Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

The embedded deformation problem for monomial ideals

T0 review · 3 major / 6 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 For rings defined by monomial ideals, embedded deformations correspond exactly to central degree-two elements of the homotopy Lie algebra and to free summands of the conormal module.

desk verdict A strong, correct resolution of Avramov's question for monomial rings; the n≤5 classification is a nice extra, with only minor computational-reproducibility caveats. read the letter →

arxiv 2506.10827 v1 pith:SWYKBKGS submitted 2025-06-12 math.AC math.RA

Benjamin Briggs , Eloísa Grifo , Josh Pollitz This is my paper · ORCID
classification math.ACmath.RA MSC 13D0913C1513D0213D0713H1014M1016E45
keywords embeddeddeformationmonomialidealhomotopyLiealgebracohomologicalsupportvarietyconormalmoduleTaylorresolutioncompleteintersectionregularsequence
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 answers, for rings defined by monomial relations in a regular sequence, a question posed in the late 1980s: is the existence of an embedded deformation of codimension $c$ exactly the same as having a $c$-dimensional space of degree-two central elements in the homotopy Lie algebra $\pi(R)$, the graded Lie algebra built from a minimal free resolution of the residue field? The authors prove the answer is yes for such monomial rings, and that three apparently different conditions are equivalent: the ring deforms as an embedded quotient by a regular sequence, its conormal module $I/I^2$ has a free summand of rank $c$, and degree two of $\pi(R)$ contains a $c$-dimensional subspace of central elements — in fact, of radical elements. The equivalence matters because embedded deformations are the first step toward being a complete intersection, and it converts a homological question into a combinatorial one about monomial generators and their gcds. The paper also establishes a lower bound on the dimension of cohomological support varieties over such rings and classifies all possible support varieties for rings defined by at most five monomials.

What carries the argument

The engine is the Taylor dg algebra $T(f)$ on the monomial generators $f_1,\dots,f_n$: its basis elements $b_J$ are indexed by subsets, with differential $\partial(b_J)=\sum_i \pm (f_J/f_{J\setminus\{i\}}) b_{J\setminus\{i\}}$ and product $b_J\cdot b_K = \operatorname{sign}(J,K)(f_J f_K/f_{J\cup K}) b_{J\cup K}$. For monomials in a regular sequence this complex is a free resolution of $R$, so it can serve as the Taylor model to which Tate variables are adjoined, and its quadratic part controls the Lie bracket on $\pi_2(R)$. Two combinatorial data determine the relevant support computations: the GCD graph $\Gamma_f$, with an edge when $\gcd(f_i,f_j)$ is a nonunit, and the Taylor graph, whose directed edges record which coefficients in the two-periodic complexes $\widehat{C}_{E,a}(T)$ are nonzero. Whether $\widehat{C}_{E,a}(T)$ is exact at a point $a$ decides whether $a$ lies in $\mathrm{V}_R(R)$, which is how hyperplane containments get translated into regular sequences and hence into embedded deformations.

What would settle it

One decisive calculation: for the five-generator exceptional case in Theorem C (GCD graph 1 or 2 with $f_3 \mid f_{24}$), the paper predicts $\mathrm{V}_R(R)=V(\chi_1\chi_5)$. Running the $\widehat{C}_{E,a}(T)$ exactness test from Section 6 on an explicit ideal realizing that graph, and comparing the resulting closed set with $V(\chi_1\chi_5)$, would settle the classification and the underlying support-computation method.

Watch

Extended reading notes

Core claim

The central claim is that for a minimal regular presentation $Q/I$ with $I$ minimally generated by monomials in a regular sequence of $Q$, conditions (1) $R$ has an embedded deformation of codimension $c$, (2) $I/I^2$ has a free summand of rank $c$, (3) $\pi_2(R)$ contains a $c$-dimensional space of central elements, and (4) $\pi_2(R)$ contains a $c$-dimensional space of radical elements, are equivalent. The genuinely new implication is (4)$\Rightarrow$(1): a subspace of radical elements, without computing the whole infinite Lie algebra, forces an embedded deformation whose deforming regular sequence is a subset of the given monomial generators, with disjoint monomial support from the other generators. The paper also shows that containment of the cohomological support variety $\mathrm{V}_R(R)$ in a codimension-$c$ linear subspace is equivalent to these conditions. The authors note in the introduction that while this answers the question for monomial rings, a counterexample to the unrestricted local version has been found and will appear in future work.

