pith. sign in

arxiv: 2208.14413 · v6 · submitted 2022-08-30 · 🧮 math.RT · math.AG· math.AT· math.KT· math.QA

The Derived Auslander-Iyama Correspondence

Gustavo Jasso , Bernhard Keller , Fernando Muro This is my paper

Pith reviewed 2026-05-24 11:19 UTC · model grok-4.3

classification 🧮 math.RT math.AGmath.ATmath.KTmath.QA
keywords twisted periodic algebrasdZ-cluster tiltingdifferential graded enhancementalgebraic triangulated categoriesAuslander correspondencecluster categoriesrepresentation theory of algebras
0
0 comments X

The pith

For each d ≥ 1, twisted (d+2)-periodic algebras correspond to algebraic triangulated categories with a dZ-cluster tilting object that admit a unique differential graded enhancement.

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

The paper extends the Derived Auslander Correspondence to higher dimensions over a perfect field. It establishes a correspondence between twisted (d+2)-periodic algebras and algebraic triangulated categories of finite type that contain a dZ-cluster tilting object. These categories are proved to possess a unique differential graded enhancement. The result supplies recognition theorems for the Amiot cluster category of a self-injective quiver with potential and for the Amiot-Guo-Keller cluster category of a d-representation finite algebra. Applications include infinitely many triangulated categories whose differential graded enhancement is unique but not strongly unique, plus a key step toward the Donovan-Wemyss Conjecture via the appendix.

Core claim

Given an integer d ≥ 1, twisted (d+2)-periodic algebras correspond to algebraic triangulated categories with a dZ-cluster tilting object; the latter admit a unique differential graded enhancement. This higher-dimensional Derived Auslander-Iyama Correspondence yields recognition theorems for the Amiot cluster category of a self-injective quiver with potential and the Amiot-Guo-Keller cluster category associated with a d-representation finite algebra.

What carries the argument

The dZ-cluster tilting object inside an algebraic triangulated category of finite type, which induces the bijection with twisted (d+2)-periodic algebras and guarantees uniqueness of the differential graded enhancement.

If this is right

  • Recognition theorems identify the Amiot cluster category of a self-injective quiver with potential as an algebraic triangulated category with a dZ-cluster tilting object.
  • Recognition theorems identify the Amiot-Guo-Keller cluster category of a d-representation finite algebra in the same way.
  • Infinitely many triangulated categories exist whose differential graded enhancement is unique but not strongly unique.
  • The appendix result supplies the missing ingredient, when combined with August and Hua-Keller, to prove the Donovan-Wemyss Conjecture for the Homological Minimal Model Program on threefolds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Explicit low-dimensional checks (d=1 recovers the known 3-periodic case) could verify the extension step by step.
  • The uniqueness of enhancement may make certain triangulated invariants independent of auxiliary choices in representation-theoretic computations.
  • The correspondence offers a route to construct further families of categories with controlled enhancements in higher-dimensional representation theory.

Load-bearing premise

The base constructions and uniqueness statements from the 3-periodic case extend to arbitrary d while preserving algebraicity and the cluster-tilting property without further restrictions on twisting or the perfect field.

What would settle it

Exhibit one twisted (d+2)-periodic algebra for some d > 1 whose associated triangulated category is either non-algebraic or admits more than one differential graded enhancement.

Figures

Figures reproduced from arXiv: 2208.14413 by Bernhard Keller, Fernando Muro, Gustavo Jasso.

Figure 1
Figure 1. Figure 1: Range of definition of the extended Bousfield–Kan spectral sequence (r = 5). The red region is where the classical Bousfield–Kan spectral sequence is defined; the extended spectral sequence is defined also in the blue region. so that we have a tower of pointed fibrations · · · ։ Xi+2 ։ · · · ։ X1 ։ X0. The above perspective on A∞-structures is advantageous in that it enables us to apply robust techniques f… view at source ↗
Figure 2
Figure 2. Figure 2: The E •,∗ d+2-page of the extended Bousfield–Kan spectral sequence can be non-trivial only in the green part (d = 4), see Proposition 5.2.9. Consider the degree (d + 2, −d) endomorphism of C •,∗ (A, {md+2}) given by the isomorphisms HHs,t(A) ∼=−→ HHs+d+2,t−d (A), x 7−→ {md+2} · x, if (s, t) 6= (d + 1, −d), see Proposition 4.7.15, and by HHd+1,−d (A) ∼=−→ HH2d+3,−2d (A), x 7−→ {md+2} · x + {δ/d} · x 2 . The… view at source ↗
read the original abstract

