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arxiv: 2511.02641 · v2 · pith:4RBVV54Cnew · submitted 2025-11-04 · 🧮 math.AG · math.RA· math.RT

Higher hereditary algebras and toric Fano stacks of Picard number one or two

Ryu Tomonaga This is my paper

Pith reviewed 2026-05-21 19:16 UTC · model grok-4.3

classification 🧮 math.AG math.RAmath.RT
keywords d-tilting bundlestoric Fano stacksPicard numberhigher representation infinite algebrasnon-commutative crepant resolutiond-APR tiltsline bundlesendomorphism algebras
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The pith

Smooth toric Fano stacks of Picard number one or two classify all d-tilting bundles consisting of line bundles and realize them as geometric models for higher representation-infinite algebras.

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

The paper proves the existence of and classifies every d-tilting bundle made from line bundles on d-dimensional smooth toric Fano stacks whose Picard number is one or two. These bundles link the geometry of the stacks to higher Auslander-Reiten theory because their endomorphism algebras have global dimension at most d. For Picard number one the bundles stand in bijection with non-trivial upper sets in the Picard group under a fixed partial order; every such bundle is d-tilting and its endomorphism algebra is a d-representation infinite algebra of type Ã, and conversely every algebra of this type arises from some stack and bundle. For Picard number two the correspondence uses pairs of non-trivial upper sets in two related posets, one tied to a non-commutative crepant resolution and the other to a cut of its quiver, producing algebras of type ÃÃ. The stacks therefore serve as geometric models for the two algebra classes and supply combinatorial proofs that each class is closed under d-APR tilts.

Core claim

All d-tilting bundles consisting of line bundles on a d-dimensional smooth toric Fano stack of Picard number one are in bijection with the non-trivial upper sets of its Picard group equipped with a certain partial order; each such bundle is d-tilting and its endomorphism algebra is a d-representation infinite algebra of type Ã, while every algebra of this type arises in exactly this way. For Picard number two the d-tilting bundles correspond to pairs (I, I') of non-trivial upper sets in two partially ordered sets, where I determines a non-commutative crepant resolution of a toric singularity and I' determines a cut of the resolution quiver; the resulting endomorphism algebras belong to the (

What carries the argument

Bijection between d-tilting bundles of line bundles and non-trivial upper sets (or pairs of such sets) in the Picard group or related posets, which forces the endomorphism algebras to have global dimension at most d and the stated representation-infinite properties.

If this is right

  • Every classified bundle is d-tilting and its endomorphism algebra belongs to the appropriate class of higher representation-infinite algebras.
  • Conversely every algebra in these classes arises from a d-tilting bundle of line bundles on one of the stacks.
  • The two classes of algebras are closed under d-APR tilts, with the tilting modules described combinatorially from the upper sets.
  • The stacks function as geometric models that make the algebraic closure properties visible through explicit correspondences.

Where Pith is reading between the lines

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

  • The same upper-set construction may supply explicit tilting objects for stacks of higher Picard number once suitable partial orders are identified.
  • The geometric models could be used to compute explicit bases for the Grothendieck groups or to track the action of derived equivalences on the bundles.
  • Because the stacks are toric, the classification may translate into lattice-point counting problems that yield new examples of algebras with prescribed global dimension.

Load-bearing premise

The partial order placed on the Picard group (or the two posets used for Picard number two) is defined so that its non-trivial upper sets correspond exactly to the d-tilting bundles of line bundles and automatically produce endomorphism algebras of global dimension at most d with the claimed representation-infinite type.

What would settle it

A single d-tilting bundle of line bundles on one of these stacks whose associated collection of degrees fails to be an upper set in the defined partial order, or a d-representation infinite algebra of type à whose quiver and relations cannot be realized as the endomorphism algebra of any such bundle.

