Pith. sign in

REVIEW 3 major objections 3 minor 3 cited by

Tilting theory for extended module categories

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper proves that in extended module categories, tilting pairs, functorially finite s-torsion pairs, and silting complexes correspond one-to-one.

desk verdict Interesting bijection in Section 4, but Lemma 3.1 is false and the AR section needs a rewrite; major revision, not acceptance as is. read the letter →

arxiv 2411.15473 v2 pith:UMZDL7MA submitted 2024-11-23 math.RT math.CTmath.RA

Yu Zhou This is my paper · ORCID
classification math.RTmath.CTmath.RA MSC 16G1016G7018E30
keywords extendedmodulecategorys-torsionpairtau[m]-tiltingsiltingcomplexAuslander-ReitentheoryHappel-Reiten-Smalotiltingextriangulatedt-structure
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 tries to prove that tilting theory can be built directly on extended module categories, the full subcategory of the derived category made of complexes whose cohomology lives in a fixed window of degrees. Its central result is a three-way bijection among tau[m]-tilting pairs, functorially finite s-torsion pairs, and (m+1)-term silting complexes. If correct, this makes classical tau-tilting theory and higher-term silting theory two views of one construction, with explicit formulas converting a silting complex into a torsion pair and back. The paper also proves that these extended module categories have Auslander-Reiten triangles, so they behave like module categories for Auslander-Reiten theory even when the ambient derived category does not.

What carries the argument

The argument is carried by the $m$-extended heart $D^{[-(m-1),0]}$, an extriangulated category with a negative first extension, together with a correspondence, imported from a cited theorem, that identifies $s$-torsion pairs in this heart with bounded t-structures satisfying $D^{\le -m} \subseteq C^{\le 0} \subseteq D^{\le 0}$. Inside $m\text{-}\mathrm{mod}\,A$, the paper uses m-factors and m-subobjects, objects built by iterated extensions from a subcategory, and the Auslander-Reiten translation $\tau_{[m]}$ defined from minimal projective presentations and the Nakayama functor. The decisive bridge is that a positive $\tau_{[m]}$-rigid pair $(X,P)$ corresponds to the $(m+1)$-term complex $p_m(X) \oplus P[m]$, and the tau-tilting condition is exactly the assertion that this complex is silting.

What would settle it

Find a finite-dimensional algebra $A$ and an $(m+1)$-term silting complex $P$ for which the explicit equality $T(P) = \mathrm{Fac}_m(H^{[-(m-1),0]}P)$ fails, or exhibit a functorially finite s-torsion pair in $m\text{-}\mathrm{mod}\,A$ not equal to $(\mathrm{Fac}_m(H^{[-(m-1),0]}P), \mathrm{Sub}_m(H^{[-(m-1),0]}(\nu P[-1])))$ for any such $P$. Either would directly contradict Theorem 4.7.

Watch

Extended reading notes

Core claim

The paper's main result, Theorem 4.7, asserts that for the $m$-extended module category $m\text{-}\mathrm{mod}\,A$ (complexes of modules with cohomology only in degrees $-(m-1),\dots,0$), the following three sets are in bijection: basic $\tau_{[m]}$-tilting pairs, functorially finite $s$-torsion pairs, and basic $(m+1)$-term silting complexes of projective $A$-modules. Under the bijection, a $\tau_{[m]}$-tilting pair $(X,P)$ is sent to the $s$-torsion pair $(\mathrm{Fac}_m(X), X^{\perp_{\le 0}})$ and to the silting complex $p_m(X) \oplus P[m]$; conversely, a silting complex $P$ gives the torsion pair $(\mathrm{Fac}_m(H^{[-(m-1),0]}P), \mathrm{Sub}_m(H^{[-(m-1),0]}(\nu P[-1])))$. The paper additionally proves a Happel-Reiten-Smalo tilting theorem for $s$-torsion pairs in arbitrary $m$-extended hearts and an Auslander-Reiten theorem for $m\text{-}\mathrm{mod}\,A$ with translations $\tau_{[m]}$.

Load-bearing premise

The paper's framework rests on a cited theorem, stated as Proposition 1.9, that every s-torsion pair in the m-extended heart corresponds to a unique bounded t-structure whose negative part lies between $D^{\le -m}$ and $D^{\le 0}$; the paper does not prove this correspondence itself, so if that theorem does not apply here, the main bijections lose their foundation.

