Pith. sign in

REVIEW 2 major objections 4 minor 54 references

Degree functions of graded families of ideals

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that Rees-style degree functions extend from single ideals to graded families of m-primary ideals, expressing multiplicities as limits of intersection products and giving closed formulas in dimension 2.

desk verdict Useful paper with a real gap: Theorem D's key inequality is reversed in Proposition 4.7; the earlier theorems and new examples are still worth referee time. read the letter →

arxiv 2506.05248 v1 pith:7M5ZR7ZG submitted 2025-06-05 math.AC

Steven Dale Cutkosky , Jonathan Montaño This is my paper
classification math.AC MSC 13H1514C1713A3013B2214E15
keywords gradedfamiliesofidealsdivisorialfiltrationsReesvaluationsdegreefunctionsmultiplicitiesintersectionproductsexcellentlocalringsHilbert-Samuelmultiplicity
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 extends Rees's classical degree function, defined for a single m-primary ideal, to graded families of m-primary ideals in excellent normal local rings. It proves that the multiplicity of the family modulo a reduced principal ideal is a limit of sums of intersection products over centers of m-valuations, and that under a finiteness condition the limit and the sum commute to give a genuine degree function. In dimension 2, for Q-divisorial filtrations, the per-valuation coefficients stabilize and are computed by intersecting an effective exceptional Q-divisor with the valuation's center. The paper also constructs examples showing that the union of all Rees valuations of a graded family can be finite or infinite, and that limits and sums need not commute in general.

What carries the argument

The load-bearing object is the intersection product (-)_R defined on schemes proper and birational over a local ring, together with the center C(Y_n,v) of an m-valuation on the normal model where I_nO_{Y_n} is invertible. For dimension 2, the filtration is represented by a Q-divisor $\Delta$ with exceptional support such that Gamma(X,O_X(-ceil(nDelta)))=I_n and -$\Delta$ is nef; the coefficient (-$\Delta$·C(X,v)) is the limit that appears in the degree formula.

What would settle it

A concrete falsifier would be an explicit Q-divisorial filtration on a 2-dimensional excellent normal local ring together with an m-valuation v for which the limit of (I_nO_{Y_n}·C(Y_n,v))/n is not equal to (-$\Delta$·C(X,v)), despite $\Delta$ satisfying the representation property; or a graded family for which the limit in Theorem B fails to exist.

Watch

Extended reading notes

Core claim

The central discovery is that degree functions, classically attached to a single m-primary ideal via its Rees valuations, make sense for entire graded families after passing to limits: for an excellent normal local ring, e(I(R/xR)) equals the limit as n tends to infinity of (1/$n^{{d-1}}$) times the sum over m-valuations v of v(x)((I_nO_{Y_n})^{d-1}·C(Y_n,v))_R. In dimension 2, for Q-divisorial filtrations, this simplifies to a linear combination with coefficients (-$\Delta$·C(X,v)) that exist and equal the limit of the normalized per-valuation intersection numbers. When the union of the Rees valuations of all members is finite, the resulting degree function is a finite sum of valuation terms weighted by these intersection numbers.

Load-bearing premise

The dimension-2 formula assumes, via the cited structural result, that every Q-divisorial filtration of m-primary ideals on a 2-dimensional excellent normal local ring can be represented as Gamma(X,O_X(-ceil(nDelta))) with $\Delta$ effective, exceptional, and -$\Delta$ nef, and the general theorems assume the correctness of the companion preprint that constructs the intersection product.

Editorial extensions

If this is right

  • Multiplicities of graded families of m-primary ideals modulo reduced principal ideals are computable as limits of intersection products, generalizing Rees's classical degree function.
  • The limit and the sum in the degree formula commute whenever the union of the Rees valuations of all members is finite and each per-valuation limit exists, giving a finitely supported degree function.
  • For Q-divisorial filtrations on 2-dimensional excellent normal local rings, the per-valuation limits always exist and equal (-Delta·C(X,v)), so the degree function has an explicit closed form.
  • Noetherian filtrations always have finitely many Rees valuations in total, so the finite-union condition is automatically satisfied in the Noetherian case.
  • There exist non-Noetherian divisorial filtrations whose union of Rees valuations is still finite, and others for which that set is infinite, showing that the finiteness condition is independent of Noetherianity.

