{"id":"acdef78a-c6a7-4ed8-9da7-682dfa7bfdbb","arxiv_id":"2502.03887","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Over a recollement of abelian categories, ICE-closed subcategories, epibricks and monobricks glue and reduce along the recollement, yielding a bijection for ICE-closed subcategories under a natural containment condition.","lead":"This pure-math paper proves that certain classes of subcategories, called ICE-closed subcategories, epibricks and monobricks, can be transferred between the middle of a recollement of abelian categories and its two ends. It gives a general tool for building or classifying such subcategories in representation theory, recovering earlier one-point extension results as special cases.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.10's bijection is literally false because A''_ice includes the empty subcategory, which has no preimage in the left-hand set; the statement should exclude the empty subcategory.","rationale":"The reader's weakest assumption concerned exactness of j^* or j_! in the transfer from A'' to A. That is a presentation issue with the abstract: the bijection in Theorem 3.10 does not require exactness, because the A''-to-A direction uses the always-exact functor j^* via the preimage construction, while exactness is needed only for the direct image functors in Theorem 3.4. The genuinely load-bearing defect in the central classification is the empty-subcategory edge case. Definition 2.1 explicitly includes the empty subcategory, and Corollary 3.9(1) relies on the false claim that a zero object belongs to every W. This makes the literal bijection statement false, although the correction is minor: restrict A''_ice to nonempty subcategories, or require all subcategories to be nonempty. This does not change the reader's CONDITIONAL verdict, since the conditionality is already appropriate; it sharpens the reason why the theorem needs amending.","tokens_in":16227,"tokens_out":26311,"duration_ms":263879,"concrete_test":"Run the trivial recollement 0 -> A' -> A -> A'' -> 0 with A' = 0 and A'' = A, where j_*, j^* and j_! are identity functors and i_*(A') is the zero-object subcategory. In this setting j_!j^*(C) ⊂ C is automatic, so the left-hand side of Theorem 3.10 consists of the nonempty ICE-closed subcategories of A containing the zero object, while A''_ice contains the empty subcategory. If the theorem's statement is read literally, the empty subcategory has no preimage, disproving the bijection; excluding the empty subcategory from A''_ice makes the check pass.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 2.1 explicitly counts the empty subcategory as ICE-closed, so the empty subcategory belongs to A''_ice. Corollary 3.9(1) asserts that the zero object \"always belongs to W\" because it is the cokernel of an identity map; this is false exactly for W = ∅. For W = ∅, the proposed preimage C = {M in A | j^*(M) in W} is the empty subcategory of A. Since i_*(A') contains the zero object whenever A' is nonzero, this C is not in i_*(A')_A_ice, so the map from A''_ice to the left-hand set is not even well-defined. A concrete witness is the recollement with A' = 0, A'' = A, and j_* = j^* = j_! = id_A: the left-hand set is all nonempty ICE-closed subcategories containing the zero object, while A''_ice also contains the empty subcategory. Thus the bijection as stated is false; it becomes correct after replacing A''_ice by the set of nonempty ICE-closed subcategories, or by adopting the convention that all subcategories are nonempty. The proposed fix is local, and the main applications use nonempty subcategories, but the literal theorem statement needs this amendment.