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Multiplicity in triangulated categories

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Pith's one-line read This paper establishes that a single numerical invariant, the s-multiplicity e_s(X,Y), governs the asymptotic growth of Hom sets in triangulated categories with a central ring action, and that it specializes to Serre's intersection…

desk verdict Solid and inventive unification of multiplicity invariants, but the central invariant is not intrinsic: it depends on an auxiliary even degree d that the paper treats as a WLOG. read the letter →

arxiv 2506.02437 v1 pith:VIESTTLF submitted 2025-06-03 math.KT math.ACmath.CT

Petter Andreas Bergh , David A. Jorgensen , Peder Thompson This is my paper
classification math.KTmath.ACmath.CT MSC 18G8013H15
keywords multiplicitytriangulatedcategoryHerbranddifferenceHilbertpolynomialKoszulhomologyintersectionvanishingofcohomologycentralringaction
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 introduces a single numerical invariant, the s-multiplicity e_s(X,Y), for pairs of objects in a triangulated category equipped with a central action of a graded-commutative ring. The invariant is built from the lengths of the Hom groups between shifts of the two objects, and the paper shows that it is governed by the leading coefficients of the associated Hilbert polynomials. If the theory is right, one number packages several established intersection-theoretic invariants, and it gives a new vanishing criterion for cohomology in settings ranging from local complete intersection rings to group algebras. The paper's central claim is that this multiplicity is additive on distinguished triangles and can be computed as an Euler characteristic of Koszul homology, exactly as classical multiplicity is computed for modules over local rings.

What carries the argument

The machinery is the generalized Herbrand difference h(X,Y)(n) together with its difference operators Delta_s. For a fixed even degree d, h(n) is the alternating sum of lengths of Hom_T(X, $Sigma^{{n+i}}$Y) for i = 0,...,d-1, and the s-multiplicity is the constant value of $\Delta$^{s-1} h(X,Y)(n) for large n. The Hilbert polynomials g_i(t), defined by g_i(n) = length_{R_0} Hom_T(X, $Sigma^{{dn+i}}$Y), carry the leading-coefficient data that determine e_s. The Koszul object Y tensor z, defined by the distinguished triangle Y -> Sigma^d Y -> Y tensor z -> Sigma Y, is the device through which the invariant becomes an Euler characteristic and through which complexity is reduced step by step.

What would settle it

Take a pair (X,Y) of complexity s where the central ring has generators of different degrees, and compute e_s(X,Y) using d and using a multiple md as in the reduction 4.2; if the two values do not differ exactly by the factor $m^{{2s-1}}$, the 'without loss of generality' reduction changes the invariant and the multiplicity is presentation-dependent rather than intrinsic.

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Extended reading notes

Core claim

The central discovery is that the s-multiplicity e_s(X,Y), defined as the (s-1)-st difference of the generalized Herbrand difference h(X,Y)(n) = sum_{i=0}^{d-1} (-1)^{n+i} length_{R_0} Hom_T(X, $Sigma^{{n+i}}$Y), is a well-behaved invariant. Proposition 4.8 identifies it with (s-1)! $d^{{s-1}}$ times the alternating sum of the degree s-1 coefficients of the Hilbert polynomials of the pair. Corollary 4.11 expresses it as an alternating sum of lengths of Hom groups into a Koszul object, namely e_s(X,Y) = sum_n (-1)^n length_{R_0} Hom_T(X, Sigma^n(Y tensor (z_1,...,z_s))). Theorem 4.12 proves additivity on distinguished triangles, and Section 6 shows that the invariant recovers Serre's intersection multiplicity, Hochster's $\theta$ invariant, and Buchweitz's Herbrand difference as special cases. Theorem 6.11 provides a vanishing criterion: if e_s(X,Y) = 0 and the Hom groups vanish for d/2 consecutive even (or odd) degrees, then they vanish for all sufficiently large degrees.