Reading between the lines

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

  • The limit formulas suggest that a graded family of ideals carries an asymptotic degree function that can be thought of as a limit of the degree functions of its members; one might expect convex-geometric analogues via Newton-Okounkov bodies in higher dimensions.
  • The question of when the per-valuation limits exist (Question 1.3) is central; the dimension-2 answer hints that such limits may fail in higher dimensions or without the Q-divisorial representation hypothesis.
  • The examples with zero analytic spread show that finiteness of the total set of Rees valuations is not a Noetherian property but rather a boundedness condition on normalized intersection numbers; one could test whether it is equivalent to uniform boundedness of these numbers.
  • Since the paper relies on an unpublished companion preprint for the intersection product, the sharpest check of the results would be a direct computation of the formulas on the explicit elliptic-curve examples given here.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies degree functions and multiplicities of graded families of m_R-primary ideals in excellent normal local rings. Using the intersection product developed by the authors in the companion preprint [20], it proves that the multiplicity of a graded family modulo a principal radical ideal can be expressed as a limit of sums of intersection numbers (Theorem B), and that this limit and the sum commute when the union of Rees valuations is finite and the individual limits exist (Corollary C). In dimension 2, for Q-divisorial filtrations, it states a closed formula for the limit of the intersection numbers in terms of a Q-divisor Delta (Theorem D), from which a degree-function formula follows (Corollary E). The paper also constructs two divisorial filtrations on 2-dimensional normal local rings, one with only finitely many Rees valuations overall and one with infinitely many, and an example showing that the limit and sum in Theorem B need not commute in general.

Significance. If the proofs are completed, the paper gives a natural extension of Rees's classical degree-function theory to graded families, with a clean geometric formula in dimension 2. The examples in Section 5 clarify an interesting distinction between Noetherian and non-Noetherian filtrations. The paper is well organized and careful with notation, and it explicitly identifies its dependence on the companion preprint [20] and on [15]. However, the proof of the lower bound in Proposition 4.7 contains a sign error in a key sheaf inclusion, so the central dimension-2 theorem is not proved as written. The remaining parts of the paper are largely coherent, and the gap appears local and likely repairable.

major comments (2)
  1. [Section 4, Proposition 4.7] In the 'remaining case' after Eq. (19), the proof asserts h^0(O_X(-⌈nΔ⌉)⊗E_n^{-1}) ≤ h^0(O_X(-⌈nΔ⌉-rΔ)⊗E_n^{-1}) and then uses this inequality in the chain e > h^0(L_C)-h^0(A) ≥ χ(L_C)-χ(B) = -deg(O_C(-rΔ)) + deg(E_n) > e. Since rΔ is an effective divisor, O_X(-rΔ) is an ideal sheaf, so O_X(-⌈nΔ⌉-rΔ)⊗E_n^{-1} is a subsheaf of O_X(-⌈nΔ⌉)⊗E_n^{-1}, not a super-sheaf. The h^0 inequality therefore has the wrong direction, and the displayed contradiction does not follow. As a result, Proposition 4.7, and with it Theorem 4.2 and Corollary 4.3, are not proved as written. The gap appears repairable: in the remaining case deg(A) ≤ 2p_a(C)-2, so h^0(A) is bounded, while Eq. (19) and the linear growth of h^0(L_C) force h^0(A) to grow linearly; this contradiction would eliminate the remaining case for n ≫ 0, but this argument is not supplied in the manuscript.
  2. [Sections 1 and 3] Theorems A and B and their corollaries rely on Theorem 1.1, which is quoted from the authors' unpublished companion preprint [20] (arXiv:2503.14429), and Theorem D depends on the structural representation of Q-divisorial filtrations from [15]. The present manuscript is therefore conditional on the correctness of those external inputs. The dependence is stated, but the authors should make explicit that the main results cannot be verified independently of [20] and should indicate whether [20] is under review or available in final form.
minor comments (4)
  1. [Section 2, Proposition 2.2] In the proof of Proposition 2.2, the sentence 'f_i h = 0 if and only if Y is not in the support of Div(σ)' is backwards; the intended statement is that the restriction map is injective if and only if Y is not in the support of Div(σ). The proposition itself is correct, but the explanatory sentence should be fixed.
  2. [Example 4.1, before Eq. (10)] The displayed expression 'v_0(x)+v_0(x)+v_1(x)+···+v_n(x)/n' is missing parentheses; it should read ((n+1)v_0(x)+∑_{i=1}^n v_i(x))/n, or equivalently v_0(x)+(v_0(x)+···+v_n(x))/n.
  3. [Example 4.1, after Eq. (9)] In the line '∑_{v∈Div(nR)} v(x)lim...' the notation 'Div(nR)' should be 'Div(m_R)'.
  4. [References] Reference [48] lists the year as 1050; this should be 1950.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem B/D limits are genuine asymptotic results; the Proposition 4.7 inequality issue is a proof gap, not a circular step.