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies transfer, gluing, and reduction of ICE-closed subcategories, epibricks, and monobricks under recollements of abelian categories. The main results assert that i_* sends ICE-closed subcategories of A' to those of A; that j^* (and, under an exactness hypothesis, j_!) sends ICE-closed subcategories of A'' to those of A; and that, under the containment condition j_!j^*(C)⊂C, there is a bijection between ICE-closed subcategories of A containing i_*(A') and ICE-closed subcategories of A''. The paper also proves preservation and gluing results for epibricks and monobricks, and applies the framework to triangular matrix algebras, one-point extensions, and Morita rings. The work extends Zhang's results on wide subcategories and explicitly identifies the non-automatic condition j_!j^*(C)⊂C as a new ingredient.","tokens_in":16428,"tokens_out":20923,"duration_ms":208422,"significance":"If the main theorems hold, the paper provides a uniform framework for transferring ICE-closed subcategories and brick-type classes across recollements, generalizing [35] and recovering the one-point-extension results of [26]. The proofs are mostly written out; the few appeals to [35] are for arguments that are genuinely analogous, and I verified the main steps of Theorems 3.3, 3.4, 4.2, and 4.4. The paper is honest that j_!j^*(C)⊂C is not automatic for ICE-closed subcategories, and it does not build the bijection by construction. The concrete applications to triangular matrix algebras, one-point extensions, and Morita rings are a useful feature. The one substantive defect is the treatment of the empty subcategory in the central bijection; this is local and repairable, and the applications in Section 5 are unaffected.","major_comments":[{"comment":"Definition 2.1 explicitly includes the empty subcategory among ICE-closed subcategories, so ∅∈A''_ice. In Theorem 3.10 the proposed inverse image of W=∅ is C={M∈A | j^*(M)∈∅}=∅. This C does not lie in i_*(A')_{A ice}, because i_*(A') contains the zero object i_*(0) whenever A' is an abelian category, and the empty subcategory contains no objects. Equivalently, the assertion in the proof of Corollary 3.9(1) that the zero object 'always belongs to W' is false exactly for W=∅. Hence the map A''_ice → {C∈i_*(A')_{A ice} | j_!j^*C⊂C} is not well-defined, and the bijection in Theorem 3.10 (hence in Theorem 1.1(3)) is false as stated; a concrete witness is the recollement with A'=0, A''=A and j_*=j^*=j_!=id_A. The theorem becomes correct if A''_ice is replaced by the set of nonempty ICE-closed subcategories of A'' (or if all subcategories are declared nonempty), and Section 5's applications are unaffected; however, the statement needs this amendment. The same empty-subcategory issue affects Corollary 3.9(1) and, if the empty torsion class is admitted under Definition 2.1(6), Corollary 3.12(5).","section":"Theorem 3.10 and Corollary 3.9(1)"}],"minor_comments":[{"comment":"The statement reads 'j!∗(ebrickA′) ⊂ ebrickA and j!∗(mbrickA′) ⊂ mbrickA'; since j_{!*} is defined on A'', the source should be A'' rather than A', as in Theorem 4.3.","section":"Theorem 4.2(2)"},{"comment":"The inverse direction of the bijection is not written out; the proof says it follows from '[35, Theorem 3.4]'. Since the hypotheses for ICE-closed subcategories differ from the wide case, please include the short verification, which uses the exact sequence in Remark 2.4(3) and the condition j_!j^*C⊂C.","section":"Proof of Theorem 3.10"},{"comment":"The proof is omitted with a reference to [35, Proposition 3.3]; a three-line proof via exactness of F (closure under images, cokernels, and extensions) would make the paper more self-contained.","section":"Lemma 3.8"},{"comment":"The first sentence could be read as asserting transfer from A'' via j_! without any exactness hypothesis; the unconditional transfer is via j^*, while the j_! variant in Theorem 3.4 requires exactness of j_!. Please rephrase to identify the transfer functor.","section":"Abstract"},{"comment":"The symbols i^* and i_* (and similarly j^* and j_*) are typeset nearly identically throughout, which makes exactness statements hard to parse; please use unambiguous notation. Also, the proof of Theorem 3.11 refers to 'Lemma 3.6 (1)', but the correct reference is Proposition 3.6(1).","section":"Remark 2.4(1) and proof of Theorem 3.11"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the mathematical content is sound after the nonempty-subcategory correction. The empty-subcategory issue in Theorem 3.10 is easy to miss but affects the central statement; I recommend insisting on the amendment to Theorem 3.10 and Corollaries 3.9(1) and 3.12(5). I see no novelty or attribution concerns; the relation to [35] is clearly stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things up front. This is a genuinely useful transfer