read the original abstract

On smooth projective varieties of dimension $d$, $d$-tilting bundles are important in both geometry and representation theory, since they provide a bridge from the geometry of such varieties to the derived McKay correspondence and to higher Auslander--Reiten theory. Here, a $d$-tilting bundle means a tilting bundle whose endomorphism algebra has global dimension at most $d$. In this paper, we prove the existence of and classify all $d$-tilting bundles consisting of line bundles on $d$-dimensional smooth toric Fano stacks of Picard number one or two. In the case of Picard number one, tilting bundles consisting of line bundles are in bijection with non-trivial upper sets in its Picard group equipped with a certain partial order. Moreover, all of them are $d$-tilting bundles and their endomorphism algebras are $d$-representation infinite algebras of type $\widetilde A$. Conversely, all such algebras arise in this way. In the case of Picard number two, $d$-tilting bundles consisting of line bundles are in bijection with pairs $(I,I')$, where $I$ and $I'$ are non-trivial upper sets in certain partially ordered sets. Here, $I$ corresponds to a non-commutative crepant resolution (NCCR) of a certain toric singularity and $I'$ corresponds to a cut of the quiver of this NCCR. We introduce higher representation infinite algebras of type $\widetilde A\widetilde A$ and show that the endomorphism algebras of these $d$-tilting bundles belong to this class. Conversely, all such algebras arise in this way. Thus smooth toric Fano stacks of Picard number one (respectively, two) can be regarded as geometric models of higher representation infinite algebras of type $\widetilde A$ (respectively, $\widetilde A\widetilde A$). Using these geometric models, we show that these classes of algebras are closed under $d$-APR tilts by giving a combinatorial description of their $d$-APR tilting modules.

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

2 major / 2 minor

Summary. The paper proves the existence of and classifies all d-tilting bundles consisting of line bundles on d-dimensional smooth toric Fano stacks of Picard number one or two. For Picard number one, tilting bundles of line bundles are in bijection with non-trivial upper sets in the Picard group under a certain partial order; all such are d-tilting and their endomorphism algebras are d-representation infinite of type Ã, with the converse holding. For Picard number two, d-tilting bundles correspond to pairs (I,I') of non-trivial upper sets in certain posets, where I gives an NCCR of a toric singularity and I' a cut of its quiver; the endomorphism algebras belong to the newly introduced class of higher representation infinite algebras of type ÃÃ, again with converse. The geometric models are used to give a combinatorial description of d-APR tilts and prove closure under them.

Significance. If the classifications and verifications hold, the work supplies explicit geometric models for the new classes of d-representation infinite algebras of types à and ÃÃ, together with combinatorial control over d-APR tilts. This strengthens the bridge between toric geometry, derived McKay correspondence, and higher Auslander-Reiten theory. The explicit bijections and the combinatorial description of d-APR tilting modules are particular strengths.

major comments (2)
  1. [§3] §3 (Picard number one case): The 'certain partial order' on the Picard group is introduced so that non-trivial upper sets correspond to collections of line bundles. The manuscript must explicitly verify (e.g., via a dedicated proposition) that for every non-trivial upper set the direct sum E satisfies gl.dim(End(E)) ≤ d and that End(E) is d-representation infinite of type Ã; without this step the bijection and the 'all of them are d-tilting' and converse statements rest on an unverified correspondence.
  2. [§5] §5 (Picard number two case): Analogously, the posets for the pairs (I,I'), the identification of I with an NCCR and I' with a cut, and the verification that the resulting endomorphism algebra lies in the class of higher representation infinite algebras of type ÃÃ (including the global-dimension bound) are load-bearing for the classification and converse claims; these verifications should be stated as separate lemmas.
minor comments (2)
  1. [Introduction] Notation for the partial orders and the distinction between à and Ãà could be introduced with a short table or diagram in the introduction for improved readability.
  2. [References] A few references to prior work on NCCRs of toric singularities and on d-tilting bundles could be added for context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We appreciate the recognition of the significance of the classifications and the combinatorial descriptions provided. Below, we address the major comments point by point and outline the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Picard number one case): The 'certain partial order' on the Picard group is introduced so that non-trivial upper sets correspond to collections of line bundles. The manuscript must explicitly verify (e.g., via a dedicated proposition) that for every non-trivial upper set the direct sum E satisfies gl.dim(End(E)) ≤ d and that End(E) is d-representation infinite of type Ã; without this step the bijection and the 'all of them are d-tilting' and converse statements rest on an unverified correspondence.

    Authors: We agree with the referee that an explicit verification is necessary to make the arguments fully rigorous and self-contained. In the revised manuscript, we will add a dedicated proposition in Section 3. This proposition will prove that for any non-trivial upper set in the Picard group under the defined partial order, the corresponding direct sum E of line bundles satisfies gl.dim(End(E)) ≤ d, and that End(E) is a d-representation infinite algebra of type Ã. This will directly support the bijection, the statement that all such bundles are d-tilting, and the converse. The partial order is constructed precisely to ensure these properties, but we will now state and prove them explicitly as requested. revision: yes

  2. Referee: [§5] §5 (Picard number two case): Analogously, the posets for the pairs (I,I'), the identification of I with an NCCR and I' with a cut, and the verification that the resulting endomorphism algebra lies in the class of higher representation infinite algebras of type ÃÃ (including the global-dimension bound) are load-bearing for the classification and converse claims; these verifications should be stated as separate lemmas.