full rationale

The claimed derivation chain does not contain a step that is equivalent by construction to its inputs. Theorem B (Theorem 3.5) is derived by writing e(I(R/xR)) = lim_n e(I_n(R/xR))/n^{d-1} via [12, Theorem 6.5] and then applying the single-ideal degree-function formula e(I_n(R/xR)) = sum_v v(x)((I_nO_{Y_n})^{d-1}·C(Y_n,v))_R from Theorem 3.2, which is [20, Corollary 4.16]. The target statement is about the limit of these single-ideal quantities; the cited results contain no such limit, so the reduction is not a restatement. The same is true for Theorem A: the Volume=Multiplicity formula of [13] gives e(I)=lim e(I_n)/n^d, and Theorem 1.1(1) evaluates each e(I_n); the asymptotic passage is the new content. Theorem D relies on [15] for the structural representation X, Delta with Gamma(X,O_X(-ceil(n Delta)))=I_n and -Delta nef, but that representation is an input supplied by prior work; the theorem's conclusion concerns intersection numbers on arbitrary normal models Y_n and is not definitionally forced by the representation. The self-citations [12], [13], [14], [15], [20] are load-bearing but they are prior theorems whose hypotheses do not include the target statements; no uniqueness theorem is imported to forbid alternatives, and no fitted parameter is relabeled as a prediction. I also flag, without treating it as circularity, a proof gap in Proposition 4.7, final paragraph: the text asserts h0(O_X(-ceil(n Delta)) tensor E_n^{-1}) <= h0(O_X(-ceil(n Delta)-r Delta) tensor E_n^{-1}); because r Delta is effective, O_X(-r Delta) tensor O_C is a subsheaf of O_C, so the inequality has the wrong direction as written. This is a correctness concern in the lower-bound argument for Theorem 4.2, not a circularity, and it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims rest almost entirely on prior theorems: the volume=multiplicity limit, the intersection-product formulas from the authors' companion preprint [20], and the structural representation of Q-divisorial filtrations from [15]. No numerical data are fitted, and no new entities are postulated.

assumptions (7)
  • domain assumption Theorem 1.1 from the companion preprint [20] (e(I)= -((I O_Y)^d)_R and the degree-function formula) is correct.
    Used as the backbone of Theorems A and B; the preprint is authored by the present authors and has not been peer-reviewed at the time of this review.
  • standard math Volume = Multiplicity formula of [13, Theorem 1.1]: the limit e(I)=lim e(I_n)/n^d exists and equals the asymptotic multiplicity.
    Invoked directly in the proof of Theorem 3.3.
  • domain assumption Existence of a nonsingular model X and effective Q-divisor Delta with Gamma(X,O_X(-ceil(n Delta)))=I_n and -Delta nef for Q-divisorial filtrations [15].
    The key structural premise for Theorem D; imported from prior literature.
  • standard math Thorup's rational equivalence and the intersection product on schemes over a Noetherian base [52,20,22] behave as stated.
    The entire intersection-theoretic formalism relies on this background.
  • standard math Riemann-Roch theorem for projective curves over arbitrary fields [35, Remark 7.3.33].
    Used in Proposition 4.7 and in the cohomology vanishing arguments in Section 5.
  • standard math Excellence and analytic unramifiedness properties: reduced excellent local rings are analytically unramified [26, Scholie IV.7.8.3].
    Needed in Theorem 3.5 to apply the degree-function formula.
  • standard math Grauert's contraction criterion and the algebraization theorems [24,28,19,2] used in Example 5.3.
    Bridges analytic and algebraic surfaces in the infinite-Rees-valuations example.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Degree functions of graded families of ideals." pith.science (2026). https://pith.science/paper/7M5ZR7ZG

@misc{pith2026250605248,
  author       = {Pith},
  title        = {Pith review of: Degree functions of graded families of ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7M5ZR7ZG}},
  note         = {Machine review of arXiv:2506.05248}
}
abstract

