ICE-closed subcategories and epibricks over recollements

Jinrui Yang; Yongyun Qin

REVIEW 1 major objections 5 minor 36 references

ICE-closed subcategories and epibricks over recollements

T0 review · 1 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2502.03887 v1 pith:BZ6YUASO submitted 2025-02-06 math.RT math.CT

Jinrui Yang , Yongyun Qin This is my paper · ORCID

classification math.RTmath.CT MSC 16E3516G1016G2018G80

keywords ICE-closedsubcategoryepibrickmonobrickrecollementabeliancategoryintermediateextensionfunctorgluingsubcategoriestorsionclass

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

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.

Credit 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.

Soft 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.

This 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.

Referee Report

1 major / 5 minor

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.

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 (1)

[Theorem 3.10 and Corollary 3.9(1)] 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).

minor comments (5)

[Theorem 4.2(2)] 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.
[Proof of Theorem 3.10] 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.
[Lemma 3.8] 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.
[Abstract] 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.
[Remark 2.4(1) and proof of Theorem 3.11] 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).

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the transfer theorems are proved from adjointness and exactness, not from their conclusions; the empty-subcategory issue in Theorem 3.10 is a boundary correctness problem, not circularity.

full rationale

The paper's central results are transfer and gluing theorems for ICE-closed subcategories, epibricks, and monobricks along a recollement. The bijection in Theorem 3.10 is not constructed by fitting parameters or by defining one side in terms of the other: the left-hand set is explicitly restricted by the non-automatic condition j!j*(C) ⊂ C, and the paper emphasizes that this condition is necessary and does not always hold for ICE-closed subcategories, unlike for wide subcategories. The forward and inverse transfers are proved from the adjointness relations, exactness assumptions, and the definition of ICE-closed subcategories rather than assumed. Citations to [35] and [36] are to prior work by other authors and serve as proof templates or analogies, not as self-referential justification; no uniqueness theorem or load-bearing claim is imported from the present authors' own prior work. There is no parameter fitting, no renamed empirical pattern, and no self-definitional reduction. The empty-subcategory issue in Corollary 3.9(1) and Theorem 3.10 — the assertion that the zero object 'always belongs to W' fails when W is empty, so the preimage construction is not well-defined on all of A''_ice — is a genuine mathematical boundary-condition problem, but it is not a circularity: the theorem does not reduce to its inputs by construction. Therefore the circularity score is 0.

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

No free parameters, fitted constants, or invented entities appear. The paper is a transfer theorem in pure categorical algebra, deriving new claims from the recollement axiom and prior published lemmas. The main burden is the set of external citations and the exactness hypotheses in the theorem statements.

assumptions (3)

domain assumption The recollement axioms of Definition 2.3 hold for the categories and functors under study.
The paper's setting is a recollement of abelian categories with four adjoint pairs and Ker j* = Im i_*. All transfer results are conditional on this structure existing.
domain assumption The exactness assumptions appearing in the hypotheses of the main theorems are satisfied for the statements claimed, e.g., j* exact in Theorem 3.4(1), i* exact in Theorem 4.4.
The abstract's unconditional wording suggests all ICE-closed subcategories in A'' transfer to A, but the body requires exactness of j* or j! for that direction. The statements must be read with these hypotheses.
standard math External lemmas used as black boxes: [35, Prop 3.2, Thm 3.4, Prop 3.6], [17, Lemma 3.1], [18, Prop 8.8], [30, Prop 2.6], [9, Lemma 2.2], [18, Prop 4.3], [25, Prop 4.6].
Several proof steps are cited from other papers by different authors rather than derived in full. Correctness of these inherited steps is assumed; this is an external dependency, not circularity.

how reviews work

0 comments

Cite this review

Pith. "Pith review of ICE-closed subcategories and epibricks over recollements." pith.science (2026). https://pith.science/paper/BZ6YUASO

@misc{pith2026250203887,
  author       = {Pith},
  title        = {Pith review of: ICE-closed subcategories and epibricks over recollements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BZ6YUASO}},
  note         = {Machine review of arXiv:2502.03887}
}

abstract

Let $( \mathcal{A^{'}},\mathcal{A},\mathcal{A^{''}},i^\ast,i_\ast,i_!,j_!,j^\ast,j_\ast)$ be a recollement of abelian categories. We proved that every ICE-closed subcategory (resp. epibrick, monobrick) in $\mathcal{A^{'}}$ or $\mathcal{A^{''}}$ can be extended to an ICE-closed subcategories (resp. epibrick, monobrick) in $\mathcal{A}$, and the assignment $\mathcal{C}\mapsto j^*(\mathcal{C})$ defines a bijection between certain ICE-closed subcategories in $\mathcal{A}$ and those in $\mathcal{A}''$. Moreover, the ICE-closed subcategory $\mathcal{C}$ of $\mathcal{A}$ containing $i_\ast(\mathcal{A^{'}})$ admits a new recollement relative to ICE-closed subcategories $\mathcal{A^{'}}$ and $j^\ast(\mathcal{C})$ which induced from the original recollement when $j_!{j^\ast(\mathcal{C})}\subset\mathcal{C}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 35 canonical work pages