Editorial extensions

If this is right

  • Every functorially finite s-torsion pair in $m\text{-}\mathrm{mod}\,A$ has the explicit form $(\mathrm{Fac}_m(H^{[-(m-1),0]}P), \mathrm{Sub}_m(H^{[-(m-1),0]}(\nu P[-1])))$ for a unique basic $(m+1)$-term silting complex $P$.
  • Counting $\tau_{[m]}$-tilting pairs is the same as counting $(m+1)$-term silting complexes, because $(X,P) \mapsto p_m(X) \oplus P[m]$ is a bijection.
  • The extended module category has Auslander-Reiten triangles, and the translation $\tau_{[m]}$ connects Ext-vanishing to torsion theory through an Auslander-Reiten formula.
  • For any functorially finite s-torsion pair, the number of indecomposable projective objects in each side equals the number of indecomposable injective objects in that side.
  • When $m=1$, the bijections reduce to the classical bijections among support tau-tilting pairs, functorially finite torsion pairs, and 2-term silting complexes.

Reading between the lines

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

  • If the main bijection holds, mutation of $(m+1)$-term silting complexes should induce a well-defined mutation theory on $\tau_{[m]}$-tilting pairs, although the paper does not develop such a theory.
  • The paper's Example 4.14 shows that the classical completion property of tau-tilting pairs fails for $m=2$; a reader might infer that any meaningful notion of maximal $\tau_{[m]}$-rigid pair needs extra hypotheses beyond maximality, possibly involving the positive condition on negative extensions.
  • Because the proof relies only on a formal correspondence between s-torsion pairs and bounded t-structures, the same three-way bijection may transfer to other Hom-finite extriangulated categories with a negative first extension and suitable projectives and injectives, not only $m\text{-}\mathrm{mod}\,A$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper develops tilting theory for m-extended module categories m-modA. It proves a generalized Happel-Reiten-Smalø theorem for s-torsion pairs in extended hearts of bounded t-structures, characterizes the torsion pairs induced by (m+1)-term silting complexes in terms of m-factors and m-subobjects, constructs Auslander-Reiten triangles in m-modA, introduces τ[m]-tilting pairs, and claims bijections between τ[m]-tilting pairs, functorially finite s-torsion pairs, and (m+1)-term silting complexes. The main structural results are Theorem 1.12, Theorem 2.15, Theorem 3.12, and Theorem 4.7.

Significance. If the main results are correct, the paper gives a natural simultaneous generalization of classical τ-tilting theory, silting theory, and torsion-pair theory, and the explicit bijections in Theorem 4.7 would be valuable. The paper contains several useful intermediate constructions, such as the m-factor/m-subobject description in Theorem 2.15 and the projective/injective structure of T(P) and F(P) in Proposition 2.14. However, Section 3 contains a false foundational lemma about projective and injective objects in m-modA, and several later results rely on that lemma. The main bijection in Theorem 4.7 may be salvageable, but the paper as written is not internally consistent.

