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Higher hereditary algebras and Calabi-Yau algebras arising from some toric singularities

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Pith's one-line read The paper constructs tilting objects in the graded singularity categories of two families of toric singularities, with endomorphism rings that are higher representation infinite and carry strict root pairs of the shifted Serre functor.

desk verdict Good Veronese half, broken Segre half: the reduction theorem's key hypothesis is impossible, so the Segre results fail as written, but the paper deserves a careful referee. read the letter →

arxiv 2412.19040 v2 pith:XSJIH242 submitted 2024-12-26 math.RT math.ACmath.RA

Norihiro Hanihara This is my paper · ORCID
classification math.RTmath.ACmath.RA MSC 13C1416E3516S3813D0216G1014A2216G60
keywords VeronesesubringSegreproductn-representationinfinitealgebrasingularitycategoryclusterstrictrootpairtwistedCalabi-Yautoric
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

This paper proves that for two families of commutative Gorenstein toric singularities—Veronese subrings of polynomial rings and Segre products of polynomial rings—the graded singularity category admits an explicit tilting object whose endomorphism ring is higher representation infinite. The tilting objects are constructed so that they carry strict root pairs of the shifted Serre functor, which yields equivalences between the singularity categories and (folded) cluster categories that preserve cluster tilting objects. These equivalences also yield explicit twisted Calabi-Yau algebras as Calabi-Yau completions, with quiver and relation presentations given in the paper.

What carries the argument

The main machinery is a reduction theorem: under Assumption 2.2, an idempotent quotient $A/(e)$ of a $d$-representation infinite algebra $A$ is $(d-1)$-representation infinite. This reduction is used together with a semi-orthogonal decomposition theorem for derived categories of coherent sheaves and singularity categories, and with the theory of strict root pairs of the shifted Serre functor, to compute the endomorphism rings of the constructed tilting objects and to organize them into orbit decompositions.

What would settle it

Compute the self-injective dimension of the standard $d$-representation infinite algebra of the paper's Example 2.9 (presented by a linear quiver with $d+1$ arrows and commutativity relations): if it equals $d$ rather than being strictly less than $d$, then Assumption 2.2(2) cannot hold for a $d$-representation infinite algebra, and the proof of Theorem 5.1 via the reduction theorem would need correction.

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Extended reading notes

Core claim

For the Veronese subring $R = k[x_1,\ldots,x_d]^{(n)}$ with $d = an$, the object $T = S \oplus \Omega S(1) \oplus \cdots \oplus \Omega^{a-1}S(a-1)$ in the graded singularity category $\mathrm{sg}^{\mathbb{Z}} R$ is tilting, its endomorphism ring $A$ is $(d-a-1)$-representation infinite, and $A$ has a strict $a$-th root pair $(U,P)$ of $\nu_{d-a-1}^{-1}$. For the Segre product $R = k[x_1,y_1]\#\cdots\#k[x_n,y_n]$, the object $T = \bigoplus_{0<v<1} M_v$ is tilting in $\mathrm{sg}^{\mathbb{Z}} R$, its endomorphism ring is $(n-2)$-representation infinite, and for $n=3$ it is hereditary with a strict square root pair of $\nu_1$. These equivalences respect (folded) cluster tilting subcategories and yield twisted Calabi-Yau algebras whose quivers and relations are described explicitly.

Load-bearing premise

The Segre product proof depends on the assumption that a certain finite-dimensional algebra can have self-injective dimension strictly smaller than the global dimension bound $d$, a condition required by the paper's reduction theorem for idempotent quotients.

Editorial extensions

If this is right

  • The singularity categories of these toric singularities are equivalent as triangulated categories to derived categories of explicit finite-dimensional algebras that are higher representation infinite.
  • The equivalences are compatible with canonical cluster tilting subcategories, so cluster tilting objects correspond under the equivalences.
  • For these rings, there are explicit quiver and relation presentations of the endomorphism rings and the associated twisted Calabi-Yau algebras.
  • The study of folded cluster categories is enriched by examples arising from actual singularities, going beyond formal constructions.
  • The results provide new examples of Gorenstein rings of hereditary representation type.

