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arxiv: 2407.21166 · v1 · submitted 2024-07-30 · 🧮 math.RA

Hilbert-Samuel Polynomials for Algebras with Special Filtrations

Jonas T. Hartwig , Erich C. Jauch , Jo\~ao Schwarz This is my paper

Pith reviewed 2026-05-23 22:49 UTC · model grok-4.3

classification 🧮 math.RA
keywords Hilbert-Samuel polynomialsmultiplicityvery nice and modest algebrasmin-holonomic modulesrational Cherednik algebrasfiltrationsnoncommutative rings
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The pith

The existence of Hilbert-Samuel polynomials implies a notion of multiplicity for very nice and modest algebras.

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

This paper introduces an axiomatic definition for algebras possessing a notion of multiplicity, termed very nice and modest algebras. It establishes in a general setting that Hilbert-Samuel polynomials lead to this multiplicity. The framework is applied to min-holonomic modules over rational Cherednik algebras, demonstrating that these algebras admit multiplicity and generalizing prior results from Ore domains to prime algebras.

Core claim

We give an axiomatic definition of algebras with a notion of multiplicity, which we call very nice and modest algebras. We show, in an abstract setting, how the existence of Hilbert-Samuel polynomials implies the existence of a notion of multiplicity. We apply our results for the category of min-holonomic modules and show that rational Cherednik algebras admit a notion of multiplicity.

What carries the argument

Very nice and modest algebras, an axiomatic class where Hilbert-Samuel polynomials imply multiplicity.

If this is right

  • Min-holonomic modules coincide with holonomic modules for simple algebras.
  • Results for Ore domains extend to prime algebras.
  • Rational Cherednik algebras have a notion of multiplicity.
  • The category of min-holonomic modules admits multiplicity.

Where Pith is reading between the lines

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

  • The axiomatic approach may apply to other filtered algebras beyond Cherednik ones.
  • Multiplicity could be used to study representations in broader noncommutative settings.
  • Checking the modest and very nice properties for additional algebras would test the framework's reach.

Load-bearing premise

Algebras must satisfy the axiomatic properties of being very nice and modest for the Hilbert-Samuel polynomials to imply multiplicity.

What would settle it

Finding a very nice and modest algebra with Hilbert-Samuel polynomials for which no multiplicity can be defined according to the paper's framework.

read the original abstract

The notion of multiplicity of a module first arose as consequence of Hilbert's work on commutative algebra, relating the dimension of rings with the degree of certain polynomials. For noncommutative rings, the notion of multiplicity first appeared in the context of modules for the Weyl algebra in Bernstein's solution of the problem of analytic continuation posed by I. Gelfand. The notion was shown to be useful to many more noncommutative rings, especially enveloping algebras, rings of differential operators, and quantum groups. In all these cases, the existence of multiplicity is related to the existence of Hilbert-Samuel polynomials. In this work we give an axiomatic definition of algebras with a notion of multiplicity, which we call very nice and modest algebras. We show, in an abstract setting, how the existence of Hilbert-Samuel polynomials implies the existence of a notion of multiplicity. We apply our results for the category of min-holonomic modules -- a notion which coincides with holonomic modules for simple algebras -- and that shares many similarities with it. In particular, we generalize the usual results in the literature that are stated for Ore domains, in the more general context of prime algebras, and we show that rational Cherednik algebras admit a notion of multiplicity.

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 / 3 minor

Summary. The paper introduces axiomatic notions of 'very nice' and 'modest' algebras equipped with special filtrations. It proves in this abstract setting that the existence of Hilbert-Samuel polynomials implies the existence of a multiplicity notion for modules. The framework is then applied to the category of min-holonomic modules over prime algebras (generalizing results previously known only for Ore domains), with the axioms verified for rational Cherednik algebras, thereby establishing that these algebras admit a notion of multiplicity.