major comments (3)
  1. [§3, Lemma 3.1] Lemma 3.1 is false. The proof asserts that for the (m+1)-term silting complex A, T(A)=m-modA, but by (2.7) T(A)={X∈m-modA | Hom(A,X[j])=0, 1≤j≤m}, and Hom(A,X[j])≅H^{-j}(X). An object with nonzero cohomology in degrees -(m-1),...,-1, for example X=k[-1] when m=2 and A=k, has H^{-1}(X)≠0 and hence is not in T(A). In fact T(A)=modA, not m-modA. Consequently the E-projective objects of m-modA are not projA: they are P[-(m-1)] for P∈projA, and E(P,k[-1])=Hom(P,k) is generally nonzero. Similarly, (injA)[m-1] is not even a subcategory of m-modA for m>1; the E-injective objects are injA. This is not a harmless misstatement: it is used in the proofs of Proposition 3.10, Proposition 3.11, Corollary 3.13, and Proposition 3.17.
  2. [§3, Proposition 3.10] Because the projective objects are misidentified, the quotient categories in Notation 3.2 are not the categories actually used in the proofs. In Proposition 3.10 the functor F is claimed to restrict to an equivalence from projA to projA and the quotient K[-m,0](projA)/add(A⊕A[m]) is identified with m-modA; with the true projectives P[-(m-1)], the kernel of F is different and the displayed equivalence does not hold as stated. Proposition 3.11(v) and Theorem 3.12 rely on this quotient equivalence, so the proof of the Auslander-Reiten theorem is currently unsupported. The theorem may be true after correcting the projective and injective objects, but the present argument is invalid.
  3. [§4, Corollary 4.13] Corollary 4.13 asserts that the number of projective objects equals the number of injective objects in T(P) and F(P). The proof for T(P) uses Proposition 2.14 and is independent of Lemma 3.1, but the proof for F(P) and the comparison with injectives of m-modA use the false statement that νQ[m-1] is injective in m-modA. The actual injectives are νQ, so the argument does not establish the claimed equality. This is a direct consequence of the Section 3 error and shows that the error propagates into the τ[m]-tilting part of the paper.
minor comments (3)
  1. [§2, (2.9)-(2.10)] The reductions of the vanishing conditions to all j≥1 and all j≤0 are asserted without proof. They are true, but a short justification using the cohomological range of X and the fact that P is an (m+1)-term complex of projectives would improve readability.
  2. [§3, Notation 3.2] The two quotient categories are denoted by nearly identical symbols, which is difficult to follow in print. Please use clearly distinguishable notation, for example overlined and underlined variants with an explicit explanation.
  3. [§3, Remark 3.9] The comparison between τ[m] and the higher Auslander-Reiten translation τm is useful, but the sentence 'it is clear that τ[m] and τm have different domains' is imprecise because both are defined on related but different categories; please spell out the domain and codomain of each functor.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation found; the central bijections are argued from definitions and external theorems, with only minor non-load-bearing self-citations.

full rationale

The load-bearing steps of the paper are not circular. The bijection from silting complexes to functorially finite s-torsion pairs is imported from Gupta [G, Theorem 4.1], and the s-torsion-pair/t-structure bijection is imported from [AET, Theorem 3.9] via Proposition 1.9; neither is a self-citation. The new results, Theorems 1.12, 2.15, 3.12, and 4.7, are proved from definitions, truncation triangles, duality, and standard external results. In particular, the τ[m]-tilting condition (4.6) is not defined in terms of silting complexes; the equality ⊥≤0(τ[m](X)) ∩ P⊥≤0 = T(P) is proved using Lemma 4.8, and the characterization of silting complexes is obtained through Proposition 2.8 and Theorem 2.15. There is no fitted parameter renamed as a prediction, and no uniqueness theorem is imported from the authors' own prior work to force the choices. The self-citations [BZ1] and [LZ] are used only for an m=1 analogue (Lemma 2.3, Proposition 2.14 context) and for an illustrative example (Example 4.14), so they do not carry the central derivation. A separate correctness concern exists: Lemma 3.1 asserts T(A)=m-modA, which appears false for m>1 since objects with nonzero cohomology in degree −(m−1) are in m-modA but not in T(A); this is an internal mathematical issue, not a circular dependence.

Assumptions & free parameters 0 free parameters · 4 assumptions · 3 invented entities

The central claim rests on standard t-structure and extriangulated category machinery, plus two imported external theorems. No numerical parameters are fitted. The new mathematical objects (tau[m]-translations, tau[m]-tilting pairs, m-factors) are definitions with no independent falsifiable content beyond the stated theorems.

assumptions (4)
  • domain assumption The category D[-(m-1),0] is an extriangulated category with a negative first extension, making [AET, Theorem 3.9] applicable.
    Used to obtain Proposition 1.9 and the s-torsion pair bijection; stated in Remark 1.4.
  • domain assumption A is a finite-dimensional algebra over a field, so mod A is Hom-finite, Krull-Schmidt, and the Nakayama duality (2.3) holds.
    Stated in Section 2; required for bijections, Auslander-Reiten theory, and the duality used in Lemmas 2.3, 2.6, 4.8.
  • domain assumption Every object of m-modA has a representative as a complex concentrated in degrees [-(m-1),0], used implicitly in the simplification (2.9)/(2.10).
    Justifies extending the vanishing conditions to all j>=1 and j<=0; not explicitly proven in the paper.
  • standard math Standard t-structure orthogonality: Hom(D<=a, D>=b)=0 for a<b.
    Used throughout, e.g., in Lemmas 1.19, 2.3, 2.6, and the proof of Theorem 3.12.
invented entities (3)
  • tau[m] and tau^-[m] translations
    purpose: Generalized Auslander-Reiten translations for m-modA; used to define tau[m]-rigid and tau[m]-tilting objects.
    Defined in Definition 3.7 via truncations of projective and injective presentations. No external falsifiable prediction; the relationship to Iyama's tau_m is discussed in Remark 3.9.
  • tau[m]-tilting pairs
    purpose: New class of pairs (X,P) that biject with (m+1)-term silting complexes and functorially finite s-torsion pairs.
    Introduced in Definition 4.5; the bijections in Theorem 4.7 are the main result. No independent experimental handle.
  • m-factors and m-subobjects (Fac_m, Sub_m)
    purpose: New closure operations on subcategories of extended hearts; characterize s-torsion pairs (Proposition 1.18) and describe T(P), F(P) (Theorem 2.15).
    Defined in Definition 1.13. Internal tools with no external falsifiable content.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Tilting theory for extended module categories." pith.science (2026). https://pith.science/paper/UMZDL7MA