Load-bearing premise

The load-bearing premise is Taylor's theorem that the Taylor complex on any list of monomials in a regular sequence resolves the quotient ring; if that acyclicity failed for these generators, the implication from support contained in a hyperplane to an embedded deformation would not go through.

Editorial extensions

If this is right

  • For any monomial ring, the rank of a free summand of $I/I^2$ equals the maximal codimension of an embedded deformation, so the conormal module carries exact deformation-theoretic information.
  • Degree-two central and radical elements of the homotopy Lie algebra coincide for these rings: a subspace of radical elements is automatically central.
  • Containment of $\mathrm{V}_R(R)$ in a linear subspace of codimension $c$ is equivalent to admitting an embedded deformation of codimension $c$, bridging support geometry and deformation theory.
  • Every nonzero complex of finite type over a monomial ring has cohomological support of dimension at least the complete intersection defect; in particular, the origin is not realizable as a support unless the ring is complete intersection.
  • For rings defined by at most five monomials, all possible support varieties are coordinate subspaces or, in one exceptional five-generator case, the union of two coordinate hyperplanes; which case occurs is read off from the GCD graph and one divisibility condition.

Reading between the lines

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

  • A direct consequence the authors do not spell out: for monomial rings, deciding embeddability is a finite combinatorial problem, since the Taylor graph is determined by the GCD and LCM lattice of the generators.
  • If the announced counterexample to the unrestricted local question is correct, the monomial setting becomes a natural boundary case: the equivalence holds exactly where a finite Taylor model computes both the Lie algebra and the support.
  • The $n\le 5$ classification suggests a testable pattern: supports are unions of coordinate subspaces for small GCD graphs, but the six-cycle example shows nonlinear supports appear at $n=6$, so the finite-varieties statement likely does not extend to a simple classification for larger $n$.
  • One could test whether the equivalence extends to ideals satisfying the broader gcd condition identified in Remark 5.4, which the authors note is the only property of monomials their lower-bound proof uses.
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

3 major / 6 minor

Summary. The paper studies local or positively graded rings defined by monomial ideals on a regular sequence, and proves an equivalence (Theorem A / Corollary 4.7) between the existence of embedded deformations of codimension c, the existence of a free summand of rank c in the conormal module, the existence of a c-dimensional subspace of degree-2 central elements in the homotopy Lie algebra, and the existence of a c-dimensional subspace of radical elements in π2(R). The proof passes through support varieties: radical elements force the support to lie in hyperplanes, and a new Theorem 4.5 shows that, for monomial ideals, containment of the support in a codimension-c linear subspace forces an embedded deformation. The paper also proves a lower bound (Theorem B) for the dimension of support varieties of complexes over such rings, and gives a finite classification (Theorem C) of the possible cohomological supports when the ideal is generated by at most five monomials.

Significance. The main theorem gives the first broad positive answer to Avramov's question for a class of rings that is not covered by the previously known cases, and it does so without computing the full homotopy Lie algebra. The strategy of using the Taylor complex and cohomological support varieties is natural and well executed. The paper is also honest about a forthcoming counterexample in the general local setting, which makes the monomial result well delineated. The classification for n ≤ 5 is concrete and the use of GCD and Taylor graphs is appealing. However, the proof of Theorem B contains a Loewy-length off-by-one error, and the proof of Lemma 4.3 applies an infinite resolution in a context where the quoted support computation is stated for bounded complexes; these issues require repair before the results can be considered fully established.