    Authors: We concur that separating these verifications into distinct lemmas will enhance the structure and readability of Section 5. In the revision, we will introduce separate lemmas that: (1) define the relevant posets for the pairs (I, I'); (2) identify I as corresponding to an NCCR of the toric singularity and I' as a cut of its quiver; and (3) verify that the endomorphism algebra has global dimension at most d and belongs to the newly introduced class of higher representation infinite algebras of type ÃÃ. These lemmas will provide the necessary foundation for the classification and the converse statements. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit combinatorial classification with independent verification

full rationale

The paper defines a specific partial order on the Picard group (or the relevant posets for Picard number two) from the toric fan data and proves that non-trivial upper sets correspond bijectively to d-tilting line-bundle sums whose endomorphism algebras satisfy the global-dimension bound and belong to the newly introduced higher representation-infinite classes. This is a direct classification result: the order is not defined in terms of the tilting property, the bijection is established by explicit computation rather than by construction, and the converses are shown by constructing the geometric models from the algebraic data. No load-bearing step reduces to a self-citation chain, fitted parameter, or tautological renaming; the derivation remains self-contained against the toric geometry and quiver representations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

This is a pure-mathematics classification theorem. It rests on standard definitions from toric geometry and tilting theory rather than numerical fitting or new physical postulates. The only invented object is the new algebraic class needed to name the endomorphism algebras that arise.

axioms (2)
  • domain assumption Smooth projective toric Fano stacks of dimension d admit a Picard group that carries a natural partial order making non-trivial upper sets correspond to tilting bundles of line bundles.
    Invoked when the abstract states the bijection for Picard number one and the pair-of-upper-sets description for Picard number two.
  • standard math The global dimension of the endomorphism algebra of a d-tilting bundle is at most d by definition.
    Used throughout the statements that the constructed bundles are d-tilting.
invented entities (1)
  • higher representation infinite algebras of type ÃÃ no independent evidence
    purpose: To name and classify the endomorphism algebras arising from the d-tilting bundles on Picard-number-two stacks.
    Newly introduced class whose definition is tied directly to the pairs (I,I') constructed from the stacks.

pith-pipeline@v0.9.0 · 5908 in / 1745 out tokens · 70190 ms · 2026-05-21T19:16:15.403976+00:00 · methodology

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · 3 internal anchors

  1. [1]

    Claire Amiot,Cluster categories for algebras of global dimension 2 and quivers with potential, Annales de l’institut Fourier59(2009), 2525–2590

    work page 2009

  2. [2]

    Aleksandr Aleksandrovich Beilinson,Coherent sheaves onP n and problems of linear algebra, Funktsional. Anal. i Prilozhen12(1978), no. 3, 68–69

    work page 1978

  3. [3]

    1, 193–215

    Lev Borisov, Linda Chen, and Gregory Smith,The orbifold Chow ring of toric Deligne-Mumford stacks, Journal of the American Mathematical Society18(2005), no. 1, 193–215

    work page 2005

  4. [4]

    1, 277–301

    Lev Borisov and Zheng Hua,On the conjecture of King for smooth toric Deligne–Mumford stacks, Advances in Math- ematics221(2009), no. 1, 277–301

    work page 2009

  5. [5]

    2, 572–618

    Aslak Bakke Buan, Bethany Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov,Tilting theory and cluster combinatorics, Advances in mathematics204(2006), no. 2, 572–618

    work page 2006

  6. [6]

    Ragnar-Olaf Buchweitz and Hille Lutz,Endomorphism rings of geometric tilting objects, in preparation

    work page

  7. [7]

    Ragnar-Olaf Buchweitz, Hille Lutz, and Osamu Iyama,Cluster tilting for projective varieties and Horrocks type theorem, in preparation

    work page

  8. [8]

    Darius Dramburg and Oleksandra Gasanova,A classification ofn-representation infinite algebras of type ˜A, arXiv:2409.06553, 2024

    work page arXiv 2024

  9. [9]

    2, 350–372

    Alexander I Efimov,Maximal lengths of exceptional collections of line bundles, Journal of the London Mathematical Society90(2014), no. 2, 350–372

    work page 2014

  10. [10]

    Victor Ginzburg,Calabi–Yau algebras, arXiv:0612139, 2006

    work page 2006

  11. [11]