paper: it carries Zhang's wide-subcategory recollement machinery over to ICE-closed subcategories, epibricks and monobricks, and it recovers the Li-Gao one-point-extension results as a special case. And the stress-test note is right: Theorem 3.10 is false as literally stated. Definition 2.1 explicitly counts the empty subcategory as ICE-closed. For W = ∅, the proposed preimage {M ∈ A | j*(M) ∈ W} is the empty subcategory of A, which does not contain i_*(A') — and i_*(A') always contains the zero object. So the map from A''_ice to the left-hand set is not well-defined, and ∅ is never hit from the other side either. The fix is local: restrict to nonempty ICE-closed subcategories. The applications, all of which use nonempty subcategories, survive unchanged; but the printed statement of the headline theorem needs the amendment.\n\nCredit where it is due. The transfer theorems (3.3, 3.4, 3.6, 3.7), the bijection modulo the empty case, the induced recollement on C (3.11), and the epibrick/monobrick gluing (4.2-4.4) are new relative to [35] and [26]. The paper is also honest where it follows Zhang's script, and it explicitly observes that j_!j*(C) ⊂ C is necessary and not automatic for ICE-closed subcategories, unlike the wide case. That is a real point.\n\nSoft spots. The abstract overstates the A'' → A extension: it presents it unconditionally, while Theorem 3.4 requires exactness of the functor involved, and Corollary 3.9(1)'s preimage construction silently fails for W = ∅. Several arguments, including the inverse direction of the bijection, are delegated to [35]; a referee should verify those citations rather than take them on faith, and the reader's flagged cross-reference issue is worth checking. These are minor next to the empty-subcategory problem.\n\nThis paper is for representation theorists who glue subcategory lattices along recollements. It will not reshape the field, but it will be the standard reference for gluing ICE-closed subcategories. Conditional accept after the empty-subcategory amendment; it deserves a serious referee.","headline":"Useful recollement transfer framework for ICE-closed subcategories, but Theorem 3.10's bijection is literally false as stated because A''_ice contains the empty subcategory.","tokens_in":17023,"tokens_out":13039,"would_cite":true,"duration_ms":111131,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E35","16G10","16G20","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"In a recollement of abelian categories, ICE-closed subcategories of the two outer categories extend to the middle, and a bijection classifies those containing the left image.","keywords":["ICE-closed subcategory","epibrick","monobrick","recollement","abelian category","intermediate extension functor","gluing subcategories","torsion class"],"falsifier":"Take a recollement of module categories over finite-dimensional algebras in which $j_*$ (or $j_!$) is not exact, set $W = A''$ (the full right-hand category, which is ICE-closed), and check whether $j_*(A'')$ (or $j_!(A'')$) is closed under extensions in the middle category $A$. If it is not, the unconditional extension claim in the abstract fails, and the same test on a well-chosen ICE-closed subcategory would settle the scope of Theorem 3.4.","tokens_in":15946,"feed_emoji":"🔗","tokens_out":11247,"duration_ms":98742,"temperature":0.7,"pith_summary":"Working with recollements of abelian categories — the six-functor setup that splits a middle category into left and right parts — this paper proves that subcategories closed under images, cokernels, and extensions (ICE-closed subcategories), as well as epibricks and monobricks, can be transferred into the middle category. The central result is a bijection between ICE-closed subcategories of the middle that contain the image of the left-hand category and satisfy a containment condition, and all ICE-closed subcategories of the right-hand category. It also shows that the intermediate extension functor glues epibricks and monobricks from both outer categories into the middle without extra hypotheses. These theorems generalize earlier one-point-extension results and give a uniform framework for reducing classification problems on the middle category to its outer pieces.","feed_headline":"Recollements transfer ICE-closed subcategories by a bijection","feed_subtitle":"Recollements glue ICE-closed subcategories, epibricks, and monobricks into the middle category.","key_machinery":"The central machine is a recollement of abelian categories: a diagram of six adjoint functors among three abelian categories satisfying $\\ker j^* = \\operatorname{im} i_*$. The transfer arguments run through the exactness properties of these functors and the intermediate extension functor $j_{!