major comments (3)
  1. [Theorem 5.3, proof] The claim that ℓℓ_Λ(A) ≤ height(I) is false. For Q = k[x,y] and I = (x^2, xy, y^2), we have n = 3 and height(I) = 2, while A = T ⊗_Q k is the exterior algebra on b1, b2, b3 with b1·b2 = b2·b3 = 0 and b1·b3 ≠ 0; hence (Λ_+)^2 A ≠ 0 and (Λ_+)^3 A = 0, so ℓℓ_Λ(A) = 3. The proof only shows that products of length greater than height(I) vanish, which gives ℓℓ_Λ(A) ≤ height(I)+1. Consequently the quoted bound from [BGP24, Theorem 2.7] yields only dim V_R(M) ≥ n − height(I) − 1, so Theorem B is not proved as written.
  2. [Lemma 4.3, proof] The proof applies the construction of 2.7 and Proposition 2.8 to the Taylor model T[X], which is generally an infinite complex, whereas Proposition 2.8 is stated for a bounded complex of finite rank free Q-modules. Since Lemma 4.3(1) is used in the proof of Theorem 4.5, the argument for Theorem A depends on this step. The gap is repairable, for example by passing to a soft truncation of T[X] in sufficiently high degrees, but the manuscript should justify why the infinite T[X] may be used or should replace it with a bounded resolution.
  3. [Lemma 4.3(2), proof] The assertion that replacing f_j by g yields a minimal generating set is not justified: the condition g ∈ I ∖ (f_i, mI) does not imply that g is outside the span of the remaining f_k modulo mI, nor that the resulting set is minimal. This lemma is not used in the main theorems, but as stated it is part of the paper's toolkit and should be corrected or given a fuller proof.
minor comments (6)
  1. [Lemma 4.3(3)] Lemma 4.3(3) is false for n = 1: for a hypersurface R = Q/(f1), the hypothesis holds vacuously but V_R(R) = {0}, not A^1. The statement should assume n ≥ 2, and the proof of Theorem 6.14 for n = 1 should be handled separately.
  2. [Proposition 2.6] The statement begins 'Let R1 = Q/I1, R1 = Q/I1'; the second identity should presumably be R2 = Q/I2.
  3. [Corollary 4.7, proof] The word 'defomation' should be 'deformation'.
  4. [Example 1.1.10] 'acchieved' should be 'achieved'.
  5. [Remark 6.15] The phrase 'The the support variety' contains a duplicated article.
  6. [Theorem 6.16, proof] The displayed matrices for deven and dodd are difficult to read in the arXiv rendering; please clarify the conventions for boxed and circled entries and ideally present the two GCD graphs on separate displays.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found: the central implication (4)⇒(1) rests on standard Taylor acyclicity and independent prior results.

full rationale

The central theorem, Corollary 4.7, proves the implication (4)⇒(1) by means of Theorem 4.5. The load-bearing external input is Taylor's theorem [Tay66] that the Taylor complex on a list of monomials in a regular sequence is a free resolution; this is a standard, parameter-free theorem whose assumptions do not include the target result. The proof of Theorem 4.5 uses Lemma 4.3 to convert non-membership of basis vectors in V_R(R) into pairwise unit gcds, then uses the Taylor decomposition T(f) ≅ T′ ⊗ T″ to deduce that f_1,…,f_c is regular on Q/(f_{c+1},…,f_n); this is a genuine reduction, not an identity with the conclusion. The self-citations to [BGP24] (radical elements give hyperplanes containing V_R(R), and the Loewy-length inequality used in Theorem 5.3) and to [Pol21] (characterization of cohomological support varieties) are published results with stated, independent hypotheses; they do not presuppose the monomial deformation theorem, so under the instructions they are real evidence and do not raise the circularity score. The appended note that the authors have discovered a counterexample to Question 1 in the general local setting explicitly bounds the scope of the theorem and does not create circularity. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors, and no known empirical pattern is merely relabeled. The classification results in Section 6 are combinatorial consequences of the Taylor graph computation, not consequences of the theorem they are used to prove. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new entities. The central claims rest on standard background in commutative algebra and on two prior papers by the same authors ([BGP24], [Pol21]) for the support variety framework. These are published results with independent proofs, so they are recorded as assumptions rather than circular inputs.