Load-bearing premise

The construction fixes an auxiliary even degree d in which the central ring R is generated and assumes this is no loss of generality, but the paper does not prove that different choices of d give the same (or consistently rescaled) numerical values for e_s(X,Y).

Editorial extensions

If this is right

  • The s-multiplicity e_s(X,Y) is determined by the top Hilbert polynomial coefficients: e_s(X,Y) = (s-1)! d^{s-1} sum_{i=0}^{d-1} (-1)^i a_i, where a_i is the coefficient of the degree (s-1) term of the i-th Hilbert polynomial.
  • For a suitable sequence of elements z_1,...,z_s in R_d, the multiplicity equals the Euler characteristic sum_n (-1)^n length_{R_0} Hom_T(X, Sigma^n(Y tensor (z_1,...,z_s))), generalizing the Serre and Auslander-Buchsbaum formula.
  • The invariant is additive on distinguished triangles: if Y_1 -> Y_2 -> Y_3 -> Sigma Y_1 is a distinguished triangle with cx(X,Y_i) <= s for all i, then e_s(X,Y_2) = e_s(X,Y_1) + e_s(X,Y_3).
  • If e_s(X,Y) = 0 and the Hom groups vanish for d/2 consecutive even (or odd) degrees, then they vanish for all sufficiently large degrees, giving a new cohomology vanishing criterion with applications to local complete intersection rings.
  • The construction recovers Serre's intersection multiplicity as e_0 in a derived category over a regular local ring, Hochster's theta invariant as e_1 for hypersurface rings, and Buchweitz's Herbrand difference as e_1 in the derived category of a hypersurface.

Reading between the lines

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

  • The paper leaves open whether e_s(X,Y) is canonically normalized: changing the auxiliary degree d to a multiple md rescales the value by m^{2s-1}, so a normalization (such as fixing a canonical d or dividing by d^{2s-1}) would be needed to call the number intrinsic to the pair (X,Y) rather than to a chosen presentation of the central ring.
  • The contravariant theory, which the paper mentions but does not develop, would likely produce a parallel invariant governing Hom_T(X tensor z, Y) instead of Hom_T(X, Y tensor z); this could be relevant in categories with a duality or a tensor product.
  • The vanishing theorem suggests a concrete test in stable module categories of finite groups, where the central ring is group cohomology: the vanishing of e_s for a pair of modules should force eventual vanishing of Ext groups once enough consecutive even or odd degrees vanish, and this could be checked computationally in small examples like the symmetric group S_4.
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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

3 major / 4 minor

Summary. The paper develops a multiplicity theory for pairs of objects in an R-linear triangulated category with a central action of a graded-commutative ring R. It defines a generalized Herbrand difference h(X,Y) and an s-multiplicity e_s(X,Y), and proves that e_s is determined by the leading coefficients of the associated Hilbert polynomials (Proposition 4.8), alternates sign under shifts (Proposition 4.9), is additive on distinguished triangles (Theorem 4.12), and can be expressed as an Euler characteristic of Koszul homology (Corollary 4.11). A parallel "negative" theory is given in Section 5, and Section 6 recovers Serre's intersection multiplicity, Hochster's theta invariant, and Buchweitz's Herbrand difference as special cases, together with a vanishing theorem (Theorem 6.11). A limit formula is proved in Section 7. The central claim is that e_s is a unified, meaningful multiplicity for triangulated categories with a central ring action.

Significance. If the main issues are repaired, the paper would provide a genuinely useful unifying framework: the Hilbert-polynomial formula, the Koszul-homology interpretation, additivity on triangles, and the recovery of several known invariants are attractive and well-motivated. The exposition is careful, the auxiliary appendix on difference operators is self-contained, and the examples are concrete. However, the definition of e_s depends on an auxiliary even degree d in a way that is not addressed, the proof of the vanishing theorem uses an invalid inference, and several load-bearing results are outsourced to an unpublished preprint. These are substantive, not cosmetic, problems for the central claims.