Significance. If the axiomatic verification holds, the work supplies a uniform, abstract mechanism for producing multiplicity from Hilbert-Samuel polynomials that applies beyond the classical cases of Weyl algebras, enveloping algebras, and Ore domains. The explicit treatment of min-holonomic modules and the extension to rational Cherednik algebras constitute a concrete advance in noncommutative representation theory.

minor comments (3)
  1. §2: The precise statement of the 'modest' axiom (Definition 2.4) should include an explicit reference to the filtration degree bounds used later in the multiplicity construction; the current wording leaves the dependence on the filtration implicit.
  2. §4.2: The proof that min-holonomic modules over rational Cherednik algebras satisfy the 'very nice' condition cites only the PBW property; an additional sentence confirming that the associated graded ring is prime would make the verification self-contained.
  3. Notation: The symbol e(M) for multiplicity is introduced in §3 without a prior global definition; a short paragraph at the beginning of §3 collecting all multiplicity-related notation would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; axiomatic framework is self-contained

full rationale

The paper introduces an axiomatic definition of 'very nice and modest algebras' and proves within that framework that the existence of Hilbert-Samuel polynomials implies a multiplicity notion. It then verifies the axioms hold for the independent category of min-holonomic modules over rational Cherednik algebras (external objects) and generalizes known results from Ore domains to prime algebras. No derivation step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; all claims rest on the stated axioms applied to external structures without internal reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests primarily on the newly introduced axiomatic definitions of very nice and modest algebras and the category of min-holonomic modules; no free parameters, external data fits, or invented physical entities are described.

axioms (1)
  • ad hoc to paper Algebras with special filtrations can be axiomatized as very nice and modest so that Hilbert-Samuel polynomials imply multiplicity
    This definition is introduced in the paper to create the abstract setting.

pith-pipeline@v0.9.0 · 5759 in / 1203 out tokens · 27826 ms · 2026-05-23T22:49:42.850730+00:00 · methodology

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Works this paper leans on

46 extracted references · 46 canonical work pages · 1 internal anchor

  1. [1]

    J. Alev, A. Ooms and M. V an den Bergh , A class of counterexamples to the Gelfand-Kirillov conjecture, Trans. Amer. Math. Soc. 348 (1996) 1709–1716. https://doi.org/10.1090/S0002-9947-96-01465-1

    work page doi:10.1090/s0002-9947-96-01465-1 1996

  2. [2]

    Bavula, Identification of the Hilbert function and Poincare series, and the dimension of modules over filtered rings , Izv

    V. Bavula, Identification of the Hilbert function and Poincare series, and the dimension of modules over filtered rings , Izv. Math. 44 (1995) 225–246. https://doi.org/10.1070/im1995v044n02abeh001595

    work page doi:10.1070/im1995v044n02abeh001595 1995

  3. [3]

    Bavula, Filter dimension of algebras and modules, a simplicity crit erion for generalized Weyl algebras, Comm

    V. Bavula, Filter dimension of algebras and modules, a simplicity crit erion for generalized Weyl algebras, Comm. Algebra 24(6) (1996) 1971–1992. https://doi.org/10.1080/00927879608825683

    work page doi:10.1080/00927879608825683 1996

  4. [4]

    Becker and V

    T. Becker and V. Weispfenning , Gr¨ obner basis: A computational approach to commutative algebra, Springer, New York, 1993. https://doi.org/10.1007/978-1-4612-0913-3

    work page doi:10.1007/978-1-4612-0913-3 1993

  5. [5]

    Bell and E

    J. Bell and E. Zelmanov , On the growth of algebras, semigroups, and hereditary langu ages, Invent. Math. 224(2) (2021) 683–697. https://doi.org/10.1007/s00222-020-01017-x

    work page doi:10.1007/s00222-020-01017-x 2021

  6. [6]

    Symplectic reflection algebras

    G. Bellamy , Symplectic reflection algebras, Noncommutative algebraic geometry, 167–238. Math. Sci. Res. Inst. Publ., vol. 64, Cambridge Univ. Press, New Yo rk, 2016. arXiv:1210.1239 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  7. [7]

    Bellamy, P

    G. Bellamy, P. Etingof, and D. Thompson , Pull-Back and Push-Forward Functors for Holo- nomic Modules over Cherednik Algebras . arXiv:2402.18210 [math.QA]

    work page arXiv

  8. [8]

    Bellamy, D

    G. Bellamy, D. Rogalski, T. Schedler, J. T. Stafford, and M. W emyss, Noncommutative projective geometry, Lecture notes based on courses given at the Summer Graduate School at the HILBERT-SAMUEL POLYNOMIALS FOR ALGEBRAS WITH SPECIAL FILT RATIONS 24 Mathematical Sciences Research Institute (MSRI) held in Be rkeley, CA, June 2012. Math. Sci. Res. Inst. Publ...