assumptions (4)
  • domain assumption The Taylor complex on a list of monomials in a regular sequence is a free resolution (acyclic) of the quotient ring.
    Used in the proofs of Theorems 4.5, 5.3, and 6.8; cited to [Tay66] but not proven in the paper. This is a classical theorem, so we mark it as an external input.
  • domain assumption Cohomological support varieties are characterized by the vanishing of Ext over hypersurfaces Q_a, via the identification V_R(M) with the set of a where projdim_{Q_a}(M) is infinite.
    Used in Section 2.5 and Proposition 2.8. The paper cites [Pol21, Theorem 5.2.4] and the Nullstellensatz but does not reprove this characterization.
  • domain assumption Radical degree-2 elements in pi(R) correspond to hyperplanes containing V_R(R), and dim V_R(M) is bounded below by n - Loewy length, both from the authors' prior work [BGP24].
    These prior results (Theorems 3.1 and 2.7 of [BGP24]) are used as black boxes in Theorems 4.1 and 5.3. They are published and have independent proofs, but they are inputs to the present argument.
  • standard math Minimal dg algebra resolutions exist, the homotopy Lie algebra bracket is well-defined, and Ext_R(k,k) is the universal envelope of pi(R).
    Standard background summarized in Section 1.1, cited to [Avr10] and [AH87]. Not proven in the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The embedded deformation problem for monomial ideals." pith.science (2026). https://pith.science/paper/SWYKBKGS

@misc{pith2026250610827,
  author       = {Pith},
  title        = {Pith review of: The embedded deformation problem for monomial ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWYKBKGS}},
  note         = {Machine review of arXiv:2506.10827}
}
abstract

This article is concerned with homological properties of local or graded rings whose defining relations are monomials on some regular sequence. The main result of the article positively answers a question of Avramov for such a ring $R$. More precisely, we establish that an embedded deformation of $R$ corresponds exactly to a degree two central element in the homotopy Lie algebra of $R$, as well as a free summand of the conormal module of $R$. A major input in the proof is an analysis of cohomological support varieties. Other main results include establishing a lower bound for the dimension of the cohomological support variety of any complex over such rings, and classifying all possible subvarieties of affine $n$-space that are the cohomological support of rings defined by $n$ monomial relations where $n$ is five or less.

Figures

Figures reproduced from arXiv: 2506.10827 by the authors.

Figure 1
Figure 1. Taylor graph for (ab, bc, cd, de). The support of R might be full even if the Taylor graph of f has no isolated vertices. Example 6.11. Consider I = (xy, yz, xz) in Q = kJx, y, zK. The ring R = Q/I does not satisfy the hypothesis of Lemma 4.3 (3), and its Taylor graph has no isolated vertices: 123 23 13 12 1 2 3 ∅ [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Taylor graph for (xy, xz, yz) Nevertheless, one can easily see that the support of R is full in many ways: by noting that b{1,2} is a cycle but not a boundary in CbEa (T) for all a ∈ A 3 k , by doing direct calculations, using Macaulay2, or noting that R is Golod and thus has full support [BGP22, Theorem 4.1] [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. SPEARBench: A Benchmark for Naturalness Evaluation in Streaming Speech-to-Speech Language Models

    cs.CL 2026-07 conditional novelty 6.0 of 10

    An open-source benchmark for speech-to-speech models shows that current systems produce intelligible audio but diverge from human conversational behavior in latency, dialect consistency, emotional entrainment, and prosody.

Reference graph

Works this paper leans on

39 extracted references · 21 canonical work pages · cited by 1 Pith paper

  1. [1]

    Avramov and Ragnar-Olaf Buchweitz

    Luchezar L. Avramov and Ragnar-Olaf Buchweitz. Homological algebra modulo a regular sequence with special attention to codimension two. J. Algebra , 230(1):24--67, 2000

    work page 2000

  2. [2]

    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

    2000

  3. [3]

    Avramov and Stephen Halperin

    Luchezar L. Avramov and Stephen Halperin. On the nonvanishing of cotangent cohomology. Comment. Math. Helv. , 62(2):169--184, 1987