major comments (3)
  1. [Section 4.2, Definition 4.7, Proposition 4.8] The invariant e_s(X,Y) is not shown to be independent of the auxiliary even degree d. Passing from the step-d presentation to the step-md presentation multiplies the leading coefficients a_i in Proposition 4.8 by m^{s-1} and multiplies the number of residue classes by m; hence e_s^{md}(X,Y)=m^{2s-1}e_s^d(X,Y). No normalization or comparison result is given. For example, in Example 6.2 the d=2 value is r, whereas the subring generated by chi^2 (d=4) gives 2r. Thus the phrase "without loss of generality" in Section 4.2 is not harmless: the numerical output is a function of the chosen subring R_0[R_d], not of the pair (X,Y) alone. This affects Definition 4.7 and all results depending on the value of e_s, including Theorems 4.10, 4.12, and 7.2. The authors should either prove independence of d after a suitable normalization (for example e_s/d^{2s-1}), or explicitly treat d as part of the data and state all scaling identities.
  2. [Theorem 6.11, proof] The proof claims that e_s(X,Y)=0 forces, for all large n, the parity balance sum_{i=0}^{d/2-1} lambda_{n+2i} = sum_{i=0}^{d/2-1} lambda_{n+2i+1}. This is not a consequence of Definition 4.7: e_s=0 only says the (s-1)-st difference Delta^{s-1}h(X,Y) vanishes for n much larger than 0, not that h(X,Y) itself vanishes. For s>1, h can be a nonzero periodic or lower-degree quasi-polynomial. A concrete failure occurs in the paper's own Example 6.5 with c=2 and d=2: lambda_{2t}=2t+1 and lambda_{2t+1}=2t+2, so e_2=0 but lambda_{2t} and lambda_{2t+1} differ for every t. Therefore the proof of Theorem 6.11 does not go through, and Corollary 6.12, which depends on it, is not established. Please either supply a correct argument or revise the statement and proof.
  3. [Theorems 4.10, 4.12, 5.6, 6.11, and Proposition 4.13] Several load-bearing results rely on [10], an unpublished arXiv preprint, specifically [10, Lemma 2.2] for the existence of eventually injective elements and [10, Theorem 2.3] for asymptotic vanishing after d consecutive zeros. These ingredients are used in Theorem 4.10, Corollary 4.11, Theorem 4.12, Theorem 5.6, Corollary 5.7, Proposition 4.13, and Theorem 6.11. Since the present manuscript is otherwise detailed and self-contained, the authors should include proofs of these facts in an appendix or give a published reference; otherwise the central claims are not independently verifiable within the manuscript.
minor comments (4)
  1. [Definition 5.4] The negative multiplicity in Section 5.4 is denoted by the same symbol e_s as the covariant multiplicity in Definition 4.7. Since both can be defined simultaneously (see Proposition 6.6), please use distinct notation or explicitly state which convention is in force in each result.
  2. [Example 6.4] The six Hilbert polynomials are stated after "some computations" using the Hilbert series, but no computation is shown. Since this example is used to obtain e_2(F_2,F_2)=0, please include enough detail for the reader to verify the polynomials and the complexity computation.
  3. [Proposition 4.9] The proof of Proposition 4.9 uses the fact that d is even in a visible way; this hypothesis is in force from Section 4.2 but should be stated explicitly in the proposition for clarity.
  4. [Introduction, Theorem 1.1] The term "eventually injective" is used in Theorem 1.1 but is defined only later in Section 4. Please define it in the introduction or move the definition earlier.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the s-multiplicity is derived from Hom-length data and Koszul-object long exact sequences, not from its own conclusions.