Reading between the lines

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

  • The orbit construction $T = \bigoplus_{i=0}^{a-1} F^i T_0$ suggests a general recipe for producing tilting objects compatible with cluster structure in other graded Gorenstein singularities with an algebraic root of the shifted Serre functor.
  • The explicit quiver presentations of the endomorphism rings and Calabi-Yau completions give concrete models that could be used to compute cohomology or invariants of these singularity categories, and to study their noncommutative geometry directly.
  • The reduction theorem for idempotent quotients may hold under weaker hypotheses than Assumption 2.2(2), which would make the construction available for a broader class of higher representation infinite algebras.
  • The equivalences with folded cluster categories suggest a combinatorial interpretation of the cluster tilting subcategories via the quivers, possibly yielding new cluster structures on these singularity categories.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies graded and ungraded singularity categories of two classes of commutative Gorenstein toric singularities: Veronese subrings of polynomial rings and Segre products of copies of k[x,y]. For Veronese subrings the author constructs an explicit tilting object T whose endomorphism ring is claimed to be higher representation infinite and equipped with a strict root pair, yielding equivalences with folded cluster categories and explicit quiver presentations of the associated Calabi-Yau completions. For Segre products the author constructs a tilting object ⊕_{0<v<1} M_v, claims its endomorphism ring is (n−2)-representation infinite, and, for n=3, gives a hereditary endomorphism ring with a strict root pair and an alternating quiver presentation. The core technical tool is a reduction theorem (Theorem 2.3) asserting that a certain idempotent quotient of a d-representation infinite algebra is (d−1)-representation infinite. The Veronese arguments appear detailed and internally coherent, but the Segre-product proof depends on an assumption that is impossible for d-representation infinite algebras, and that part of the paper is not established as written.

Significance. If the Veronese results are correct, they provide explicit, noncommutative-resolution-style descriptions of graded singularity categories, with tilting objects whose endomorphism rings are higher representation infinite and whose root pairs are strict; the quiver-and-relations descriptions of the twisted Calabi-Yau algebras are concrete and valuable. The proposed reduction theorem for idempotent quotients would also be of independent interest if a correct hypothesis could be supplied. The Segre-product claims, however, rest on Theorem 2.3, whose Assumption 2.2(2) is inconsistent with the definition of higher representation infiniteness. Since the abstract and Section 5 present the Segre results as a main contribution, the paper in its current form cannot be accepted, although the Veronese half may survive a substantial repair.

major comments (2)
  1. [Section 2, Assumption 2.2(2) and Theorem 2.3] Assumption 2.2(2), "inj.dim_A A < d", cannot hold for any d-representation infinite algebra A with d≥1. By Definition 2.1, ν_d^{-1}A = RHom_A(DA,A)[d] lies in mod A; it is nonzero because Hom_A(DA,A)≠0 (A is an injective cogenerator), so Ext^d_A(DA,A)≠0. Hence inj.dim_A A ≥ d, while gl.dim A ≤ d gives inj.dim_A A ≤ d, forcing equality. Thus the hypothesis of Theorem 2.3 is empty, and the proof's key step — that ν_d^{-1}(ν^{-i}_{d-1}A) has cohomology only in degree ≤−1 — uses exactly the false inequality. Consequently Example 2.9's assertion that the d-Beilinson algebra satisfies Assumption 2.2(2) is false. The theorem must be replaced or repaired with a viable hypothesis; the surrounding argument suggests the author may have intended an inequality involving the quotient A/(e), but that corrected statement and proof are not what appears.
  2. [Section 5.2, proof of Theorem 5.1 and Theorem 5.2] The proof that End^Z_{sg R}(T) is (n−2)-representation infinite applies Theorem 2.3 to B = (Kronecker)^{⊗n} and then Theorem 2.12 to B/(e0). But B is n-representation infinite, so by the argument in the previous comment inj.dim_B B = n, not < n; Theorem 2.3 therefore cannot be applied to (B,e0). Similarly, B/(e0), if it is (n−1)-representation infinite, satisfies proj.dim_{B/(e0)}D(B/(e0)) = n−1, not < n−1, so the dual Theorem 2.12 cannot be applied either. The claim that these applications are valid is the load-bearing step for Theorem 5.1(2), Theorem 5.2(3)–(4), and the corresponding statements in the abstract. The tilting-object part of Theorem 5.1 and the computations in Theorem 5.2(1)–(2) do not depend on Section 2, but the higher-representation-infinite and strict-root-pair conclusions are unsupported as written.
minor comments (3)
  1. [Throughout] There are numerous typographical slips, e.g. "monimial" in Theorem 1.1, "in genral" and "exsistence" in the introduction, "oridnary" in Section 3, "In particuar" in Section 5.3, and "potention" in reference [1]. These should be corrected.
  2. [Abstract vs. Theorem 4.1(2)] The abstract and Theorem 4.1(2) refer to a strict root pair of ν^{-1}_{d-a-1}, while the theorem statement says "for ν_{d-a-1}"; the notation should be made consistent, since the root pair is for the inverse shifted Serre functor.
  3. [Section 5.2] The sentence "It is easy to see that the injective dimension of the simple B-module corresponding to the summand M_v is n−∑ v_i" should be expanded into a proof and, more importantly, connected explicitly to whatever corrected version of Assumption 2.2(2) is used after Theorem 2.3 is repaired.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the tilting objects and endomorphism rings are constructed independently; cited prior work supplies external framework, not the target claims.