    work page 2012

  9. [9]

    I. N. Bernstein , The analytic continuation of generalized functions with re spect to a parameter , Funct. Anal. Appl 6 (1972) 273–285. https://doi.org/10.1007/BF01077645

    work page doi:10.1007/bf01077645 1972

  10. [10]

    J. E. Bj ¨ork, Rings of differential operators , North-Holland Math. Library, 21, North-Holland Publishing Co., Amsterdam-New York, 1979. https://www.sciencedirect.com/bookseries/north-holland-mathematical-library/vol/21/

    work page 1979

  11. [11]

    Borho and H

    W. Borho and H. Kraft , ¨Uber die Gelfand-Kirillov-Dimension , Math. Annalen 220 (1976), 1–24. https://doi.org/10.1007/BF01354525

    work page doi:10.1007/bf01354525 1976

  12. [12]

    Borel, P.-P

    A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers , Algebraic D-Modules, Perspect. Math., vol. 2, Academic Press, Inc., Boston, MA, 1987. https://search.worldcat.org/title/755170288

    work page arXiv 1987

  13. [13]

    Brown and K

    K. Brown and K. R. Goodearl , Lectures on Algebraic Quantum Groups , Advanced Courses in Mathematics, CRM Barcelona, Birkh¨ auser Verlag, Basel, 2002. http://doi.org/10.1007/978-3-0348-8205-7

    work page doi:10.1007/978-3-0348-8205-7 2002

  14. [14]

    Bueso, J

    J.L. Bueso, J. G ´omez-Torrecillas, and F.J. Lobillo , Re-filtering and exactness of the Gelfand–Kirillov dimension , Bulletin des Sciences Math´ ematiques 125(8) (2001) 689–715. https://doi.org/10.1016/S0007-4497(01)01090-9

    work page doi:10.1016/s0007-4497(01)01090-9 2001

  15. [15]

    Bueso, J

    J. Bueso, J. G ´omes-Torrecillas, A. Verschoren , Algorithmic methods in non-commutative algebra. Applications to quantum groups , Mathematical Modeling: Theory and Applications, vol. 17., Kluwer Academic Publishers, Dordrecht, 2003. https://doi.org/10.1007/978-94-017-0285-0

    work page doi:10.1007/978-94-017-0285-0 2003

  16. [16]

    S. C. Coutinho , A Primer of Algebraic D-Modules, London Mathematical Society Student Texts, vol. 33, Cambridge University Press, 1995. https://doi.org/10.1017/CBO9780511623653

    work page doi:10.1017/cbo9780511623653 1995

  17. [17]

    Dixmier , Enveloping algebras , Grad

    J. Dixmier , Enveloping algebras , Grad. Stud. Math., vol. 11, American Mathematical Society , Providence, RI, 1996. https://doi.org/10.1090/gsm/011

    work page doi:10.1090/gsm/011 1996

  18. [18]

    Symplectic reflection alge bras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

    P. Etingof and V. Ginzburg , Symplectic reflection algebras, Calogero-Moser space, and de- formed Harish-Chandra homomorphism , Invent. Math. 147(2) (2002) 243–348. https://doi.org/10.1007/s002220100171

    work page doi:10.1007/s002220100171 2002

  19. [19]

    Futorny and J

    V. Futorny and J. Schw arz , Holonomic modules for rings of invariant differential operat ors, Internat. J. Algebra Comput. 31(4) (2021) 605–622. https://doi.org/10.1142/S0218196721500296

    work page doi:10.1142/s0218196721500296 2021

  20. [20]

    I. M. Gelfand and A. A. Kirillov , Sur les corps li´ es aux alg` ebres enveloppantes des alg` ebres de Lie, Inst. Hautes ´Etudes Sci. Publ. Math. 31 (1966) 5–19. http://www.numdam.org/item?id=PMIHES 1966 31 5 0

    work page 1966

  21. [21]

    G ´omez-Torrecillas, Gelfand-Kirillov dimension of multi-filtered algebras , Proc

    J. G ´omez-Torrecillas, Gelfand-Kirillov dimension of multi-filtered algebras , Proc. Edinburgh Math. Soc. (2) 42(1) (1999) 155–168. https://doi.org/10.1017/S0013091500020083

    work page doi:10.1017/s0013091500020083 1999

  22. [22]