full rationale

The derivation chain is self-contained. The invariant e_s(X,Y) is defined directly from the lengths of Hom sets (Definitions 3.1 and 4.7), and the main structural results—Proposition 4.8, Theorem 4.10, Corollary 4.11, and Theorem 4.12—are proved from the long exact sequences attached to Koszul objects and standard Hilbert-polynomial facts, not assumed as inputs. The citations to prior work by the same authors ([10, Lemma 2.2; Theorem 2.3]) provide auxiliary technical ingredients whose statements do not presuppose the s-multiplicity; they are not machine-checked but are parameter-free, previously stated results, and the proofs here do not reduce to a self-citation chain. The Section 6 recoveries of Serre intersection multiplicity, Hochster's theta invariant, and Buchweitz's Herbrand difference are consistency checks and special cases, not fitted predictions. A non-circular caveat remains: the value of e_s depends on the auxiliary even degree d chosen in Section 4.2, and replacing d by md scales the value by m^{2s-1} according to Proposition 4.8, so the numerical invariant is not shown to be independent of that choice; this is a well-definedness/canonicality issue rather than a circular derivation.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standing finite-length and eventual Noetherian conditions, the infinite residue field assumption for the Koszul step, and two lemmas from the authors' unpublished preprint [10]. No new physical or algebraic entities are introduced; the multiplicity is a new invariant, not an entity.

free parameters (1)
  • Even degree d used for the reduction R = R_0[R_d] = d = lcm(|x_i|) if lcm is even, else 2*lcm
    The value of e_s(X,Y) depends on d; replacing d by md scales e_s by m^{2s-1}. The paper does not prove independence from this choice, despite calling the reduction 'without loss of generality' in Section 4.2.
assumptions (5)
  • domain assumption Hom^*_T(X,Y) belongs to noeth_fl(R), meaning eventually Noetherian and degreewise finite length.
    This is the standing assumption for the covariant theory in Section 4, needed for Hilbert polynomial growth. It holds for the examples listed in Remark 4.1.
  • domain assumption R_0 contains an infinite field or is local with infinite residue field.
    Necessary for the existence of an eventually injective element z in Theorem 4.10, via [10, Lemma 2.2].
  • domain assumption R is graded-commutative Noetherian and T is an R-linear triangulated category with central ring action.
    This is the overall setting defined in Section 2.
  • domain assumption Lemma 2.2 and Theorem 2.3 of the authors' preprint [10] (arXiv:2405.12763).
    Theorem 4.10 and Theorem 6.11 rely on these external results from an unpublished preprint by the same authors.
  • standard math Hilbert polynomial theory for graded modules (Atiyah-Macdonald, Kirby).
    Used to assert that lengths are given by polynomials or quasi-polynomials in Sections 4.3 and 5.

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Pith. "Pith review of Multiplicity in triangulated categories." pith.science (2026). https://pith.science/paper/VIESTTLF

@misc{pith2026250602437,
  author       = {Pith},
  title        = {Pith review of: Multiplicity in triangulated categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VIESTTLF}},
  note         = {Machine review of arXiv:2506.02437}
}
abstract

We lay out the theory of a multiplicity in the setting of a triangulated category having a central ring action from a graded-commutative ring $R$, in other words, an $R$-linear triangulated category. The invariant we consider is modelled on those for graded modules over a commutative graded ring. We show that this invariant is determined by the leading coefficients of the Hilbert polynomials expressing the lengths of certain Hom sets. Our theory is a natural analogue of Hochster's theta invariant for homology and Buchweitz's Herbrand difference for cohomology. Moreover, we give applications to vanishing of cohomology and modules over local complete intersection rings, group algebras of a finite group, and certain finite dimensional algebras.