full rationale

The paper's claimed derivations do not reduce to their inputs. The Veronese tilting object T = S ⊕ ΩS(1) ⊕ ... ⊕ Ω^{a-1}S(a-1) is not fitted to its endomorphism ring; the higher-representation-infiniteness of A is proved directly from Ext-vanishing computed in Lemma 4.6 and the form of the Serre functor, not assumed. The Segre result derives End(T) as a quotient of the n-fold Kronecker tensor product algebra, using Theorem 2.3; although that theorem is invoked, the reduction is an algebraic argument and the source algebra is an independent standard object. The self-citations to [19] and [20] supply the folded cluster category and root-pair formalism and the equivalence diagram once a tilting object is found; they are prior parameter-free results whose stated assumptions do not contain the theorem being proved, so they are independent support rather than circularity. The substantive defect identified by the skeptic — Assumption 2.2(2) requiring inj.dim_A A < d for a d-representation infinite algebra, whereas standard theory forces inj.dim_A A = d — is a correctness/validity problem in Theorem 2.3 and the Segre proof, not a case of the paper's conclusions being equivalent to its hypotheses. Under the circularity rubric, a false or untenable assumption is not circularity; accordingly no circular step is recorded and the score is 0.

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

The central claims rest on the standard framework of singularity categories, cluster categories, and higher representation infinite algebras, plus the author's own preprints for root pairs. The only ad hoc assumption is the impossible-looking Assumption 2.2(2). No free parameters are fitted to data, and no new entities are introduced.

assumptions (3)
  • ad hoc to paper Assumption 2.2(2): inj.dim_A A < d for a d-representation infinite algebra A.
    This assumption is used in the proof of Theorem 2.3, which is applied in the proof of Theorem 5.1. It appears impossible for any d-representation infinite algebra over a field because standard theory gives gl.dim A = d and inj.dim_A A = d, making the assumption contradictory as stated.
  • domain assumption The theory of folded cluster categories and strict root pairs from [19] (Hanihara, arXiv:2412.18753) and the equivalence diagram from [20] (Hanihara-Iyama, arXiv:2209.14090).
    The paper relies on these preprints by the same author (and Iyama) for the existence of cluster tilting objects in folded cluster categories and for the diagram (1.1). They are cited as established results but are not independently verified in this paper.
  • standard math Orlov's theorem (Theorem 5.3) describing the singularity category as a subcategory of a derived category of graded modules.
    Used in Section 5.1 to identify singularity categories and to obtain the semi-orthogonal decomposition used in the proof of Theorem 5.1.