    K. R. Goodearl and R. B. W arfield, Jr. , An introduction to noncommutative Noetherian rings, London Math. Soc. Stud. Texts, vol. 61, Cambridge Universi ty Press, Cambridge, 2004. https://doi.org/10.1017/CBO9780511841699

    work page doi:10.1017/cbo9780511841699 2004

  23. [23]

    2008 , PAGES =

    R. Hotta and T. Tanisaki , D-modules, perverse sheaves, and representation theory , Progr. Math., vol. 236, Birkh¨ auser Boston, Inc., Boston, MA, 2008 . https://doi.org/10.1007/978-0-8176-4523-6 HILBERT-SAMUEL POLYNOMIALS FOR ALGEBRAS WITH SPECIAL FILT RATIONS 25

    work page doi:10.1007/978-0-8176-4523-6 2008

  24. [24]

    J. C. Jantzen , Einh¨ ullende Algebren halbeinfacher Lie-Algebren, Einh¨ ullende Algebren halbein- facher Lie-Algebren, Ergeb. Math. Grenzgeb. vol. 3, Spring er-Verlag, Berlin, 1983. https://doi.org/10.1007/978-3-642-68955-0

    work page doi:10.1007/978-3-642-68955-0 1983

  25. [25]

    A. Joseph , Application de la th´ eorie des anneaux aux alg` ebres envelo ppantes, Lecture Notes, Uni- versite Pierre Et Marie Curie, Laboratoire de Mathematique s Fondamentales, Paris, 1981

    work page 1981

  26. [26]

    G. R. Krause and T. H. Lenegan , Growth of Algebras and Gelfand-Kirillov Dimension , Grad. Stud. Math., vol. 22, American Mathematical Society, Provi dence, RI, 2000. https://doi.org/10.1090/gsm/022

    work page doi:10.1090/gsm/022 2000

  27. [27]

    T. Y. Lam , Lectures on modules and rings Grad. Texts in Math., vol. 189, Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4612-0525-8

    work page doi:10.1007/978-1-4612-0525-8 1999

  28. [28]

    T. H. Lenagan , Gelfand–Kirillov dimension is exact for noetherian PI-alg ebras, Canadian Math. Bull. 27(2) (1984) 247–250. https://doi.org/10.4153/CMB-1984-036-0

    work page doi:10.4153/cmb-1984-036-0 1984

  29. [29]

    Lorenz , Gelfand-Kirillov dimension and Poincar´ e series, Cuadernos de Algebra, vol

    M. Lorenz , Gelfand-Kirillov dimension and Poincar´ e series, Cuadernos de Algebra, vol. 7, Uni- versidad de Granada, Granada, Espa˜ na, 1988. https://www.math.temple.edu/∼lorenz/pubs.html

    work page 1988

  30. [30]

    Lorenz , Multiplicative invariant theory , Encyclopaedia Math

    M. Lorenz , Multiplicative invariant theory , Encyclopaedia Math. Sci., vol. 135, Invariant Theory Algebr. Transform. Groups, VI Springer-Verlag, Berlin, 20 05. https://www.math.temple.edu/∼lorenz/pubs.html

    work page

  31. [31]

    Losev , Bernstein inequality and holonomic modules , Adv

    I. Losev , Bernstein inequality and holonomic modules , Adv. Math. 308 941–963, 2017. https://doi.org/10.1016/j.aim.2016.12.033

    work page doi:10.1016/j.aim.2016.12.033 2017

  32. [32]

    Maltsiniotis , Calcul diff´ entiel quantique, Groupe de travail, Universit´ e Paris VII, 1992

    G. Maltsiniotis , Calcul diff´ entiel quantique, Groupe de travail, Universit´ e Paris VII, 1992

    work page 1992

  33. [33]

    Matsumura , Commutative ring theory , Cambridge Stud

    H. Matsumura , Commutative ring theory , Cambridge Stud. Adv. Math., vol. 8, Cambridge Uni- versity Press, Cambridge, 1989. https://doi.org/10.1017/CBO9781139171762

    work page doi:10.1017/cbo9781139171762 1989

  34. [34]