@misc{pith2026241115473,
  author       = {Pith},
  title        = {Pith review of: Tilting theory for extended module categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMZDL7MA}},
  note         = {Machine review of arXiv:2411.15473}
}
abstract

In extended hearts of bounded $t$-structures on a triangulated category, we provide a Happel-Reiten-Smalo tilting theorem and a characterization for $s$-torsion pairs. Applying these to $m$-extended module categories, we characterize torsion pairs induced by $(m+1)$-term silting complexes. After establishing Auslander-Reiten theory in extended module categories, we introduce $\tau_{[m]}$-tilting pairs and show bijections between $\tau_{[m]}$-tilting pairs, $(m+1)$-term silting complexes, and functorially finite $s$-torsion pairs.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

  1. Extended heart construction (I): The heart of $n$-cotorsion pairs on triangulated categories

    math.CT 2026-08 accept novelty 8.0 of 10

    Every n-cotorsion pair on a triangulated category has a heart that is an abelian n-truncated category, carrying compatible pretriangulated and extriangulated structures.

  2. Extriangulated factorization systems, $s$-torsion pairs and recollements

    math.CT 2025-07 conditional novelty 7.0 of 10

    Extriangulated factorization systems are shown to be equivalent to s-torsion pairs, offering a unified framework that recovers classical torsion pairs and t-structures.

  3. Higher-dimensional generalization of abelian categories via DG-categories

    math.CT 2025-01 conditional novelty 6.0 of 10

    Abelian n-truncated DG-categories are introduced as a higher-dimensional analogue of abelian categories, with extriangulated homotopy categories and unique factorization of morphisms.

Reference graph

Works this paper leans on

32 extracted references · 29 canonical work pages · cited by 3 Pith papers

  1. [1]

    Adachi, H

    T. Adachi, H. Enomoto and M. Tsukamoto, Intervals of s -torsion pairs in extriangulated categories with negative first extensions, Mathematical Proceedings of the Cambridge Philosophical Society 174 (2023), 451--469

    work page 2023

  2. [2]

    Adachi, O

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

    work page 2014

  3. [3]

    Aihara and O

    T. Aihara and O. Iyama, Silting mutation in triangulated categories, Journal of the London Mathematical Society 85 (2012), 633--668

    work page 2012

  4. [4]

    A characterisation of higher torsion classes

    J. August, J. Haugland, K. M. Jacobsen, S. Kvamme, Y. Palu and H. Treffinger. A characterisation of higher torsion classes, arXiv:2301.10463

    work page Pith review arXiv

  5. [5]

    Auslander, M

    M. Auslander, M. I. Platzeck and I. Reiten, Coxeter functions without diagrams, Trans. Amer. Math. Soc. 250 (1979) 1--12

    work page 1979

  6. [6]

    Auslander and S

    M. Auslander and S. O. Smal , Almost split sequences in subcategories, J. Algebra 69 (1981), 426--454. Addendum: J. Algebra 71 (1981), 592--594

    work page 1981

  7. [7]

    I. N. Bernstein, I. M. Gelfand and V. A. Ponomarev, Coxeter functors and Gabriel's theorem, Russ. Math. Surv. 28 (1973), 17--32

    work page 1973

  8. [8]