    Norihiro Hanihara,Calabi–Yau completions for roots of dualizing dg bimodules, arXiv:2412.18753, 2024

    work page arXiv 2024

  12. [12]

    Norihiro Hanihara and Osamu Iyama,Silting correspondences and Calabi–Yau dg algebras, arXiv:2508.12836, 2025

    work page arXiv 2025

  13. [13]

    285, American Mathematical Society, 2023

    Martin Herschend, Osamu Iyama, Hiroyuki Minamoto, and Steffen Oppermann,Representation theory of Geigle- Lenzing complete intersections, vol. 285, American Mathematical Society, 2023

    work page 2023

  14. [14]

    2, 292–342

    Martin Herschend, Osamu Iyama, and Steffen Oppermann,n–Representation infinite algebras, Advances in mathematics 252(2014), no. 2, 292–342

    work page 2014

  15. [15]

    6, 1507–1521

    Lutz Hille and Markus Perling,A counterexample to King’s conjecture, Compositio Mathematica142(2006), no. 6, 1507–1521

    work page 2006

  16. [16]

    ,Tilting bundles on rational surfaces and quasi-hereditary algebras, Annales de l’Institut Fourier64(2014), 625–644

    work page 2014

  17. [17]

    Akira Ishii and Kazushi Ueda,Dimer models and exceptional collections, arXiv:0911.4529, 2009

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  18. [18]

    1, 51–82

    Osamu Iyama,Auslander correspondence, Advances in mathematics210(2007), no. 1, 51–82

    work page 2007

  19. [19]

    1, 22–50

    ,Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Advances in mathematics 210(2007), no. 1, 22–50

    work page 2007

  20. [20]

    ,Cluster tilting for higher Auslander algebras, Advances in mathematics226(2011), no. 1, 1–61

    work page 2011

  21. [21]

    12, 6575–6614

    Osamu Iyama and Steffen Oppermann,n–representation–finite algebras andn–APR tilting, Transactions of the Amer- ican Mathematical Society363(2011), no. 12, 6575–6614

    work page 2011

  22. [22]

    5, 1641–1659

    Osamu Iyama and Dong Yang,Quotients of triangulated categories and equivalences of Buchweitz, Orlov, and Amiot– Guo–Keller, American Journal of Mathematics142(2020), no. 5, 1641–1659

    work page 2020

  23. [23]

    Kapranov,On the derived categories of coherent sheaves on some homogeneous spaces, Inventiones mathe- maticae92(1988), no

    Mikhail M. Kapranov,On the derived categories of coherent sheaves on some homogeneous spaces, Inventiones mathe- maticae92(1988), no. 3, 479–508

    work page 1988

  24. [24]

    3, 517–536

    Yujiro Kawamata,Derived categories of toric varieties, Michigan Mathematical Journal54(2006), no. 3, 517–536

    work page 2006

  25. [25]

    654, 125–180

    Bernhard Keller and Michel Van den Bergh,Deformed Calabi–Yau completions, Journal f¨ ur die reine und angewandte Mathematik (2011), no. 654, 125–180

    work page 2011

  26. [26]

    Alastair King,Tilting bundles on some rational surfaces, preprint athttps://people.bath.ac.uk/masadk/papers/, 1997

    work page 1997

  27. [27]

    5, 4061–4095

    Hiroyuki Minamoto and Izuru Mori,The structure of AS-Gorenstein algebras, Advances in Mathematics226(2011), no. 5, 4061–4095

    work page 2011

  28. [28]

    Ryu Tomonaga,Non-commutative crepant resolutions of toric singularities with divisor class group of rank one, arXiv:2510.26252

    work page internal anchor Pith review Pith/arXiv arXiv

  29. [29]

    26 RYU TOMONAGA

    ,On silting mutations preserving global dimension, arXiv:2510.26206. 26 RYU TOMONAGA

    work page arXiv

  30. [30]

    ,Weak del Pezzo surfaces yield2-hereditary algebras and3-Calabi-Yau algebras, arXiv:2510.26199

    work page internal anchor Pith review Pith/arXiv arXiv

  31. [31]

    Berlin, Heidelberg: Springer Berlin Heidelberg (2004), 749–770

    Michel Van den Bergh,Non-commutative crepant resolutions, The Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, 2002. Berlin, Heidelberg: Springer Berlin Heidelberg (2004), 749–770. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan Email address:ryu-tomonaga@g.ecc.u-tokyo.ac.jp

    work page 2002