We express multiplicities and degree functions of graded families of $\mathfrak{m}_R$-primary ideals in an excellent normal local ring $(R,\mathfrak{m}_R)$ as limits of intersection products. Moreover, in dimension 2, we show more refined results for divisorial filtrations. Finally, also in dimension 2, we give an example of a non-Noetherian divisorial filtration $\{I_n\}_{n\geqslant 0}$ of $\mathfrak{m}_R$-primary ideals such that the union of all the sets of Rees valuations of all the $I_n$ is a finite set, and another example of a (necessarily non-Noetherian) divisorial filtration of $\mathfrak{m}_R$-primary ideals such that the set of all Rees valuations is infinite.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

54 extracted references · 52 canonical work pages

  1. [5]

    Rational powers, invariant ideals, and the summation formula

    S. Bisui, S. Das, T. Huy Hà, and J. Montaño, Rational powers, invariant ideals, and the summation formula, preprint arXiv:2402.12350 (2025)

    work page Pith review arXiv 2025

  2. [20]

    Multiplicities and degree functions in local rings via intersection products

    S.D Cutkosky and J. Montaño, Multiplicities and degree functions in local rings via intersection products, preprint arXiv:2503.14429 (2025)

    work page Pith review arXiv 2025

  3. [15]

    Cutkosky, Analytic Spread of filtrations on two dimensional normal local rings, Nagoya Math

    S.D. Cutkosky, Analytic Spread of filtrations on two dimensional normal local rings, Nagoya Math. J. 249 (2023), 239 - 268

    work page 2023

  4. [1]

    Abhyankar, On the valuations centered in a local domain, Amer

    S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321 - 348

    work page 1956

  5. [2]

    Artin, On the solutions of analytic equations, Invent

    M. Artin, On the solutions of analytic equations, Invent. Math. 5, (1968), 277 - 291

    work page 1968

  6. [3]

    Artin, Lipman’s proof of resolution of singularities for surfaces, pp

    M. Artin, Lipman’s proof of resolution of singularities for surfaces, pp. 267–287 in Arithmetic geometry (Storrs, CT, 1984), edited by G. Cornell and J. H. Silverman, Springer, 1986

    work page 1984

  7. [4]

    Barth, C

    W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Springer-Verlag, 1984. DEGREE FUNCTIONS OF GRADED FAMILIES OF IDEALS 29

    work page 1984

  8. [6]

    H. Blum, Y . Liu, and L. Qi, Convexity of multiplicities of filtrations on local rings, Compos. Math. 160 (2024), 878 - 914

    work page 2024

Show all 54 references
  1. [7]

    Bourbaki, Commutative Algebra, Chapters 1-7, Springer Verlag, 1989

    N. Bourbaki, Commutative Algebra, Chapters 1-7, Springer Verlag, 1989

    1989

  2. [8]

    Cid-Ruiz, Relative mixed multiplicities and mixed Buchsbaum-Rim multiplicities, preprint arXiv:2311.15105 (2024)

    Y . Cid-Ruiz, Relative mixed multiplicities and mixed Buchsbaum-Rim multiplicities, preprint arXiv:2311.15105 (2024)

    2024 arXiv

  3. [9]

    Cid-Ruiz, C

    Y . Cid-Ruiz, C. Polini, and B. Ulrich, Multidegrees, families, and integral dependence, preprint arXiv:2405.07000 (2024)

    2024 arXiv

  4. [10]

    Cossart, U

    V . Cossart, U. Jannsen, S. Saito, Desingularization: Invariants and strategy, Lecture Notes in Mathematics 2270, Springer, 2020

    2020

  5. [11]

    Cutkosky, On unique and almost unique factorization of complete ideals II, Inventiones Math

    S.D. Cutkosky, On unique and almost unique factorization of complete ideals II, Inventiones Math. 98 (1989), 59 - 74

    1989

  6. [12]

    Cutkosky, Asymptotic multiplicities of graded families of ideals and linear series, Adv

    S.D. Cutkosky, Asymptotic multiplicities of graded families of ideals and linear series, Adv. Math. 264 (2014), 55 - 113

    2014

  7. [13]

    Cutkosky, A general volume=multiplicity formula, Acta Math

    S.D. Cutkosky, A general volume=multiplicity formula, Acta Math. Vietnam. 40 (2015), 139 – 147

    2015

  8. [14]

    Cutkosky, Mixed multiplicities of divisorial filtrations, Adv

    S.D. Cutkosky, Mixed multiplicities of divisorial filtrations, Adv. Math. 358 (2019), 106842

    2019

  9. [16]