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Works this paper leans on

35 extracted references · 33 canonical work pages

  1. [10]

    Jorgensen, and Peder Thompson

    Petter Andreas Bergh, David A. Jorgensen, and Peder Thompson. Asymptotic vanishing of cohomology in triangulated categories.Arxiv preprint 2405.12763, 2024

    work page arXiv 2024

  2. [1]

    Remarks on a depth formula, a grade inequality and a conjecture of Auslander.Comm

    Tokuji Araya and Yuji Yoshino. Remarks on a depth formula, a grade inequality and a conjecture of Auslander.Comm. Algebra, 26(11):3793–3806, 1998

    work page 1998

  3. [2]

    M. F. Atiyah and I. G. Macdonald.Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969

    1969

  4. [3]

    Buchsbaum

    Maurice Auslander and David A. Buchsbaum. Homological dimension in noetherian rings. II. Trans. Amer. Math. Soc., 88:194–206, 1958

    work page 1958

  5. [4]

    Codimension and multiplicity

    Maurice Auslander and David A. Buchsbaum. Corrections to “Codimension and multiplicity”. Ann. of Math. (2), 70:395–397, 1959

    work page 1959

  6. [5]

    L. L. Avramov. Modules of finite virtual projective dimension.Invent. Math., 96(1):71–101, 1989

    work page 1989

  7. [6]

    Avramov and Ragnar-Olaf Buchweitz

    Luchezar L. Avramov and Ragnar-Olaf Buchweitz. Support varieties and cohomology over complete intersections.Invent. Math., 142(2):285–318, 2000

    work page 2000

  8. [7]

    Avramov and Srikanth B

    Luchezar L. Avramov and Srikanth B. Iyengar. Constructing modules with prescribed coho- mological support.Illinois J. Math., 51(1):1–20, 2007

    work page 2007

Show all 35 references
  1. [8]

    Local cohomology and support for triangulated categories.Ann

    Dave Benson, Srikanth Iyengar, and Henning Krause. Local cohomology and support for triangulated categories.Ann. Sci. ´Ec. Norm. Sup´ er., 41(4):573–619, 2008

    2008

  2. [9]

    Iyengar, Henning Krause, and Steffen Oppermann

    Petter Andreas Bergh, Srikanth B. Iyengar, Henning Krause, and Steffen Oppermann. Di- mensions of triangulated categories via Koszul objects.Math. Z., 265(4):849–864, 2010

    2010

  3. [11]

    Jorgensen, and Peder Thompson

    Petter Andreas Bergh, David A. Jorgensen, and Peder Thompson. Dimension and depth inequalities over complete intersections.Arkiv f¨ or Matematik, 63:31–60, 2025

    2025

  4. [12]

    Cohomology of twisted tensor products.J

    Petter Andreas Bergh, and Steffen Oppermann. Cohomology of twisted tensor products.J. Algebra, 320:3327–3338, 2008

    2008

  5. [13]

    Bou¸ ca, T

    V. Bou¸ ca, T. F. C. Cabral, S. H. Hassanzadeh, and J. Naeliton. Buchsbaum-Rim multiplicity: a Koszul homology description.J. Algebra, 633:754–772, 2023

    2023

  6. [14]

    Brown, Claudia Miller, Peder Thompson, and Mark E

    Michael K. Brown, Claudia Miller, Peder Thompson, and Mark E. Walker. Adams operations on matrix factorizations.Algebra Number Theory, 11(9):2165–2192, 2017

    2017

  7. [15]

    Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings.Available from http://hdl.handle.net/1807/16682, page 155, 1986

    Ragnar-Olaf Buchweitz. Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings.Available from http://hdl.handle.net/1807/16682, page 155, 1986

    1986

  8. [16]

    On the vanishing of the theta invariant and a conjecture of Huneke and Wiegand.Pacific J

    Olgur Celikbas. On the vanishing of the theta invariant and a conjecture of Huneke and Wiegand.Pacific J. Math., 309(1):103–144, 2020

    2020

  9. [17]

    Asymptotic behavior of Ext functors for modules of finite complete intersection dimension.Math

    Olgur Celikbas and Hailong Dao. Asymptotic behavior of Ext functors for modules of finite complete intersection dimension.Math. Z., 269(3-4):1005–1020, 2011

    2011

  10. [18]

    Conway and Richard K

    John H. Conway and Richard K. Guy.The book of numbers. Copernicus, New York, 1996. 24 PETTER ANDREAS BERGH, DA VID A. JORGENSEN, AND PEDER THOMPSON