We work over a perfect field. Recent work of the third-named author established a Derived Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted $3$-periodic to algebraic triangulated categories of finite type. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. In this article we prove a higher-dimensional version of this result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster tilting object. We also show that the latter triangulated categories admit a unique differential graded enhancement. Our result yields recognition theorems for interesting algebraic triangulated categories, such as the Amiot cluster category of a self-injective quiver with potential in the sense of Herschend and Iyama and, more generally, the Amiot-Guo-Keller cluster category associated with a $d$-representation finite algebra in the sense of Iyama and Oppermann. As an application of our result, we obtain infinitely many triangulated categories with a unique differential graded enhancement that is not strongly unique. In the appendix, B. Keller explains how -- combined with crucial results of August and Hua-Keller -- our main result yields the last key ingredient to prove the Donovan-Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript establishes a higher-dimensional Derived Auslander-Iyama Correspondence over perfect fields. Given an integer d ≥ 1, it relates twisted (d+2)-periodic algebras to algebraic triangulated categories with a dZ-cluster tilting object and proves that the latter admit a unique differential graded enhancement. The work yields recognition theorems for Amiot cluster categories of self-injective quivers with potential and for Amiot-Guo-Keller cluster categories associated to d-representation finite algebras, together with an application to the Donovan-Wemyss conjecture via the appendix by B. Keller.

Significance. If the stated proofs hold, the result extends the third author's prior Derived Auslander Correspondence from the 3-periodic case to arbitrary dimensions, supplying recognition theorems for important classes of algebraic triangulated categories and providing a key ingredient for the Donovan-Wemyss conjecture in the Homological Minimal Model Program. The uniqueness of dg enhancements is a concrete strength of the correspondence.

minor comments (1)
  1. [Introduction] The introduction could include a brief explicit statement of how the twisting parameter is preserved under the correspondence (cf. the definition of twisted (d+2)-periodic algebras).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending acceptance. We are pleased that the work is viewed as a natural extension of the third author's prior Derived Auslander Correspondence and as providing useful recognition theorems and an ingredient for the Donovan-Wemyss conjecture.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes a higher-dimensional extension of the Derived Auslander Correspondence from prior work by the third author, relating twisted (d+2)-periodic algebras to algebraic triangulated categories with dZ-cluster tilting objects and proving unique dg enhancements. No quoted steps in the abstract or description exhibit self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the new claims to the base case by construction. The recognition theorems and Donovan-Wemyss application are presented as consequences of the independent extension over perfect fields, with the base case serving as external support rather than an internal tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities appear; the work rests on the standard axioms of triangulated and dg categories together with the base case established in prior work by one of the authors.

axioms (1)
  • standard math Standard axioms of triangulated categories, dg enhancements, and algebraicity over a perfect field.
    Invoked throughout the statement of the correspondence and the uniqueness of enhancements.

pith-pipeline@v0.9.0 · 5787 in / 1379 out tokens · 44572 ms · 2026-05-24T11:19:31.426371+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

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

  1. Higher exact dg-categories

    math.CT 2026-04 unverdicted novelty 7.0

    n-exact dg-categories are introduced so that their homotopy categories carry n-exangulated structures when Hom-cohomologies vanish, and n-cluster tilting subcategories of exact dg-categories naturally become n-exact d...

  2. Lectures on singularity categories

    math.RT 2026-03 unverdicted

    Lecture notes providing an introduction to singularity categories.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · cited by 2 Pith papers · 2 internal anchors

  1. [1]

    On the uniqueness of infinity-categ orical enhancements of trian- gulated categories, arXiv:1812.01526 [math.AG]

    [Ant] Benjamin Antieau. On the uniqueness of infinity-categ orical enhancements of trian- gulated categories, arXiv:1812.01526 [math.AG]. [AR91] Maurice Auslander and Idun Reiten. On a theorem of E. G reen on the dual of the transpose. In Representations of finite-dimensional algebras (Tsukuba, 1990), vol- ume 11 of CMS Conf. Proc. , pages 53–65. Amer. Math...

    work page arXiv 1990

  2. [2]