    Cutkosky, Rees algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings, Commun

    S.D. Cutkosky, Rees algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings, Commun. Algebra 53 (2025), 3364–3387

    2025

  10. [17]

    Cutkosky, The Rees algebra and analytic spread of a divisorial filtration, preprint arXiv:2407.19585 (2024)

    S.D. Cutkosky, The Rees algebra and analytic spread of a divisorial filtration, preprint arXiv:2407.19585 (2024)

    2024 arXiv

  11. [18]

    Cutkosky and P

    S.D. Cutkosky and P. Sarkar, Multiplicities and mixed multiplicities of arbitrary filtrations, Res. Math. Sci. 9 (2022), Paper No. 14

    2022

  12. [19]

    Cutkosky and H

    S.D. Cutkosky and H. Srinivasan, An intrinsic criterion for isomorphism of singularities, Amer. J. Math. 115 (1993), 789 - 821

    1993

  13. [21]

    L. Ein, R. Lazarsfeld, and K. Smith, Uniform Approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003), 409 - 440

    2003

  14. [22]

    Fulton, Intersection Theory, second edition, Springer, 1998

    W. Fulton, Intersection Theory, second edition, Springer, 1998

    1998

  15. [23]

    Goel, R.V

    K. Goel, R.V . Gurjar, and J.K. Verma, Minkowski inequality and equality of ideals in finite length Noetherian rings, Contemp. Math. 738 (2019), 33 - 56

    2019

  16. [24]

    Grauert, uber modifikationen und exceptilonelle analytischen mengen, Math

    H. Grauert, uber modifikationen und exceptilonelle analytischen mengen, Math. Ann. 146 (1962), 331 - 368

    1962

  17. [25]

    Grothendieck, Éléments de géométrie algébrique II, Étude globale élémentaire de quelques classes de morphisms, Pub

    A. Grothendieck, Éléments de géométrie algébrique II, Étude globale élémentaire de quelques classes de morphisms, Pub. Math. de l’IHÉS, vol 8 (1961)

    1961

  18. [26]

    Grothendieck, Éléments de géométrie algébrique IV , Étude locale des Schémas et des Morphismes des Schémas (Seconde Partie), Pub

    A. Grothendieck, Éléments de géométrie algébrique IV , Étude locale des Schémas et des Morphismes des Schémas (Seconde Partie), Pub. Math. IHES, vol 24 (1965)

    1965

  19. [27]

    Hartshorne, Algebraic Geometry, Springer, 1978

    R. Hartshorne, Algebraic Geometry, Springer, 1978

    1978

  20. [28]

    Hironaka, On the equaivalence of singularities, I, Arithmetical Algebraic Geometry, Proceedings

    H. Hironaka, On the equaivalence of singularities, I, Arithmetical Algebraic Geometry, Proceedings. (Purdue Univeristy, 1963) (O.F.G. Schilling ed.) Harper and Row, New York 1965, 153 -200

    1963

  21. [29]

    Katz and J

    D. Katz and J. Validashti, Multiplicities and Rees valuations, Collect. Math. 61 (2010), 1 - 24

    2010

  22. [30]

    Kleiman, Toward a numerical theory of ampleness, Ann

    S. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. 84 (1966), 293 - 344

    1966

  23. [31]

    Kleiman and A

    S. Kleiman and A. Thorup, Mixed Buchbaum–Rim multiplicities, Amer. J. Math. 118 (1996), 529 - 569. 30 STEVEN DALE CUTKOSKY AND JONATHAN MONTAÑO

    1996

  24. [32]

    Laufer, On deformations of resolutions of two-dimensional singularities, Rice University studies in complex analysis, 1972

    H. Laufer, On deformations of resolutions of two-dimensional singularities, Rice University studies in complex analysis, 1972

    1972

  25. [33]

    Lazarsfeld, Positivity in Algebraic Geometry I, Springer, 2000

    R. Lazarsfeld, Positivity in Algebraic Geometry I, Springer, 2000

    2000

  26. [34]

    Lazarsfeld and M

    R. Lazarsfeld and M. Musta¸ t˘a, Convex bodies associated to linear series, Ann. Sci. Ec. Norm. Super 42 (2009), 783 - 835

    2009

  27. [35]

    Lin, Algebraic Geometry and Arithmetic Curves, Oxford graduate texts in mathematics 6, translated by Reinie Erné, Oxford University Press, 2002