    1987

  4. [4]

    Avramov and Srikanth B

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

    2007

  5. [5]

    Avramov and Srikanth B

    Luchezar L. Avramov and Srikanth B. Iyengar. Restricting homology to hypersurfaces. In Geometric and topological aspects of the representation theory of finite groups , volume 242 of Springer Proc. Math. Stat. , pages 1--23. Springer, Cham, 2018

    2018

  6. [6]

    Avramov, Srikanth B

    Luchezar L. Avramov, Srikanth B. Iyengar, Saeed Nasseh, and Sean Sather-Wagstaff. Homology over trivial extensions of commutative DG algebras. Comm. Algebra , 47(6):2341--2356, 2019

    2019

  7. [7]

    Luchezar L. Avramov. Homology of local flat extensions and complete intersection defects. Math. Ann. , 228(1):27--37, 1977

    1977

  8. [8]

    Luchezar L. Avramov. Obstructions to the existence of multiplicative structures on minimal free resolutions. Amer. J. Math. , 103(1):1--31, 1981

    work page 1981

Show all 39 references
  1. [9]

    Luchezar L. Avramov. Golod homomorphisms. In Algebra, algebraic topology and their interactions ( S tockholm, 1983) , volume 1183 of Lecture Notes in Math. , pages 59--78. Springer, Berlin, 1986

    1983

  2. [10]

    Luchezar L. Avramov. Homological asymptotics of modules over local rings. In Commutative algebra ( B erkeley, CA , 1987) , volume 15 of Math. Sci. Res. Inst. Publ. , pages 33--62. Springer, New York, 1989

    1987

  3. [11]

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

    1989

  4. [12]

    Luchezar L. Avramov. Modules with extremal resolutions. Math. Res. Lett. , 3(3):319--328, 1996

    1996

  5. [13]

    Luchezar L. Avramov. Locally complete intersection homomorphisms and a conjecture of Q uillen on the vanishing of cotangent homology. Ann. of Math. (2) , 150(2):455--487, 1999

    1999

  6. [14]

    a user Class., pages 1--118. Birkh \

    Luchezar L. Avramov. Infinite free resolutions. In Six lectures on commutative algebra , Mod. Birkh \"a user Class., pages 1--118. Birkh \"a user Verlag, Basel, 2010

    2010

  7. [15]

    On support varieties for modules over complete intersections

    Petter Andreas Bergh. On support varieties for modules over complete intersections. Proc. Amer. Math. Soc. , 135(12):3795--3803, 2007

    2007

  8. [16]

    Visualize: interactive visualization and manipulation of combinatorial objects in a browser

    Brett Barwick, Thomas Enkosky, Branden Stone, Jim Vallandingham, and Doug Torrance. Visualize: interactive visualization and manipulation of combinatorial objects in a browser. Version 1.7 . A Macaulay2 package available at https://github.com/Macaulay2/M2/tree/master/M2/Macaul...

  9. [17]

    Constructing nonproxy small test modules for the complete intersection property

    Benjamin Briggs, Elo\' sa Grifo, and Josh Pollitz. Constructing nonproxy small test modules for the complete intersection property. Nagoya Math. J. , 246:412--429, 2022

    2022

  10. [18]

    Bounds on cohomological support varieties

    Benjamin Briggs, Elo\'isa Grifo, and Josh Pollitz. Bounds on cohomological support varieties. Trans. Amer. Math. Soc. Ser. B , 11:703--726, 2024

    2024

  11. [19]

    Local Commutative Algebra and H ochschild Cohomology Through the Lens of K oszul Duality

    Benjamin Briggs. Local Commutative Algebra and H ochschild Cohomology Through the Lens of K oszul Duality . PhD thesis, University of Toronto, (2018). https://www.math.utah.edu/ briggs/briggsthesis.pdf

    2018

  12. [20]

    Vasconcelos' conjecture on the conormal module

    Benjamin Briggs. Vasconcelos' conjecture on the conormal module. Invent. Math. , 227(1):415--428, 2022

    2022

  13. [21]

