{"id":"35e3e91b-fe07-4761-9582-063ab4802c35","arxiv_id":"1908.02245","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For recollements of module categories, semibricks on the two outer algebras glue via the intermediate extension functor to semibricks on the middle algebra, yielding a construction of support tau-tilting modules over tau-tilting finite algebras.","lead":"The paper shows how to combine semibricks, sets of representation-theoretic building blocks with no maps between distinct members, when a module category is glued from two smaller ones. It then uses this gluing to produce support tau-tilting modules over algebras that have only finitely many of them.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The gluing theorem rests on the unproved full faithfulness of j_{!*} (Prop 2.8(2)); if that fails for a recollement of module categories, Theorems 3.3 and 4.1 collapse.","rationale":"I agree with the reader that the weakest load-bearing premise is the unproved full faithfulness of j_{!*} in Proposition 2.8(2). Theorems 3.3, 3.6, and 4.1 all pass through Lemma 3.1 and this proposition: full faithfulness is needed both to preserve bricks and to control Hom spaces inside the image of j_{!*}. Unlike j_!, j_*, and i_*, whose full faithfulness is part of the recollement axioms in Definition 2.6, j_{!*} is only the image of a natural transformation and requires a separate theorem. The paper cites [BBD,FP] rather than proving this, and no example in the paper directly tests it. I do not assert that the property is false; I only claim that this external, unproved premise is the single point where an error would invalidate the headline result. Since an explicit computation in the paper's own Examples 4.3 and 4.4 can settle the question, I would condition acceptance on verifying Proposition 2.8(2), rather than rejecting or unconditionally accepting the paper.","tokens_in":8326,"tokens_out":25711,"duration_ms":275068,"concrete_test":"Test Proposition 2.8(2) in the explicit idempotent recollement of Example 4.3: A = K(1→2→3), e = e1+e2, so C = eAe = K(1→2). Compute j_{!*} by j_{!*} = Im(j_! → j_*) on the bricks S1, S2, and P1 of C, using the standard formulas j_! = −⊗_{eAe} eA and j_* = Hom_{eAe}(Ae, −). Compare Hom_B(j_{!*}X, j_{!*}Y) with Hom_C(X,Y) for all ordered pairs (X,Y) among {S1, S2, P1}. In particular, Hom_C(P1, S1) = 0, so if Hom_B(j_{!*}P1, j_{!*}S1) ≠ 0, Proposition 2.8(2) is false for module categories and the central construction collapses. If all nine Hom-spaces match, repeat the same computation in Example 4.4 (preprojective A3) before accepting the general recollement statement.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 4.1 reduces the entire construction to the claim in Theorem 3.3 that i_*(S_A) ⊔ j_{!*}(S_C) is a semibrick. The only non-formal input in that claim is Proposition 2.8(2), quoted from [BBD,FP]: j_{!*} is a full embedding. This is not proved in the text, and it is not a formal consequence of Definition 2.6: j_{!*} is defined as the image of the natural transformation j_! → j_*, and an image functor can fail to be full even when j^* j_{!*} ≅ id_C. Lemma 3.1 uses full faithfulness twice: once to ensure End_B(j_{!*}S) is a division algebra (brick property) and once to ensure Hom_B(j_{!*}S_r, j_{!*}S'_r) ≅ Hom_C(S_r, S'_r). If either fails, the union in Theorem 3.3 is not known to be a semibrick, the bijection in Theorem 2.4 cannot be applied, and Theorem 4.1 has no constructed M_B. The examples in Tables 1 and 2 only display outputs of the claimed procedure; they do not independently verify that j_{!*} is full on the involved bricks, so they do not settle the premise.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes to glue semibricks along a recollement of module categories of finite-dimensional algebras, using the intermediate extension functor j_{!*}. The main results are: (1) Theorem 3.3, which states that the union i_*(sbrick A) ⊔ j_{!*}(sbrick C) is a semibrick in mod B; (2) Theorem 3.6, which states that if B is τ-tilting finite then so are A and C; and (3) Theorem 4.1, which, using Asai's bijection between support τ-tilting modules and semibricks for τ-tilting finite algebras, constructs a unique support τ-tilting B-module from support τ-tilting modules in mod A and mod C. The paper also provides examples for the path algebra of type A3 and the preprojective algebra of type A3.","tokens_in":8638,"tokens_out":48658,"duration_ms":437995,"significance":"If the main theorems are correct, the paper gives a clean and useful construction principle: support τ-tilting modules over a τ-tilting finite algebra can be obtained by gluing those of the outer algebras in a recollement. The idea of transporting the BBD intermediate-extension construction to semibricks is natural, and the paper's examples honestly indicate that the construction is not surjective. The proof structure is transparent, and the dependence on Asai's bijection is explicit. The main risk is the unproved full faithfulness of j_{!