    Bongartz, Tilted Algebras, Proc

    K. Bongartz, Tilted Algebras, Proc. ICRA III (Puebla 1980), Lecture Notes in Math. No. 903, Springer-Verlag 1981, 26–38

    work page 1980

Show all 32 references
  1. [9]

    Brenner and M

    S. Brenner and M. C. R. Butler, Generalization of the Bernstein-Gelfand-Ponomarev reflection functors, Lecture Notes in Math. 839, Springer-Verlag (1980), 103–169

    1980

  2. [10]

    A. B. Buan and Y. Zhou, A silting theorem, J. Pure Appl. Algebra 220 (2016), 2748--2770

    2016

  3. [11]

    A. B. Buan and Y. Zhou, Silted algebras, Adv. Math. 303 (2016), 859--887

    2016

  4. [12]

    Gorsky, H

    M. Gorsky, H. Nakaoka and Y. Palu, Positive and negative extensions in extriangulated categories, arXiv:2103.12482

    arXiv

  5. [13]

    Gupta, d -term silting objects, torsion classes, and cotorsion classes, arXiv:2407.10562

    E. Gupta, d -term silting objects, torsion classes, and cotorsion classes, arXiv:2407.10562

  6. [14]

    Happel and C

    D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399--443

    1982

  7. [15]

    J. He, Y. Hu and P. Zhou, Torsion pairs and recollement of extriangulated categories, Communications in Algebra, 50 (2021), 2018--2036

    2021

  8. [16]

    Iyama, Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories, Adv

    O. Iyama, Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), 22-50

    2007

  9. [17]

    O. Iyama. Auslander–Reiten theory revisited, Trends in representation theory of algebras and related topics, Nicolaus Univ., Torun, 2007 (2008): 349--398

    2008

  10. [18]

    Iyama, H

    O. Iyama, H. Nakaoka and Y. Palu, Auslander-Reiten theory in extriangulated categories, Trans. Amer. Math. Soc. Ser. B 11 (2024), 248--305

    2024

  11. [19]

    M.Jacobsen and P

    K. M.Jacobsen and P. J rgensen, Maximal _d -rigid pairs, J. Algebra, 546 (2020), 119--134

    2020

  12. [20]

    J rgensen, Auslander-Reiten triangles in subcategories, J

    P. J rgensen, Auslander-Reiten triangles in subcategories, J. K-Theory 3 (2009), 583--601

    2009

  13. [21]

    Keller and D

    B. Keller and D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. S\' e r. A 40 (1988), 239--253

    1988

  14. [22]

    Krause and J

    H. Krause and J. Le, The Auslander-Reiten formula for complexes of modules, Adv. Math. 207 (2006), 133--148

    2006

  15. [23]

    K\" o nig and D

    S. K\" o nig and D. Yang. Silting objects, simple-minded collections, t -structures and co- t -structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403-438

    2014

  16. [24]

    Liu and H

    S. Liu and H. Niu, Almost split sequences in tri-exact categories, J. Pure Appl. Algebra 226 (2022), 1--31

    2022

  17. [25]

    Liu and Y

    Y-Z. Liu and Y. Zhou, A negative answer to Complement Question for presilting complexes, arXiv:2302.12502

    arXiv

  18. [26]

    Mart \'i nez and O

    L. Mart \'i nez and O. Mendoza, n -term silting complexes in K^b( ) , J. Algebra, 622 (2023), 98--133

    2023

  19. [27]

    McMahon, Support _2 -tilting and 2-torsion pairs, arXiv:2102.08254

    J. McMahon, Support _2 -tilting and 2-torsion pairs, arXiv:2102.08254

    arXiv

  20. [28]

    Nakaoka and Y

    H. Nakaoka and Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah Topol G\' e om Diff\' e r Cat\' e g, 60 (2019), 117--193

    2019

  21. [29]

    E. S. Rundsveen and L. Vaso, _d -tilting theory for linear Nakayama algebras, arXiv:2410.19505

    arXiv

  22. [30]

    S. O. Smal , Torsion theories and tilting modules, Bull. London Math. Soc. 16 (1984), 518--522

    1984

  23. [31]

    Zhou and B

    P. Zhou and B. Zhu, Triangulated quotient categories revisited, J. Algebra 502 (2018), 196–232

    2018

  24. [32]

    Zhou and B

    P. Zhou and B. Zhu, Support _n -tilting pairs, J. Algebra, 616 (2023), 193--211

    2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.