*}(M) = \\operatorname{Im}(j_!M \\to j_*M)$, which is fully faithful and preserves monomorphisms and epimorphisms. The containment condition $j_!j^*(C) \\subseteq C$ is the additional input needed for ICE-closed subcategories, where the automatic containment that holds for wide subcategories fails.","core_discovery":"The paper's main theorem is a bijection $$\\{C \\in i_*(A')_{A_{\\mathrm{ice}}} \\mid j_!j^*(C) \\subseteq C\\} \\xrightarrow{\\sim} A''_{\\mathrm{ice}}$$ sending $C$ to $j^*(C)$ and a right-hand subcategory $W$ to $\\{M \\in A \\mid j^*(M) \\in W\\}$. Around this bijection, the authors prove that $i_*$ carries ICE-closed subcategories of $A'$ into $A$, and that the fully faithful functors $j_*$ and $j_!$ carry ICE-closed subcategories of $A''$ into $A$ when they are exact. For a subcategory $C$ of $A$ that contains $i_*(A')$ and satisfies $j_!j^*(C) \\subseteq C$, the restricted functors form a new recollement $(A', C, j^*(C))$. For epibricks and monobricks, the paper shows that $i_*$ and the intermediate extension functor $j_{!*}$ preserve them, that $j_*$ preserves monobricks and $j_!$ preserves epibricks, and that gluing from both sides works via $j_{!*}$ in general and via $j_!$ or $j_*$ when $i_*$ or $i_!$ is exact.","pith_inferences":["The condition $j_!j^*(C) \\subseteq C$ is precisely what separates ICE-closed subcategories from wide subcategories in this gluing problem; identifying natural subfamilies that automatically satisfy it would make the bijection unconditional for those families.","Because the bijection is built from the quotient functor $j^*$, it should be compatible with iterations of recollements or ladders, suggesting an inductive description of ICE-closed subcategories in algebras assembled from multiple layers.","In module categories of finite-dimensional algebras, the condition $j_!j^*(C) \\subseteq C$ can be checked by testing the action of the middle ring on the right-hand part, which may yield a practical algorithm for enumerating ICE-closed subcategories from those of corner algebras.","The exactness hypotheses on $i_*$ or $i_!$ for gluing epibricks via $j_!$ or $j_*$ translate into familiar module-theoretic conditions, so the theorem may give concrete criteria for when corner algebras capture all epibricks of a Morita ring."],"forward_implications":["For a recollement, an ICE-closed subcategory of the middle that contains $i_*(A')$ and satisfies $j_!j^*(C) \\subseteq C$ is completely determined by its image under $j^*$, so classifying such subcategories reduces to classifying ICE-closed subcategories of $A''$.","The preimage construction $\\{M \\in A \\mid j^*(M) \\in W\\}$ gives a concrete way to build new ICE-closed subcategories of $A$ from any ICE-closed subcategory of $A''$.","When the containment condition holds, the subcategory $C$ itself is the middle term of a new recollement of ICE-closed subcategories, so the gluing respects the recollement structure rather than merely producing a subcategory.","For epibricks and monobricks, gluing via $j_{!*}$ requires no exactness assumptions; with exactness of $i_*$ or $i_!$, additional gluings via $j_!$ or $j_*$ become available, matching the known semibrick gluing picture.","The special case of one-point extension algebras recovers the previously known reduction results for ICE-closed subcategories and epibricks, showing that the new theorems unify those examples."],"supporting_citations":[{"why":"Introduces ICE-closed subcategories and their basic closure properties, the central objects of the paper.","marker":"[12]"},{"why":"Introduces monobricks and epibricks, the two other families of subcategories studied and transferred.","marker":"[13]"},{"why":"Supplies the method of reduction across recollements and the bijection pattern that the ICE-closed results