*} (Proposition 2.8(2)), which is load-bearing for Theorem 3.3, and hence for Theorems 3.6 and 4.1.","major_comments":[{"comment":"The assertion that j_{!*} is a full embedding (as well as the identities j^{*} j_{!*} ≅ id, i^{*} j_{!*} = 0, and i^{!} j_{!*} = 0) is not proved in the text; the citation to [BBD, FP] is too vague, giving no proposition or theorem number. This is not a formal consequence of Definition 2.6: j_{!*} is defined as the image of a natural transformation j_! → j_*, and an image functor can fail to be full. Lemma 3.1 uses full faithfulness of j_{!*} to conclude that j_{!*} sends bricks and semibricks to bricks and semibricks, and Case 2 of Theorem 3.3 uses it to identify Hom_B(j_{!*}S_r, j_{!*}S'_r) with Hom_C(S_r, S'_r). If Prop 2.8(2) fails, Theorem 3.3, Theorem 3.6, and Theorem 4.1 do not follow. The authors should either provide a proof of Prop 2.8(2) in the setting of recollements of module categories, or give a precise reference with the exact statement.","section":"Section 2.3, Proposition 2.8(2)"},{"comment":"The proof of Theorem 3.6 uses the full faithfulness of i_* and j_{!*} to deduce that the sets sbrick A and sbrick C are finite from the finiteness of sbrick B. This deduction requires that the induced maps on semibricks are injective, which holds only if i_* and j_{!*} are fully faithful. Since the full faithfulness of j_{!*} is not established (see the previous comment), the proof of Theorem 3.6 is conditional on the same unproved premise. If the premise is supplied, the argument is sound.","section":"Section 3, Theorem 3.6"}],"minor_comments":[{"comment":"The heading contains a typo: 'τ-titing theory' should read 'τ-tilting theory'.","section":"Section 2.2, heading"},{"comment":"There is a typo: 'funtor' should be 'functor'.","section":"Section 2.3, after Definition 2.6"},{"comment":"The word 'cateogry' should be 'category'.","section":"Section 3, after Definition 3.4"},{"comment":"The sentence 'it followings that' should be 'it follows that'.","section":"Corollary 3.7"},{"comment":"The reference 'Definition 2.5(1)' in Case 3 should be 'Definition 2.6(1)'.","section":"Proof of Theorem 3.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is well organized and the intended construction is appealing. The main issue is the unproved and vaguely cited full faithfulness of j_{!*} (Prop 2.8(2)); this is a load-bearing point that needs to be solidified, either by a proof or by a precise reference. I did not find evidence that the statement is false for recollements of module categories, so I do not recommend rejection, but the manuscript in its current form leaves a gap in the central argument. The examples are helpful but do not test the premise directly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a short, honest paper that formalizes a natural gluing construction for semibricks along a recollement and uses Asai's bijection to land on support τ-tilting modules over τ-tilting finite algebras. It proves the expected containment i_*(sbrick A) ⊔ j_{!*}(sbrick C) ⊆ sbrick B and, under τ-tilting finiteness of B, constructs a unique MB for each pair MA, MC. The proofs are direct and, modulo one imported fact, correct. The examples (A3 and preprojective of type A3) are worked out in tables showing the construction covers a proper subset of all support τ-tilting modules, which is honest and keeps the claim appropriately modest.\n\nWhat's actually new: while gluing simples via intermediate extension is classical, I don't know of a stated semibrick analogue. The transfer through Asai's bijection is a reasonable extension. The author gives no new mechanism — fully faithful functors preserving bricks is Lemma 3.1, and the rest is elementary adjunction bookkeeping — but the formulation itself is worth having.\n\nThe soft spot: the whole theorem leans on Proposition 2.8(2), the claim that j_{!*} is fully faithful, quoted from [BBD,FP] with no proof. The stress-test is right that this is load-bearing: without it, Lemma 3.1 doesn't apply, and the semibrick containment and hence the MB construction don't follow. The paper also doesn't give a precise page/theorem location for the fact. I have no counterexample in the module-category setting; my reading is that this is an expository gap rather than an error. But before I'd use Theorem 4.1 in my own work, I'd want to see either a proof of that full faithfulness or a clear pointer to the exact statement in [FP]. The author should be asked to add that in revision.