariantly finite, $Y$ covariantly finite). The argument is carried by the recollement adjunctions and by a dimension-shifting isomorphism that identifies higher extension groups across the three categories; this converts the given orthogonality in $A$ and $C$ into orthogonality in $B$. Finiteness is transferred by Lemma 3.9, which the paper states without proof and refers to parallel arguments in the cited references.","core_discovery":"The central claim is Theorem 3.10: in a recollement $(A,B,C)$ of extriangulated categories with $i^*$ and $i^!$ exact, any $n$-cotorsion pairs $(X',Y')$ in $A$ and $(X'',Y'')$ in $C$ glue to an $n$-cotorsion pair $(X,Y)$ in $B$, where $X = \\{ B \\mid j^*B \\in X'',\\ i^*B \\in X' \\}$ and $Y = \\{ B \\mid j^*B \\in Y'',\\ i^!B \\in Y' \\}$. The original pairs are recovered by restriction: $(i^*X,\\,i^!Y)=(X',Y')$ and $(j^*X,\\,j^*Y)=(X'',Y'')$. Theorem 3.15 proves the converse: an $n$-cotorsion pair in $B$, under explicit closure hypotheses, induces $n$-cotorsion pairs in $A$ and $C$, and is itself recovered from them by the same defining formulas.","pith_inferences":["The paper leaves Lemma 3.9 without proof, so the cleanest reading of the main theorem is conditional on that finiteness transfer; one should verify the lemma in any particular recollement before applying the theorem.","The gluing recipe is likely to transfer to settings beyond the paper, such as pairs defined by a different notion of higher extension, because the proof itself only uses dimension shifting, orthogonality, and approximations.","A natural open question, not addressed in the paper, is whether the closure hypotheses in Theorem 3.15 are also necessary; if they are, the construction would be a bijective correspondence between suitable pairs of $n$-cotorsion pairs in the outer categories and those in the middle category."],"forward_implications":["If Theorem 3.10 is correct, every compatible pair of $n$-cotorsion pairs in the outer categories produces one in the middle category, and the recollement functors send the outer pairs into the glued pair.","The recovery equalities in the theorem mean no information is lost: applying the restriction functors to the glued pair gives back exactly the original pairs.","For $n=1$, the construction recovers the known gluing theorem for ordinary cotorsion pairs in extriangulated categories.","When the input pairs have the form $(X',X')$ and $(X'',X'')$, the glued pair gives an $(n+1)$-cluster tilting subcategory in the middle category, recovering the triangulated cluster-tilting result as a special case.","The converse direction also gives a reconstruction formula: under the closure hypotheses, an $n$-cotorsion pair in the middle category is completely determined by its images in the two outer categories."],"supporting_citations":[{"why":"Defines n-cotorsion pairs in extriangulated categories and gives the characterization used in the proofs.","marker":"[6]"},{"why":"Provides the extriangulated category framework and the original cotorsion pair notion being generalized.","marker":"[25]"},{"why":"Defines recollements of extriangulated categories and supplies the adjunction and exactness properties used throughout.","marker":"[27]"},{"why":"Supplies higher extension groups and the dimension-shifting exact sequences used to transfer orthogonality conditions.","marker":"[21]"},{"why":"Gives the adjunction isomorphism for higher extensions and an earlier cotorsion pair gluing result extended here.","marker":"[24]"},{"why":"Cited as the source of the argument for the finiteness lemma (Lemma 3.9) whose proof is omitted.","marker":"[20]"},{"why":"Also cited for the omitted finiteness lemma, alongside [20].","marker":"[22]"},{"why":"Proves the triangulated-category analogue that the main theorem generalizes and recovers when n=1.","marker":"[4]"},{"why":"Gives the cluster-tilting recollement results that appear as corollaries of the new construction.","marker":"[26]"}],"fun_headline_variants":["n-cotorsion pairs glue and split via recollement","Recollement induces and recovers n-cotorsion pairs","Gluing n-cotorsion pairs in extriangulated recollements","Explicit recipe: n-cotorsion pairs in recollement","Two-way construction of n-cotorsion pairs in recollement"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on the unproved assertion that certain adjoint functors between the categories of the recollement preserve finite subcategories in the approximation sense; if that transfer fails, the subcategories built in the main theorem may fail the finiteness condition required of an n-cotorsion pair.","fun_headline_variants_meta":{"raw":{"variants":["n-cotorsion pairs glue and split via recollement","Recollement induces and recovers n-cotorsion pairs","Gluing n-cotorsion pairs in extriangulated recollements","Explicit recipe: n-cotorsion pairs in recollement","Two-way construction of n-cotorsion pairs in recollement"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00034,"raw_usage":{"total_tokens":1836,"prompt_tokens":870,"completion_tokens":966,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":873}},"tokens_in":486,"tokens_out":966,"duration_ms":8010,"temperature":1.0,"reasoning_tokens":873,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:46:17.665373+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two subcategories $X$ and $Y$ defined in Theorem 3.10 for a concrete recollement with $i^*$ and $i^!$ exact, using known $n$-cotorsion pairs in the outer categories, and check the three defining properties: the mutual orthogonality equalities and the existence of right $X$-approximations and left $Y$-approximations for every object. A failure of any one of these in such an example would disprove the theorem; a natural first place to look is a recollement induced by a torsion pair in a module category, where the functors and extension groups can be computed by hand.","supporting_citations":[{"cited_title":"Nakaoka, Y","cited_arxiv_id":null,"evidence_quote":"Provides the extriangulated category framework and the original cotorsion pair notion being generalized."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines recollements of extriangulated categories and supplies the adjunction and exactness properties used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies higher extension groups and the dimension-shifting exact sequences used to transfer orthogonality conditions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the adjunction isomorphism for higher extensions and an earlier cotorsion pair gluing result extended here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cited as the source of the argument for the finiteness lemma (Lemma 3.9) whose proof is omitted."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Also cited for the omitted finiteness lemma, alongside [20]."},{"cited_title":"Chen, Cotorsion pairs in a recollement of triangulated categories , Comm","cited_arxiv_id":null,"evidence_quote":"Proves the triangulated-category analogue that the main theorem generalizes and recovers when n=1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the cluster-tilting recollement results that appear as corollaries of the new construction."}],"review_version":1}