[35]

Y. Y. Zhang. Reduction of wide subcategories and recollements , Algebra Colloq. 30 (2023), no. 4, 713–720

work page 2023
[26]

ICE-closed subcategories and epibricks over one-point extensions

X. Li and H. P. Gao, ICE-closed subcategories and epibricks ov er one- point extensions, arXiv:2401.05645

work page Pith review arXiv
[1]

Adachi, O

T. Adachi, O. Iyama and I. Reiten, tau-tilting theory, Compos. M ath. 150 (2014), no. 3, 415–452

work page 2014
[2]

Angeleri H¨ ugel, S

L. Angeleri H¨ ugel, S. Koenig and Q. H. Liu, Recollements and tilting objects, J. Pure Appl. Algebra. 215 (2011), 420–438

work page 2011
[3]

Angeleri H¨ ugel and F

L. Angeleri H¨ ugel and F. Sentieri, Wide coreﬂective subcatego ries and torsion pairs, J. Algebra 664 (2024), 164–205

work page 2024
[4]

Asai and C

S. Asai and C. Pfeifer, Wide subcategories and lattices of torsio n classes, Algebr. Represent. Theory 25 (2022), no. 6, 1611–1629. 19

work page 2022
[5]

Baumann, J

P. Baumann, J. Kamnitzer and P. Tingley, Aﬃne Mirkovi´ c-Vilonenpoly- topes, Publ. Math. Inst. Hautes ´Etudes Sci. 120 (2014), 113–205

work page 2014
[6]

A. A. Be ˘ilinson, 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

Show all 36 references

[7]

Brenner and M

S. Brenner and M. C. R. Butler, Generalizations of the Bernstein - Gelfand-Ponomarev reﬂection functors, Representation theor y, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), pp. 103– 169, Lecture Notes in Math., 832, Springer, Berlin-New York, 1980

1979
[8]

X. W. Chen, A recollement of vector bundles, Bull. Lond. Math. So c. 44 (2012), 271–284

2012
[9]

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

2017
[10]

Demonet, O

L. Demonet, O. Iyama and G. Jasso, tau-tilting ﬁnite algebras, bricks, and g-vectors, Int. Math. Res. Not. IMRN 2019, no. 3, 852–892

2019
[11]

S. E. Dickson, A torsion theory for Abelian categories, Trans. Am. Math. Soc. 121 (1966), 223–235

1966
[12]

Enomoto, Rigid modules and ICE-closed subcategories in quive r rep- resentations, J

H. Enomoto, Rigid modules and ICE-closed subcategories in quive r rep- resentations, J. Algebra 594 (2022), 364–388

2022
[13]

Enomoto, Monobrick, a uniform approach to torsion-free c lasses and wide subcategories, Adv

H. Enomoto, Monobrick, a uniform approach to torsion-free c lasses and wide subcategories, Adv. Math. 393 (2021), Paper No. 108113, 4 1 pp

2021
[14]

Enomoto, From the lattice of torsion classes to the posets o f wide subcategories and ICE-closed subcategories, Algebr

H. Enomoto, From the lattice of torsion classes to the posets o f wide subcategories and ICE-closed subcategories, Algebr. Represen t. Theory 26 (2023), 3223–3253

2023
[15]

Enomoto and A

H. Enomoto and A. Sakai, Image-extension-closed subcatego ries of mod- ule categories of hereditary algebras, J. Pure Appl. Algebra 227 (2 023), no. 9, 107372
[16]

Enomoto and A

H. Enomoto and A. Sakai, ICE-closed subcategories and wide ta u-tilting modules, Math. Z. 300 (2022), no. 1, 541–577

2022
[17]

Feng and P

J. Feng and P. Zhang, Types of Serre subcategories of Groth endieck categories, J. Algebra 508 (2017), 16–34. 20

2017
[18]

Franjou and T

V. Franjou and T. Pirashvili, Comparison of Abelian categories re colle- ments, Doc. Math. 9 (2004), 41–56

2004
[19]

N. Gao, S. K¨ onig and C. Psaroudakis, Ladders of recollements of abelian categories, J. Algebra 579 (2021), 256–302

2021
[20]