\n\nAlso worth a small mention: the construction is not complete (it realizes only part of sτ-tilt), so the title's 'a construction' is apt; don't expect a classification.\n\nOverall: the paper is a solid, small contribution. It deserves a serious referee — the property of j_{!*} needs checking, but the surrounding logic is clean and the examples appear correct. I'd accept the paper if the j_{!*} point is settled or properly referenced.","headline":"A clean, short gluing construction for semibricks and support τ-tilting modules; sound under a cited, unproved full-faithfulness claim for j_{!*} that deserves scrutiny.","tokens_in":9132,"tokens_out":14006,"would_cite":true,"duration_ms":144469,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G10","18A22","18A40","18E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"In a recollement of module categories, semibricks on the outer categories glue, via the intermediate extension functor, into semibricks of the middle; if the middle algebra is $\\tau$-tilting finite, support $\\tau$-tilting modules on the…","keywords":["semibricks","bricks","support tau-tilting modules","tau-tilting finite algebras","recollements","intermediate extension functor","gluing construction","module categories"],"falsifier":"Find or construct a recollement $R(A,B,C)$ of finite-dimensional algebras in which the intermediate extension functor $j_{!*}$ is not fully faithful, or a semibrick in $\\operatorname{mod} C$ whose image under $j_{!*}$ contains two distinct summands with a nonzero map between them; such an example would break Lemma 3.1 and with it Theorems 3.3 and 4.1.","tokens_in":8147,"feed_emoji":"🧩","tokens_out":10304,"duration_ms":90595,"temperature":0.7,"pith_summary":"The paper proves that semibricks—sets of pairwise Hom-orthogonal bricks—can be glued along a recollement of module categories, and that when the middle category is $\\tau$-tilting finite this gluing produces a unique support $\\tau$-tilting module from support $\\tau$-tilting modules on the two outer categories. The construction matters because support $\\tau$-tilting modules are a compact combinatorial catalog of the representation theory, and gluing gives a way to build them from smaller pieces. The paper also proves a finiteness transfer: if the middle algebra is $\\tau$-tilting finite, so are both outer algebras. Examples over an $A_3$ path algebra and an $A_3$ preprojective algebra illustrate the construction, and in those examples the glued modules form a proper subset of all support $\\tau$-tilting modules.","feed_headline":"Gluing semibricks builds support tau-tilting modules","feed_subtitle":"Support tau-tilting modules on two outer algebras glue to exactly one support tau-tilting module on the middle algebra.","key_machinery":"The load-bearing object is the recollement $R(A,B,C)$ of module categories with its seven functors, especially the intermediate extension functor $j_{!*}: \\operatorname{mod} C \\to \\operatorname{mod} B$, defined as the image of the natural transformation $j_! \\to j_*$. The key property, quoted from standard recollement theory, is that $j_{!*}$ is fully faithful; Lemma 3.1 then shows any fully faithful functor sends bricks to bricks and semibricks to semibricks. Theorem 3.3 proves the two images $i_*(\\mathcal{S}_A)$ and $j_{!*}(\\mathcal{S}_C)$ are mutually Hom-orthogonal, and the known bijection for $\\tau$-tilting finite algebras converts this glued semibrick into a unique support $\\tau$-tilting module.","core_discovery":"The central claim is Theorem 3.3: for a recollement $R(A,B,C)$, the union $i_*(\\operatorname{sbrick} A) \\sqcup j_{!*}(\\operatorname{sbrick} C)$ consists of semibricks in $\\operatorname{mod} B$, so semibricks from the left and right categories can be pushed into the middle and joined without creating nonzero Homs between the two pieces. Theorem 4.1 then uses the bijection between support $\\tau$-tilting modules and semibricks over $\\tau$-tilting finite algebras to conclude that any support $\\tau$-tilting $A$-module and any support $\\tau$-tilting $C$-module determine a unique support $\\tau$-tilting $B$-module associated with the glued semibrick. The paper further proves that $\\tau$-tilting finiteness descends from the middle algebra to both outer algebras (Theorem 3.6).","pith_inferences":["Because the proof of Theorem 3.3 uses only full faithfulness and the adjunction identities, the same gluing should work in any recollement of abelian categories whose bricks and semibricks are defined by Hom-orthogonality and division endomorphism rings, not just finite-dimensional algebras.","The examples leave open a precise description of which support $\\tau$-tilting modules in the middle category are gluable; a natural next step is to express the glued module's g-vector, or its support, in terms of the outer modules and the recollement functors.","Since semibricks correspond to wide subcategories in related settings, the glued semibrick $i_*(\\mathcal{S}_A) \\sqcup j_{!