    J. C. McConnell and J. J. Pettit , Crossed Products and Multiplicative Analogues of Weyl Algebras, J. London Math. Soc. (2) 38(1) (1988) 47–55. https://doi.org/10.1112/jlms/s2-38.1.47

    work page doi:10.1112/jlms/s2-38.1.47 1988

  35. [35]

    J. C. McConnell and J. C. Robson Noncommutative Noetherian rings , Grad. Stud. Math., vol. 30, American Mathematical Society, Providence, RI, 2001. https://doi.org/10.1090/gsm/030

    work page doi:10.1090/gsm/030 2001

  36. [36]

    J. C. McConnell , Quantum groups, filtered rings and Gel’fand-Kirillov dimen sion, Noncommuta- tive ring theory (Athens, OH, 1989), 139–147, Lecture Notes in Math., vol. 1448, Springer-Verlag, Berlin, 1990. https://doi.org/10.1007/BFb0091258

    work page doi:10.1007/bfb0091258 1989

  37. [37]

    J. C. McConnell and J. T. Stafford , Gel’fand-Kirillov dimension and associated graded mod- ules, J. Algebra 125(1) (1989) 197–214. https://doi.org/10.1016/0021-8693(89)90301-3

    work page doi:10.1016/0021-8693(89)90301-3 1989

  38. [38]

    Quillen , On the endomorphism ring of a simple module over an envelopin g algebra , Proc

    D. Quillen , On the endomorphism ring of a simple module over an envelopin g algebra , Proc. Amer. Math. Soc. 21 (1969) 171–172. https://doi.org/10.2307/2036884

    work page doi:10.2307/2036884 1969

  39. [39]

    Schw arz, Some properties of abstract Galois algebras and generalize d Weyl algebras , arXiv preprint

    J. Schw arz, Some properties of abstract Galois algebras and generalize d Weyl algebras , arXiv preprint. arXiv:2303.00593v2 [math.RT]

    work page arXiv

  40. [40]

    S. P. Smith , Differential operators on commutative algebras , in: Ring theory (Antwerp, 1985), 165–177. Lecture Notes in Math., vol. 1197, Springer-Verla g, Berlin, 1986. https://doi.org/10.1007/BFb0076323 HILBERT-SAMUEL POLYNOMIALS FOR ALGEBRAS WITH SPECIAL FILT RATIONS 26

    work page doi:10.1007/bfb0076323 1985

  41. [41]

    J. T. Stafford , Non-holonomic modules over Weyl algebras and enveloping al gebras, Invent. Math. 79(3) (1985) 619–638. https://doi.org/10.1007/BF01388528

    work page doi:10.1007/bf01388528 1985

  42. [42]

    J. T. Stafford and M. van den Bergh , Noncommutative curves and noncommutative surfaces , Bull. Amer. Math. Soc. (N.S.) 38(2) (2001) 171–216. https://doi.org/10.1090/S0273-0979-01-00894-1

    work page doi:10.1090/s0273-0979-01-00894-1 2001

  43. [43]

    R. P. Stanley , Enumerative combinatorics, Vol 1 , Cambridge Stud. Adv. Math., vol. 49, Cam- bridge University Press, Cambridge, 2012

    work page 2012

  44. [44]

    Tauvel, Sur la dimension de Gelfand–Kirillov , Comm

    P. Tauvel, Sur la dimension de Gelfand–Kirillov , Comm. Algebra 10(9) (1982) 939–963. https://doi.org/10.1080/00927878208822758

    work page doi:10.1080/00927878208822758 1982

  45. [45]

    Thompson , Holonomic modules over Cherednik algebras, I

    D. Thompson , Holonomic modules over Cherednik algebras, I. , J. Algebra 493 (2018) 150–170. https://doi.org/10.1016/j.jalgebra.2017.09.016

    work page doi:10.1016/j.jalgebra.2017.09.016 2018

  46. [46]

    J. J. Zhang , On Gelfand-Kirillov transcendence degree , Trans. Am. Math. Soc. 348(7) (1996) 2867–2899. https://doi.org/10.1090/S0002-9947-96-01702-3 Department of Mathematics, Iow a State University, Ames IA 5 0011, USA Email address : jth@iastate.edu URL: http://jthartwig.net Department of Mathematics & Physics, Westminster College ( Missouri), Fulton M...

    work page doi:10.1090/s0002-9947-96-01702-3 1996