Gao and J

N. Gao and J. Ma, (Gorenstein) silting modules in recollements, arXiv:2209.00531

arXiv
[21]

E. L. Green and C. Psaroudakis, On Artin Algebras Arising from M orita Contexts, Algebr. Represent. Theory 17 (2014), no. 5, 1485–1 525

2014
[22]

Hafezi, A

R. Hafezi, A. Nasr-Isfahani and J. Q. Wei, tau-tilting theory v ia the morphism category of projective modules I: ICE-closed subcateg ories, arXiv:2410.17965

arXiv
[23]

E. J. Hanson, Sequences of ICE-closed subcategories via pre ordered tau- rigid modules, arXiv:2410.01963

arXiv
[24]

Hovey, Classifying subcategories of modules, Tran

M. Hovey, Classifying subcategories of modules, Tran. Amer. M ath. Soc. 353 (2001), 3181–3191

2001
[25]

N. J. Kuhn, Generic representations of the ﬁnite general linea r groups and the Steenrod algebra: II, K. Theory 8 (1944), no. 4, 395–42 8

1944
[27]

Q. H. Liu, J. Vit´ oria and D. Yang, Gluing silting objects, J. Nagoy a Math. 216 (2014), 117–151

2014
[28]

Lu, Gorenstein defect categories of triangular matrix algeb ras, J

M. Lu, Gorenstein defect categories of triangular matrix algeb ras, J. Algebra 480 (2017), 346–367

2017
[29]

Marks and J

F. Marks and J. ˇSt’ov ´ ıˇ cek, Torsion classes, wide subcategories and lo- calisations, Bull. London Math. Soc. 49 (2017), no. 3, 405–416

2017
[30]

Psaroudakis, Homological theory of recollements of abelian c ate- gories, J

C. Psaroudakis, Homological theory of recollements of abelian c ate- gories, J. Algebra 398 (2014), 63–110

2014
[31]

Psaroudakis, A Representation-Theoretic Approach to Re collements of Abelian Categories, in Surveys in Representation Theory of Algeb ras, Contemporary Mathematics, 716, Amer

C. Psaroudakis, A Representation-Theoretic Approach to Re collements of Abelian Categories, in Surveys in Representation Theory of Algeb ras, Contemporary Mathematics, 716, Amer. Math. Soc., Providence, RI, 2018, 67–154. 21

2018
[32]

Psaroudakis and J

C. Psaroudakis and J. Vit´ oria, Recollements of module categor ies, Appl. Categ. Struct. 22 (2014), no. 4, 579–593

2014
[33]

Sakai, Classifying t-structures via ICE-closed subcategor ies and a lattice of torsion classes, arXiv:2307.11347

A. Sakai, Classifying t-structures via ICE-closed subcategor ies and a lattice of torsion classes, arXiv:2307.11347

arXiv
[34]

Tachikawa and K

H. Tachikawa and K. Ohtake, Colocalization and localization in Abelia n categories, J. Algebra 56 (1979), 1–23

1979
[36]

Y. Y. Zhang. A construction of support tau-tilting modules ove tau- tilting ﬁnite algebras, arXiv:1908.02245. 22

1908 arXiv

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [35]

Y. Y. Zhang. Reduction of wide subcategories and recollements , Algebra Colloq. 30 (2023), no. 4, 713–720

work page 2023

[2] [26]

ICE-closed subcategories and epibricks over one-point extensions

X. Li and H. P. Gao, ICE-closed subcategories and epibricks ov er one- point extensions, arXiv:2401.05645

work page Pith review arXiv

[3] [1]

Adachi, O

T. Adachi, O. Iyama and I. Reiten, tau-tilting theory, Compos. M ath. 150 (2014), no. 3, 415–452

work page 2014

[4] [2]

Angeleri H¨ ugel, S

L. Angeleri H¨ ugel, S. Koenig and Q. H. Liu, Recollements and tilting objects, J. Pure Appl. Algebra. 215 (2011), 420–438

work page 2011

[5] [3]

Angeleri H¨ ugel and F

L. Angeleri H¨ ugel and F. Sentieri, Wide coreﬂective subcatego ries and torsion pairs, J. Algebra 664 (2024), 164–205

work page 2024

[6] [4]

Asai and C

S. Asai and C. Pfeifer, Wide subcategories and lattices of torsio n classes, Algebr. Represent. Theory 25 (2022), no. 6, 1611–1629. 19

work page 2022

[7] [5]

Baumann, J

P. Baumann, J. Kamnitzer and P. Tingley, Aﬃne Mirkovi´ c-Vilonenpoly- topes, Publ. Math. Inst. Hautes ´Etudes Sci. 120 (2014), 113–205

work page 2014

[8] [6]

A. A. Be ˘ilinson, 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