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Pith. "Pith review of Higher hereditary algebras and Calabi-Yau algebras arising from some toric singularities." pith.science (2026). https://pith.science/paper/XSJIH242

@misc{pith2026241219040,
  author       = {Pith},
  title        = {Pith review of: Higher hereditary algebras and Calabi-Yau algebras arising from some toric singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSJIH242}},
  note         = {Machine review of arXiv:2412.19040}
}
read the original abstract

We study graded and ungraded singularity categories of some commutative Gorenstein toric singularities, namely, Veronese subrings of polynomial rings, and Segre products of some copies of polynomial rings. We show that the graded singularity category has a tilting object whose endomorphism ring is higher representation infinite. Moreover, we construct the tilting object so that the endomorphism ring has a strict root pair of its higher Auslander-Reiten translation, which allows us to give equivalences between singularity categories and (folded) cluster categories in a such a way that their cluster tilting objects correspond to each other. Our distinguished form of tilting objects also allows us to construct (twisted) Calabi-Yau algebras as the Calabi-Yau completions of the root pairs. We give an explicit description of these twisted Calabi-Yau algebras as well as the higher representation infinite algebras in terms of quivers and relations. Along the way, we prove that certain idempotent quotients of higher representation infinite algebras remain higher representation infinite.

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Forward citations

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Reference graph

Works this paper leans on

43 extracted references · 35 canonical work pages · cited by 2 Pith papers

  1. [28]

    Iyama and R

    O. Iyama and R. Takahashi, Tilting and cluster tilting f or quotient singularities, Math. Ann. 356 (2013), 1065-110 5

    work page 2013

  2. [23]

    Iyama, Higher-dimensional Auslander-Reiten theor y on maximal orthogonal subcategories, Adv

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

    2007

  3. [1]

    Amiot, Cluster categories for algebras of global dime nsion 2 and quivers with potentional, Ann

    C. Amiot, Cluster categories for algebras of global dime nsion 2 and quivers with potentional, Ann. Inst. Fourier, Gr enoble 59, no.6 (2009) 2525-2590

    work page 2009

  4. [2]

    Amiot, O

    C. Amiot, O. Iyama, and I. Reiten, Stable categories of Co hen-Macaulay modules and cluster categories, Amer. J. Math , 137 (2015) no.3, 813-857

    work page 2015

  5. [3]

    Artin and W

    M. Artin and W. Schelter, Graded algebras of global dimen sion 3, Adv. Math. 66 (1987), no. 2, 171-216

    1987

  6. [4]

    Artin and J

    M. Artin and J. J. Zhang, Noncommutative projective sche mes, Adv. Math. 109 (1994) 228-287

    work page 1994

  7. [5]

    Auslander, On the purity of the branch locus, Amer

    M. Auslander, On the purity of the branch locus, Amer. J. M ath. 84 (1962), 116–125

    work page 1962

  8. [6]

    M. Auslander, Functors and morphisms determined by obje cts, in: Representation Theory of Algebras, Lecture Notes i n Pure and Applied Mathematics 37, Marcel Dekker, New York, 19 78, 1-244

    work page

Show all 43 references
  1. [7]

    Bocklandt, T

    R. Bocklandt, T. Schedler, and M. W emyss, Superpotentia ls and higher order derivations, J. Pure Appl. Algebra 214 (2 010), no. 9, 1501-1522

  2. [8]

    Bondal and A

    A. Bondal and A. Polishchuk, Homological properties of a ssociative algebras: the method of helices, (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 2, 3–50; translation in Ru ssian Acad. Sci. Izv. Math. 42 (1994), no. 2, 219–260

    1993

  3. [9]

    A. B. Buan, O. Iyama, I. Reiten, and D. Smith, Mutation of c luster-tilting objects and potentials, Amer. J. Math. 133 ( 2011), no. 4, 835–887

    2011

  4. [10]

    A. B. Buan, R. Marsh, M. Reineke, I. Reiten, and G. Todoro v, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006) 572-618

    2006

  5. [11]

