A construction of support $\tau$-tilting modules over $\tau$-tilting finite algebras

Yingying Zhang

REVIEW 2 major objections 5 minor 1 cited by

A construction of support $\tau$-tilting modules over $\tau$-tilting finite algebras

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 1908.02245 v2 pith:6F6HKQGR submitted 2019-08-06 math.RT

Yingying Zhang This is my paper · ORCID

classification math.RT MSC 16G1018A2218A4018E10

keywords semibricksbrickssupporttau-tiltingmodulesfinitealgebrasrecollementsintermediateextensionfunctorgluingconstructionmodulecategories

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What'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.

The 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.

Also 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.

Overall: 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[Section 2.3, Proposition 2.8(2)] 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 3, Theorem 3.6] 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.

minor comments (5)

[Section 2.2, heading] The heading contains a typo: 'τ-titing theory' should read 'τ-tilting theory'.
[Section 2.3, after Definition 2.6] There is a typo: 'funtor' should be 'functor'.
[Section 3, after Definition 3.4] The word 'cateogry' should be 'category'.
[Corollary 3.7] The sentence 'it followings that' should be 'it follows that'.
[Proof of Theorem 3.3] The reference 'Definition 2.5(1)' in Case 3 should be 'Definition 2.6(1)'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction feeds outer semibricks through standard recollement functors and applies external bijections; no fitted parameter or self-citation is load-bearing.

full rationale

The paper's derivation chain is self-contained in the relevant sense. Theorem 3.3 takes as input semibricks in mod A and mod C and produces i_*(S_L) ⊔ j_{!*}(S_R) in mod B; the proof uses only the full faithfulness of i_* (from Definition 2.6) and the full faithfulness of j_{!*} (quoted from [BBD,FP]), together with the vanishing i^* j_{!*} = 0 and i^! j_{!*} = 0 from Proposition 2.8(1). The semibrick output is not fed back into the inputs, nor is any parameter fitted to force the conclusion. Theorem 4.1 then applies Asai's bijection (Theorem 2.4) twice: once to pass from M_A and M_C to their associated semibricks, and once to pass from the glued semibrick back to a unique support τ-tilting B-module. This is an application of an external theorem, not a renaming of the inputs. The examples illustrate the construction but are not used to justify the general claim. The unproved-in-text full faithfulness of j_{!*} is an external cited premise; if it were false the theorem would fail, but that is a correctness concern, not circularity. There are no self-citations, no uniqueness imported from the author's own prior work, and no fitted input called a prediction. The result is therefore not circular.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The paper rests on standard background in recollements and on two external classification results (Asai, DIJ). The only assumption that deserves scrutiny is the full faithfulness of j_{!*}, which is stated but not proved in the text.

assumptions (3)

domain assumption The intermediate extension functor j_{!*} is fully faithful and satisfies i^*j_{!*}=i^!j_{!*}=0 (Proposition 2.8).
Imported from [BBD,FP]; used in Lemma 3.1 and Theorem 3.3 to guarantee semibricks are preserved.
domain assumption Asai's bijection (Theorem 2.4) identifies support tau-tilting modules with semibricks for tau-tilting finite algebras.
External result from [As]; the bridge between the gluing of semibricks and support tau-tilting modules in Theorem 4.1.
domain assumption A recollement of module categories satisfies the standard adjunction and orthogonality relations in Definition 2.6 and Remark 2.7.
Background from [BBD,FP]; the orthogonality computations in Theorem 3.3 use these relations.

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Pith. "Pith review of A construction of support $\tau$-tilting modules over $\tau$-tilting finite algebras." pith.science (2026). https://pith.science/paper/6F6HKQGR

@misc{pith2026190802245,
  author       = {Pith},
  title        = {Pith review of: A construction of support $\tau$-tilting modules over $\tau$-tilting finite algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6F6HKQGR}},
  note         = {Machine review of arXiv:1908.02245}
}

read the original abstract

The notion of (semi)bricks, regarded as a generalization of (semi)simple modules, appeared in a paper of Ringel in 1976. In recent years, there has been several new developments motivated by links to {\tau}-tilting theory studied by Demonet-Iyama-Jasso and Asai. In this paper, we will discuss how to glue semibricks along a recollement with the intermediate extension functor similar to gluing simple modules by Beilinson-Bernstein-Deligne. As an application, we investigate the behavior of {\tau}-tilting finite under recollements of module categories of algebras. Moreover, we give some examples to show the construction of support {\tau}-tilting modules over {\tau}-tilting finite algebras by gluing semibricks via recollements.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

ICE-closed subcategories and epibricks over recollements
math.RT 2025-02 conditional novelty 5.0 of 10

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.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Asai, Semibricks , to appear in Int

S. Asai, Semibricks , to appear in Int. Math. Res. Not

work page

[2] [2]

Adachi, O

T. Adachi, O. Iyama and I. Reiten, -tilting theory , Compos. Math. 150(2014) (3), 415-452

work page 2014

[3] [3]

Auslander, I

M. Auslander, I. Reiten and S. Smalo, Representation theory of Artin algebras , Cambridge Studies in Advanced Mathematics 36. Cambridge University Press, Cambrige, 1995

work page 1995

[4] [4]

M. V. Bondarko, Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes(for motives and in general) , J.K-Theory 6(2010),no.3, 387-504

work page 2010

[5] [5]

A. A. Be linson, J. Bernstein\ and\ P. Deligne, Faisceaux pervers, in Analysis and topology on singular spaces, I (Luminy, 1981) , 5--171, Ast\'erisque, 100, Soc. Math. France, Paris, 1982

work page 1981

[6] [6]

Crawley-Boevey and J

W. Crawley-Boevey and J. Sauter, On quiver Grassmannians and orbit closures for representation-finite algebras , Math. Z. 285 (2017), 367-395

work page 2017

[7] [7]

Demonet, O

L. Demonet, O. Iyama and G. Jasso, -tilting finite algebras, bricks and g-vector , Int. Math. Res. Not.IMRN 2019,(3), 852-892

work page 2019

[8] [8]

Franjou and T

V. Franjou and T. Pirashvili, Comparison of abelian categories recollements , Doc. Math. (2004) 9, 41-56

work page 2004