    With appendices and an introduction by Luchezar L

    ©2021. With appendices and an introduction by Luchezar L. Avramov, Benjamin Briggs, Srikanth B. Iyengar and Janina C. Letz. [BW18] Gavin Brown and Michael W emyss. Gopakumar-Vafa inva riants do not determine flops. Comm. Math. Phys. , 361(1):143–154,

    work page 2021

  3. [3]

    Periodic trivial ex- tension algebras and fractionally Calabi–Yau algebras, ar Xiv:2012.11927 [math.RT]

    [CDIM] Aaron Chan, Erik Darp¨ o, Osamu Iyama, and Ren´ e Marcz inzik. Periodic trivial ex- tension algebras and fractionally Calabi–Yau algebras, ar Xiv:2012.11927 [math.RT]. [CF17] J. Daniel Christensen and Martin Frankland. Higher T oda brackets and the Adams spectral sequence in triangulated categories. Algebr. Geom. Topol. , 17(5):2687–2735,

    work page arXiv 2012

  4. [4]

    Stable homotopy

    [Fre66] Peter Freyd. Stable homotopy. In Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), pages 121–172. Springer, New York,

    work page 1965

  5. [5]

    Calabi-Yau algebras

    rnab019. [Gin] Victor Ginzburg. Calabi–Yau algebras, math/0612139 . [GKO13] Christof Geiss, Bernhard Keller, and Steffen Opperm ann. n-angulated categories. J. Reine Angew. Math. , 675:101–120,

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Cluster tilting objects in generalize d higher cluster categories

    [Guo11] Lingyan Guo. Cluster tilting objects in generalize d higher cluster categories. J. Pure Appl. Algebra, 215(9):2055–2071,

    work page 2055

  7. [7]

    Auslander correspondence for t riangulated categories

    [Han20] Norihiro Hanihara. Auslander correspondence for t riangulated categories. Algebra Number Theory, 14(8):2037–2058,

    work page 2037

  8. [8]

    JASSO AND F

    116 G. JASSO AND F. MURO [HI11] Martin Herschend and Osamu Iyama. Selfinjective quiv ers with potential and 2- representation-finite algebras. Compos. Math. , 147(6):1885–1920,

    work page 1920

  9. [9]

    Rep- resentation theory of Geigle–Lenzing complete intersecti ons, 1409.0668

    [HIMO] Martin Herschend, Osamu Iyama, Hiroyuki Minamoto, a nd Steffen Oppermann. Rep- resentation theory of Geigle–Lenzing complete intersecti ons, 1409.0668. To appear in Mem. Amer. Math. Soc. [HIO14] Martin Herschend, Osamu Iyama, and Steffen Opperman n. n-representation infinite algebras. Adv. Math. , 252:292–342,

    work page arXiv

  10. [10]

    Cluster categories and r ational curves, 1810.00749 [math.AG]

    [HK] Zheng Hua and Bernhard Keller. Cluster categories and r ational curves, 1810.00749 [math.AG]. [HKK17] F. Haiden, L. Katzarkov, and M. Kontsevich. Flat sur faces and stability structures. Publ. Math. Inst. Hautes ´Etudes Sci. , 126(1):247–318,

    work page arXiv

  11. [11]

    [Kad82] T

    International Topology Confer ence (Moscow State Univ., Moscow, 1979). [Kad82] T. V. Kadeishvili. The algebraic structure in the ho mology of an A(∞ )-algebra. Soob- shch. Akad. Nauk Gruzin. SSR , 108(2):249–252 (1983),

    work page 1979

  12. [12]

    Non-commutative deformations of pe rverse coherent sheaves and rational curves, arXiv:2006.09547 [math.AG]

    [Kaw] Yujiro Kawamata. Non-commutative deformations of pe rverse coherent sheaves and rational curves, arXiv:2006.09547 [math.AG]. [Kel65] G. M. Kelly. Chain maps inducing zero homology maps. Proc. Cambridge Philos. Soc., 61:847–854,

    work page arXiv 2006

  13. [13]

    Relative singularity categ ories III: Cluster resolutions, arXiv:2006.09733 [math.RT]

    [KY] Martin Kalck and Dong Yang. Relative singularity categ ories III: Cluster resolutions, arXiv:2006.09733 [math.RT]. [KY11] Bernhard Keller and Dong Yang. Derived equivalences from mutations of quivers with potential. Adv. Math. , 226(3):2118–2168,

    work page arXiv 2006

  14. [14]