    R. O. Buchweitz, Maximal Cohen-Macaulay modules and Ta te cohomology, Mathematical Surveys and Monographs, 262. American Mathematical Society, Providence, RI, [2021], ©2021. xii+175 pp

    2021

  6. [12]

    Fomin and A

    S. Fomin and A. Zelevinsky, Cluster algebras. I. Founda tions, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529

    2002

  7. [13]

    Ginzburg, Calabi-Yau algebras, arXiv:0612139

    V. Ginzburg, Calabi-Yau algebras, arXiv:0612139

  8. [14]

    Goto and K

    S. Goto and K. W atanabe, On graded rings I, J. Math. Soc. J apan 30 (1978), no. 2, 179–213

    1978

  9. [15]

    Guo, Cluster tilting objects in generalized higher c luster categories, J

    L. Guo, Cluster tilting objects in generalized higher c luster categories, J. Pure Appl. Algebra 215 (2011), no. 9, 2 055–2071

    2011

  10. [16]

    Hanihara, Cluster categories of formal DG algebras a nd singularity categories, Forum of Mathematics, Sigma (20 22), Vol

    N. Hanihara, Cluster categories of formal DG algebras a nd singularity categories, Forum of Mathematics, Sigma (20 22), Vol. 10:e35 1–50

  11. [17]

    Hanihara, Morita theorem for hereditary Calabi-Yau categories, Adv

    N. Hanihara, Morita theorem for hereditary Calabi-Yau categories, Adv. Math. 395 (2022) 108092

    2022

  12. [18]

    Hanihara, Non-commutative resolutions for Segre pr oducts and Cohen-Macaulay rings of hereditary representat ion type, Trans

    N. Hanihara, Non-commutative resolutions for Segre pr oducts and Cohen-Macaulay rings of hereditary representat ion type, Trans. Amer. Math. Soc. 378 (2025), pp. 2429-2475

    2025

  13. [19]

    Hanihara, Calabi-Yau completions for roots of duali zing dg bimodules, arXiv:2412.18753

    N. Hanihara, Calabi-Yau completions for roots of duali zing dg bimodules, arXiv:2412.18753

    arXiv

  14. [20]

    Hanihara and O

    N. Hanihara and O. Iyama, Enhanced Auslander-Reiten du ality and Morita theorem for singularity categories, arXiv:2209.14090

    arXiv

  15. [21]

    Herschend, O

    M. Herschend, O. Iyama, and S. Oppermann, n-representation infinite algebras, Adv. Math. 252 (2014) 29 2-342

    2014

  16. [22]

    Higashitani and Y

    A. Higashitani and Y. Nakajima, Conic divisorial ideal s of Hibi rings and their applications to non-commutative cr epant resolutions, Selecta Math. (N.S.) 25 (2019), no. 5, Paper No . 78, 25 pp

    2019

  17. [24]

    Iyama, Auslander correspondence, Adv

    O. Iyama, Auslander correspondence, Adv. Math. 210 (20 07) 51-82

  18. [25]

    Iyama, Cluster tilting for higher Auslander algebra s, Adv

    O. Iyama, Cluster tilting for higher Auslander algebra s, Adv. Math. 226 (2011) 1-61

    2011

  19. [26]

    Iyama, Tilting Cohen-Macaulay representations, Pr oceedings of the International Congress of Mathematicians –Rio de Janeiro 2018

    O. Iyama, Tilting Cohen-Macaulay representations, Pr oceedings of the International Congress of Mathematicians –Rio de Janeiro 2018. Vol. II. Invited lectures, 125-162, W orld Sci . Publ., Hackensack, NJ, 2018. 22 NORIHIRO HANIHARA

    2018

  20. [27]

    Iyama and I

    O. Iyama and I. Reiten, Fomin-Zelevinsky mutation and t ilting modules over Calabi-Yau algebras, Amer. J. Math. 130 (2008), no. 4, 1087-1149

    2008

  21. [29]

    Iyama and D

    O. Iyama and D. Yang, Quotients of triangulated categor ies and equivalences of Buchweitz, Orlov and Amiot–Guo–Kel ler, Amer. J. Math. 142 (2020), no. 5, 1641–1659