    Jesse Burke and Mark E. Walker. Matrix factorizations in higher codimension. Trans. Amer. Math. Soc. , 367(5):3323--3370, 2015

    2015

  14. [22]

    de Ridder et al

    H.N. de Ridder et al. 2001-2014. Information system on graph classes and their inclusions ( ISGCI )

    2001

  15. [23]

    Un diviseur de z\' e ro induisant un \' e l\' e ment d'homotopie central

    Nicolas Dupont. Un diviseur de z\' e ro induisant un \' e l\' e ment d'homotopie central. Bull. Soc. Math. France , 125(3):337--344, 1997

    1997

  16. [24]

    The radical of the homotopy L ie algebra

    Yves F\' e lix, Stephen Halperin, Carl Jacobsson, Clas L\" o fwall, and Jean-Claude Thomas. The radical of the homotopy L ie algebra. Amer. J. Math. , 110(2):301--322, 1988

    1988

  17. [25]

    Rational homotopy theory , volume 205 of Graduate Texts in Mathematics

    Yves F \'e lix, Stephen Halperin, and Jean-Claude Thomas. Rational homotopy theory , volume 205 of Graduate Texts in Mathematics . Springer-Verlag, New York, 2001

    2001

  18. [26]

    A study of graded extremal rings and of monomial rings

    Ralf Fr \"o berg. A study of graded extremal rings and of monomial rings. Math. Scand. , 51(1):22--34, 1982

    1982

  19. [27]

    Grayson and Michael E

    Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/

  20. [28]

    Gulliksen

    Tor H. Gulliksen. A proof of the existence of minimal R -algebra resolutions. Acta Math. , 120:53--58, 1968

    1968

  21. [29]

    Free summands of conormal modules and central elements in homotopy L ie algebras of local rings

    Srikanth Iyengar. Free summands of conormal modules and central elements in homotopy L ie algebras of local rings. Proc. Amer. Math. Soc. , 129(6):1563--1572, 2001

    2001

  22. [30]

    Jorgensen

    David A. Jorgensen. Support sets of pairs of modules. Pacific J. Math. , 207(2):393--409, 2002

    2002

  23. [31]

    R -sequences and homological dimension

    Irving Kaplansky. R -sequences and homological dimension. Nagoya Math. J. , 20:195--199, 1962

    1962

  24. [32]

    Vollst\" a ndige D urchschnitte und p - B asen

    Reinhardt Kiehl and Ernst Kunz. Vollst\" a ndige D urchschnitte und p - B asen. Arch. Math. , 16:348--362, 1965

    1965

  25. [33]

    Central elements and deformations of local rings

    Clas L \" o fwall. Central elements and deformations of local rings. J. Pure Appl. Algebra , 91(1-3):183--192, 1994

    1994

  26. [34]

    The derived category of a locally complete intersection ring

    Josh Pollitz. The derived category of a locally complete intersection ring. Adv. Math. , 354:106752, 18, 2019

    2019

  27. [35]

    Cohomological supports over derived complete intersections and local rings

    Josh Pollitz. Cohomological supports over derived complete intersections and local rings. Math. Z. , 299(3-4):2063--2101, 2021

    2021

  28. [36]

    Homologie des anneaux locaux noeth\' e riens

    Colette Schoeller. Homologie des anneaux locaux noeth\' e riens. C.R.Acad.Sci.Paris S\' e r.\@ A-B , 265:A768--A771, 1967

    1967

  29. [37]

    Subcategories of singularity categories via tensor actions

    Greg Stevenson. Subcategories of singularity categories via tensor actions. Compositio Mathematica , 150(2):229–272, 2014

    2014

  30. [38]

    Homology of N oetherian rings and local rings

    John Tate. Homology of N oetherian rings and local rings. Illinois J. Math. , 1:14--27, 1957

    1957

  31. [39]

    Ideal generated by monomials in an R -sequence

    Diana Kahn Taylor. Ideal generated by monomials in an R -sequence . ProQuest LLC, Ann Arbor, MI, 1966. Thesis (Ph.D.)--The University of Chicago

    1966

Pith tools

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