    1996

  11. [19]

    Asymptotic behavior of Tor over complete intersections and applications

    Hailong Dao. Asymptotic behavior of Tor over complete intersections and applications. Preprint, arxiv:0710.5818, 2007

    2007 arXiv

  12. [20]

    Sup- port varieties for selfinjective algebras.K-Theory, 33(1):67–87, 2004

    Karin Erdmann, Miles Holloway, Rachel Taillefer, Nicole Snashall, and Øyvind Solberg. Sup- port varieties for selfinjective algebras.K-Theory, 33(1):67–87, 2004

    2004

  13. [21]

    The cohomology ring of a finite group.Trans

    Leonard Evens. The cohomology ring of a finite group.Trans. Amer. Math. Soc., 101:224–239, 1961

    1961

  14. [22]

    E. Golod. The cohomology ring of a finitep-group.Dokl. Akad. Nauk SSSR, 125:703–706, 1959

    1959

  15. [23]

    Gulliksen

    Tor H. Gulliksen. A change of ring theorem with applications to Poincar´ e series and inter- section multiplicity.Math. Scand., 34:167–183, 1974

    1974

  16. [24]

    Ueber die Theorie der algebraischen Formen.Math

    David Hilbert. Ueber die Theorie der algebraischen Formen.Math. Ann., 36(4):473–534, 1890

  17. [25]

    The dimension of an intersection in an ambient hypersurface

    Melvin Hochster. The dimension of an intersection in an ambient hypersurface. InAlgebraic geometry (Chicago, Ill., 1980), volume 862 ofLecture Notes in Math., pages 93–106. Springer, Berlin-New York, 1981

    1980

  18. [26]

    Jorgensen, Liana M

    David A. Jorgensen, Liana M. S ¸ega, and Peder Thompson. Asymptotic behavior of Ext for pairs of modules of large complexity over graded complete intersections.Math. Z., 302(3):1761–1784, 2022

    2022

  19. [27]

    D. Kirby. Artinian modules and Hilbert polynomials.Quart. J. Math. Oxford Ser. (2), 24:47– 57, 1973

    1973

  20. [28]

    Cambridge University Press, Cambridge, second edition, 1989

    Hideyuki Matsumura.Commutative ring theory, volume 8 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1989. Translated from the Japanese by M. Reid

    1989

  21. [29]

    Nakaoka,Note on cohomology algebras of symmetric groups, J

    M. Nakaoka,Note on cohomology algebras of symmetric groups, J. Math. Osaka City Univ. 13 (1962), 45–55

    1962

  22. [30]

    La notion de multiplicit´ e en alg` ebre et en g´ eom´ etrie alg´ ebrique.J

    Pierre Samuel. La notion de multiplicit´ e en alg` ebre et en g´ eom´ etrie alg´ ebrique.J. Math. Pures Appl. (9), 30:159–205, 1951

    1951

  23. [31]

    Springer Monographs in Mathematics

    Jean-Pierre Serre.Local algebra. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000. Translated from the French by CheeWhye Chin and revised by the author

    2000

  24. [32]

    Stanley.Enumerative combinatorics

    Richard P. Stanley.Enumerative combinatorics. Volume 1, volume 49 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2012

    2012

  25. [33]

    B. B. Venkov. Cohomology algebras for some classifying spaces.Dokl. Akad. Nauk SSSR, 127:943–944, 1959

    1959

  26. [34]

    On the vanishing of Hochster’sθinvariant.Ann

    Mark Walker. On the vanishing of Hochster’sθinvariant.Ann. K-Theory, 2(2):131–174, 2017

    2017

  27. [35]

    Mark E. Walker. Chern characters for twisted matrix factorizations and the vanishing of the higher Herbrand difference.Selecta Math. (N.S.), 22(3):1749–1791, 2016. (P. A. Bergh)Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway Email address:petter.bergh@ntnu.no URL...

    2016

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