    2020

  22. [30]

    Kalck, Derived categories of singular varieties and finite dimensional algebras, in: Representation Theory of Q uivers and Finite-Dimensional Algebras, Oberwolfach Rep

    M. Kalck, Derived categories of singular varieties and finite dimensional algebras, in: Representation Theory of Q uivers and Finite-Dimensional Algebras, Oberwolfach Rep. 20 (2023), no. 1, pp. 432–434

    2023

  23. [31]

    Keller, Deriving DG categories, Ann

    B. Keller, Deriving DG categories, Ann. scient. ´Ec. Norm. Sup. (4) 27 (1) (1994) 63-102

    1994

  24. [32]

    Keller, On triangulated orbit categories, Doc

    B. Keller, On triangulated orbit categories, Doc. Math . 10 (2005), 551-581

    2005

  25. [33]

    Keller, Deformed Calabi-Yau completions, with an ap pendix by M

    B. Keller, Deformed Calabi-Yau completions, with an ap pendix by M. Van den Bergh, J. Reine Angew. Math. 654 (2011) 125-180

    2011

  26. [34]

    G. J. Leuschke and R. Wiegand, Cohen-Macaulay represen tations, vol. 181 of Mathematical Surveys and Monographs, American Mathematical Society, Province, RI, (2012)

    2012

  27. [35]

    Minamoto and I

    H. Minamoto and I. Mori, The structure of AS-Gorenstein algebras, Adv. Math. 226 (2011) 4061-4095

    2011

  28. [36]

    Orlov, Triangulated categories of singularities an d D-branes in Landau–Ginzburg modules, Tr

    D. Orlov, Triangulated categories of singularities an d D-branes in Landau–Ginzburg modules, Tr. Mat. Inst. Stekl ova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh, 240-262

    2004

  29. [37]

    Orlov, Derived categories of coherent sheaves and tr iangulated categories of singularities, Algebra, arithme tic, and geom- etry: in honor of Yu

    D. Orlov, Derived categories of coherent sheaves and tr iangulated categories of singularities, Algebra, arithme tic, and geom- etry: in honor of Yu. I. Manin. Vol. II, 503-531, Progr. Math. , 270, Birkhauser Boston, Inc., Boston, MA, 2009

    2009

  30. [38]

    M. L. Reyes and D. Rogalski, Graded twisted Calabi-Yau a lgebras are generalized Artin-Schelter regular, Nagoya Ma th. J., (2021), 1–54

    2021

  31. [39]

    Rickard, Derived categories and stable equivalence , J

    J. Rickard, Derived categories and stable equivalence , J. Pure Appl. Algebra 61 (1989) 303-317

    1989

  32. [40]

    R. P. Stanley, Combinatorics and commutative algebra, Second edition. Progress in Mathematics 41, Birkh¨ auser Bo ston, Inc., Boston, MA, 1996. x+164 pp

    1996

  33. [41]

    Van den Bergh, Cohen-Macaulayness of modules of cova riants, Invent

    M. Van den Bergh, Cohen-Macaulayness of modules of cova riants, Invent. Math. 106 (1991), no. 2, 389–409

    1991

  34. [42]

    Van den Bergh, Non-commutative crepant resolutions , The legacy of Niels Henrik Abel, 749–770, Springer, Berlin , 2004

    M. Van den Bergh, Non-commutative crepant resolutions , The legacy of Niels Henrik Abel, 749–770, Springer, Berlin , 2004

    2004

  35. [43]

    Yoshino, Cohen-Macaulay modules over Cohen-Macaul ay rings, London Mathematical Society Lecture Note Series 1 46, Cambridge University Press, Cambridge, 1990

    Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaul ay rings, London Mathematical Society Lecture Note Series 1 46, Cambridge University Press, Cambridge, 1990. F aculty of Mathematics, Kyushu University, 744 Motooka, Nish i-ku, Fukuoka, 819-0395, Japan Email address : haniha...

    1990

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