    Q. Lin, Algebraic Geometry and Arithmetic Curves, Oxford graduate texts in mathematics 6, translated by Reinie Erné, Oxford University Press, 2002

    2002

  28. [36]

    Lipman, Rational singularities with applications to algebraic surfaces and unique factorization, Publ

    J. Lipman, Rational singularities with applications to algebraic surfaces and unique factorization, Publ. Math. IHES 36 (1969), 195 - 279

    1969

  29. [37]

    Lipman, Desingularization of 2-dimensional schemes, Annals of Math

    J. Lipman, Desingularization of 2-dimensional schemes, Annals of Math. 107 (1978), 115 - 207

    1978

  30. [38]

    Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8, Cambridge University Press, 1989

    H. Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8, Cambridge University Press, 1989

    1989

  31. [39]

    Milne, Étale cohomology, Princeton University Press, 1980

    J.S. Milne, Étale cohomology, Princeton University Press, 1980

    1980

  32. [40]

    Mumford, Lectures on curves on an algebraic surface, Annals of Math

    D. Mumford, Lectures on curves on an algebraic surface, Annals of Math. Studies 59, Princeton Univ. Press, 1966

    1966

  33. [41]

    Polstra, Zariski-Nagata theorem for singularities and the uniform Izumi-Rees property, preprint arXiv:2406.00759 (2024)

    T. Polstra, Zariski-Nagata theorem for singularities and the uniform Izumi-Rees property, preprint arXiv:2406.00759 (2024)

    2024

  34. [42]

    Ramanujam, A geometric interpretation of multiplicity, Inventiones Math

    C.P. Ramanujam, A geometric interpretation of multiplicity, Inventiones Math. 22 (1973), 85 - 147

    1973

  35. [43]

    Rees, Valuations associated with ideals (II), J

    D. Rees, Valuations associated with ideals (II), J. London Math. Soc. 6 (1956), 221 - 228

    1956

  36. [44]

    Rees, A note on analytically unramified local rings, J

    D. Rees, A note on analytically unramified local rings, J. Lond. Math. Soc. 36 (1961), 24 - 28

    1961

  37. [45]

    Rees, Degree functions in local rings, Proceedings of the Cambridge Phil

    D. Rees, Degree functions in local rings, Proceedings of the Cambridge Phil. Soc. 57 (1961), 1-7

    1961

  38. [46]

    Rees and Y

    D. Rees and Y . Sharp, On a theorem of B. Teissier on multiplicities of ideals in local rings, J. London Math. Soc. 18 (1978), 449-463

    1978

  39. [47]

    Rees, Lectures on the asymptotic theory of ideals, London Math Soc

    D. Rees, Lectures on the asymptotic theory of ideals, London Math Soc. Lecture Note Series 113, Cambridge Univ. Press, 1988

    1988

  40. [48]

    Samuel, Sur les variétés algébroides, Ann

    P. Samuel, Sur les variétés algébroides, Ann. Inst. Fourier 2 (1050), 147 - 160

  41. [49]

    Snapper, Polynomials associated with divisors, J

    E. Snapper, Polynomials associated with divisors, J. Math. and Mech. 8 (1959), 967 - 992

    1959

  42. [50]

    The stacks project

    The Stacks project authors. The stacks project. https://stacks.math.columbia.edu, 2023

    2023

  43. [51]

    Swanson and C

    I. Swanson and C. Huneke, Integral closure of Ideals, Rings and Modules, London Math. Soc. Lecture Note Ser

  44. [52]

    Thorup, Rational equivalence theory on arbitrary Noetherian schemes, In Enumerative Geometry, Sebastian Xambó-Descamps Ed, LNM 1436, Springer, 1990

    A. Thorup, Rational equivalence theory on arbitrary Noetherian schemes, In Enumerative Geometry, Sebastian Xambó-Descamps Ed, LNM 1436, Springer, 1990

    1990

  45. [53]

    Zariski and P

    O. Zariski and P. Samuel, Commutative Algebra V olume II, Van Nostrand, 1960. STEVENDALECUTKOSKY, DEPARTMENT OFMATHEMATICS, UNIVERSITY OFMISSOURI, COLUMBIA, MO 65211, USA.Email:cutkoskys@missouri.edu JONATHANMONTAÑO, SCHOOL OFMATHEMATICAL ANDSTATISTICALSCIENCES, ARIZONASTATEUN...

    1960

  46. [336]

    Cambridge University Press, Cambridge, 2006

    2006

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.