adapt.","marker":"[35]"},{"why":"Provides the analogous gluing results for semibricks that the epibrick and monobrick theorems extend.","marker":"[36]"},{"why":"The one-point extension reduction results that this paper recovers and generalizes to arbitrary recollements.","marker":"[26]"},{"why":"Original framework for recollements and the intermediate extension functor used throughout.","marker":"[6]"},{"why":"Homological properties of recollements of abelian categories, including exactness of the relevant functors and adjunction identities.","marker":"[30]"},{"why":"Supplies properties of the intermediate extension functor and conditions such as vanishing of certain composition functors under exactness assumptions.","marker":"[18]"}],"fun_headline_variants":["Recollements yield a bijection of ICE-closed subcategories","ICE-closed subcategories glue across recollements","Bijection maps ICE-closed subcategories via recollement","Epibricks and monobricks survive recollement gluing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The transfer of ICE-closed subcategories from the right-hand category $A''$ into the middle via the fully faithful functors $j_*$ (or $j_!$) requires those functors to be exact; the abstract's first sentence omits this hypothesis, and the proof uses exactness to obtain closure under extensions.","fun_headline_variants_meta":{"raw":{"variants":["Recollements yield a bijection of ICE-closed subcategories","ICE-closed subcategories glue across recollements","Bijection maps ICE-closed subcategories via recollement","Epibricks and monobricks survive recollement gluing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000192,"raw_usage":{"total_tokens":1400,"prompt_tokens":1051,"completion_tokens":349,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":281}},"tokens_in":667,"tokens_out":349,"duration_ms":3583,"temperature":1.0,"reasoning_tokens":281,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T00:20:57.681054+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a recollement of module categories over finite-dimensional algebras in which $j_*$ (or $j_!$) is not exact, set $W = A''$ (the full right-hand category, which is ICE-closed), and check whether $j_*(A'')$ (or $j_!(A'')$) is closed under extensions in the middle category $A$. If it is not, the unconditional extension claim in the abstract fails, and the same test on a well-chosen ICE-closed subcategory would settle the scope of Theorem 3.4.","supporting_citations":[{"cited_title":"Enomoto, Rigid modules and ICE-closed subcategories in quive r rep- resentations, J","cited_arxiv_id":null,"evidence_quote":"Introduces ICE-closed subcategories and their basic closure properties, the central objects of the paper."},{"cited_title":"Enomoto, Monobrick, a uniform approach to torsion-free c lasses and wide subcategories, Adv","cited_arxiv_id":null,"evidence_quote":"Introduces monobricks and epibricks, the two other families of subcategories studied and transferred."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the method of reduction across recollements and the bijection pattern that the ICE-closed results adapt."},{"cited_title":"A construction of support $\\tau$-tilting modules over $\\tau$-tilting finite algebras","cited_arxiv_id":"1908.02245","evidence_quote":"Provides the analogous gluing results for semibricks that the epibrick and monobrick theorems extend."},{"cited_title":"ICE-closed subcategories and epibricks over one-point extensions","cited_arxiv_id":"2401.05645","evidence_quote":"The one-point extension reduction results that this paper recovers and generalizes to arbitrary recollements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Original framework for recollements and the intermediate extension functor used throughout."},{"cited_title":"Psaroudakis, Homological theory of recollements of abelian c ate- gories, J","cited_arxiv_id":null,"evidence_quote":"Homological properties of recollements of abelian categories, including exactness of the relevant functors and adjunction identities."},{"cited_title":"Franjou and T","cited_arxiv_id":null,"evidence_quote":"Supplies properties of the intermediate extension functor and conditions such as vanishing of certain composition functors under exactness assumptions."}],"review_version":1}