*}(\\mathcal{S}_C)$ may directly construct a wide subcategory of $\\operatorname{mod} B$ from wide subcategories of $\\operatorname{mod} A$ and $\\operatorname{mod} C$.","A direct formula for the glued support $\\tau$-tilting module would let one check exhaustiveness; the fact that the construction is not surjective in the type $A_3$ examples suggests that the remaining modules are obstructed by a condition visible in the recollement diagram."],"forward_implications":["For every recollement $R(A,B,C)$ with $B$ $\\tau$-tilting finite, every semibrick of $A$ and every semibrick of $C$ combine into a semibrick of $B$.","Every pair consisting of a support $\\tau$-tilting $A$-module and a support $\\tau$-tilting $C$-module determines a unique support $\\tau$-tilting $B$-module, and the association is injective on pairs.","If $B$ is $\\tau$-tilting finite, then both $A$ and $C$ are $\\tau$-tilting finite; consequently no recollement can have a $\\tau$-tilting finite middle algebra and infinite outer algebras.","For idempotent recollements, taking an idempotent $e$ of a $\\tau$-tilting finite algebra yields that both $eAe$ and $A/\\langle e\\rangle$ are $\\tau$-tilting finite.","The construction is not exhaustive: in the $A_3$ path algebra example only ten of fourteen support $\\tau$-tilting modules arise by gluing, and in the $A_3$ preprojective example only twelve of twenty-four."],"supporting_citations":[{"why":"Supplies the bijection between support $\\tau$-tilting modules and semibricks for $\\tau$-tilting finite algebras, the conversion that turns the glued semibrick into a unique module.","marker":"[As]"},{"why":"Provides the gluing-of-simples picture and the intermediate extension functor whose full faithfulness is used to preserve brick and semibrick properties.","marker":"[BBD]"},{"why":"Gives the recollement facts about full faithfulness and vanishing of the intermediate extension functor that Proposition 2.8 and Lemma 3.1 rely on.","marker":"[FP]"},{"why":"Defines $\\tau$-tilting finiteness and provides the characterization by finitely many bricks used to push finiteness from the middle algebra to the outer algebras.","marker":"[DIJ]"},{"why":"Introduces support $\\tau$-tilting modules, the objects whose gluing the paper constructs.","marker":"[AIR]"}],"fun_headline_variants":["Semibrick gluing constructs support tau-tilting modules","Glue semibricks across recollements to get tau-tilting","Support tau-tilting modules from glued semibricks","Recollement gluing yields unique tau-tilting modules","Brick gluing builds tau-tilting modules exactly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on the quoted, unproved fact that the intermediate extension functor from the right-hand category into the middle category is fully faithful; if that property failed, bricks and semibricks would not be preserved and the gluing theorems would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Semibrick gluing constructs support tau-tilting modules","Glue semibricks across recollements to get tau-tilting","Support tau-tilting modules from glued semibricks","Recollement gluing yields unique tau-tilting modules","Brick gluing builds tau-tilting modules exactly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000187,"raw_usage":{"total_tokens":1298,"prompt_tokens":881,"completion_tokens":417,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":331}},"tokens_in":497,"tokens_out":417,"duration_ms":4186,"temperature":1.0,"reasoning_tokens":331,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:52:20.821942+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find or construct a recollement $R(A,B,C)$ of finite-dimensional algebras in which the intermediate extension functor $j_{!*}$ is not fully faithful, or a semibrick in $\\operatorname{mod} C$ whose image under $j_{!*}$ contains two distinct summands with a nonzero map between them; such an example would break Lemma 3.1 and with it Theorems 3.3 and 4.1.","supporting_citations":[],"review_version":1}