    Finite-dimensional algebras are ( m > 2)-calabi-yau-tilted, arXiv:1603.09709 [math.RT]

    [Lad] Sefi Ladkani. Finite-dimensional algebras are ( m > 2)-calabi-yau-tilted, arXiv:1603.09709 [math.RT]. [LH] Kenji Lef` evre-Hasegawa. Sur les A-infini cat´ egories, math/0310337. [Lin19] Zengqiang Lin. A general construction of n-angulated categories using periodic injec- tive resolutions. J. Pure Appl. Algebra , 223(7):3129–3149,

    work page arXiv

  15. [15]

    Formality and strongly unique enha ncements, arXiv:2204.09527 [math.KT]

    [Lor] Antonio Lorenzin. Formality and strongly unique enha ncements, arXiv:2204.09527 [math.KT]. [LP11] Yankı Lekili and Timothy Perutz. Fukaya categories o f the torus and Dehn surgery. Proc. Natl. Acad. Sci. USA , 108(20):8106–8113,

    work page arXiv

  16. [16]

    Tilting objects in periodic triangul ated categories, 2020, arXiv:2011.14096 [math.RT]

    [Sai20] Shunya Saito. Tilting objects in periodic triangul ated categories, 2020, arXiv:2011.14096 [math.RT]. [Sch01] Stefan Schwede. The stable homotopy category has a u nique model at the prime

    work page arXiv 2020

  17. [17]

    Addendum ` a Invariants Additifs de dg-Cat´ egories.Int

    [Tab06] Gon¸ calo Tabuada. Addendum ` a Invariants Additifs de dg-Cat´ egories.Int. Math. Res. Not., 2006:Art. ID 75853, 3,

    work page 2006

  18. [18]

    Corrections ` a Invariants Addi tifs de DG-cat´ egories.Int

    [Tab07a] Gon¸ calo Tabuada. Corrections ` a Invariants Addi tifs de DG-cat´ egories.Int. Math. Res. Not. IMRN , 2007(24):Art. ID rnm149, 17,

    work page 2007

  19. [19]

    Lectures on dg-categories

    THE DERIVED AUSLANDER–IYAMA CORRESPONDENCE 119 [To¨ e11] Bertrand To¨ en. Lectures on dg-categories. In Topics in algebraic and topological K- theory, volume 2008 of Lecture Notes in Math. , pages 243–302. Springer, Berlin,

    work page 2008

  20. [20]

    Des cat´ egories d´ eriv´ ees de s cat´ egories ab´ eliennes.Ast´ erisque, (239):xii+253 pp

    [Ver96] Jean-Louis Verdier. Des cat´ egories d´ eriv´ ees de s cat´ egories ab´ eliennes.Ast´ erisque, (239):xii+253 pp. (1997),

    work page 1997

  21. [21]

    [vG] Okke van Garderen

    With a preface by Luc Illusi e, Edited and with a note by Georges Maltsiniotis. [vG] Okke van Garderen. Donaldson–Thomas invariants of len gth 2 flops, arXiv:2008.02591 [math.AG]. [W ei94] Charles A. W eibel. An introduction to homological algebra , volume 38 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge,

    work page arXiv 2008

  22. [22]

    A lockdown survey on cdv singularitie s, arXiv:2103.16990 [math.AG]

    [W em] Michael W emyss. A lockdown survey on cdv singularitie s, arXiv:2103.16990 [math.AG]. [W em18] Michael W emyss. Flops and clusters in the homologic al minimal model programme. Invent. Math. , 211(2):435–521,

    work page arXiv

  23. [23]

    Cotorsion pairs and t-structures in a $2-$Calabi-Yau triangulated category

    A homological algebra point of view. [ZZ] Yu Zhou and Bin Zhu. Cotorsion pairs and t-structures in a 2 − Calabi-Yau triangu- lated category, arXiv:1210.6424 [math.RT]. (G. Jasso) Lund University, Centre for Mathematical Sciences, S ¨olvegatan 18A, 221 00 Lund, Sweden Email address : gustavo.jasso@math.lu.se URL: https://www.maths.lu.se/staff/gustavo-jasso...

    work page internal anchor Pith review Pith/arXiv arXiv