Homogeneous Liaison and the Sequentially Bounded Licci Property

Jesse Keyton

REVIEW 3 major objections 4 minor 29 references

Homogeneous Liaison and the Sequentially Bounded Licci Property

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Some licci ideals can never be linked by degree-bounded chains

desk verdict A promising negative answer to Chong's question, but the proof has a self-acknowledged gap in Proposition 3.10 and some loose ends around the N=3 reduction. read the letter →

arxiv 1908.00676 v1 pith:WWB2P56L submitted 2019-08-02 math.AC

Jesse Keyton This is my paper · ORCID

classification math.AC MSC 13C4013D02

keywords homogeneousliaisonlicciidealssequentiallyboundedminimalKoszulrelationsBettitablesEisenbud-Green-HarrisconjecturemonomialHilbertfunctions

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

In a polynomial ring, an ideal is homogeneously licci when a finite chain of homogeneous complete-intersection links leads it to a complete intersection. A stronger, numerically controlled version, the sequentially bounded licci (SBL) property, requires the degrees of the regular sequences used for successive links never to increase; a 2015 thesis question asked whether every homogeneously licci ideal is SBL. This paper answers no by constructing, for each n at least 4, a grade-3 ideal with an explicit Betti table that is homogeneously licci but not SBL. In three variables, it also shows each of these ideals has the same Hilbert function as a zero-dimensional monomial licci ideal, so the obstruction cannot be seen from Hilbert functions alone.

What carries the argument

The workhorse is the mapping-cone resolution for a homogeneous link, together with the invariant lambda(I) that counts the minimal Koszul relations among the minimal generators of a grade-3 ideal. When a complete intersection C sits inside an ideal I, the mapping cone writes down a (not necessarily minimal) resolution of the link C : I; the unit entries in the cone differentials occur precisely when generators of C are weak associates of minimal generators of I, or when two generators of C yield a minimal Koszul relation in I. Those unit entries decide which ghost summands trim from the resolution after a double link, which is what lets the paper control the Betti table after repeated or slightly reduced complete-intersection types. The proof packages this control into three Betti-table classes, (diamond), (star), and (star-star), and shows sequentially bounded double links stay inside them.

What would settle it

For n = 4, use the explicit ideal constructed in the proof of Theorem 5.4 that realizes the resolution (3.4), and run a computer algebra search through all homogeneous complete intersections with type bounded above by (2, 5, 12), computing each link with the mapping cone and iterating. The theorem predicts the search never reaches a complete intersection or an ideal of CM type 2; a single sequentially bounded chain that does would refute the main theorem. A second check: compute all minimal Koszul relations of an ideal with Betti table (3.2); any minimal relation other than the one between degrees n and n+2 would disprove the unproved claim inside Proposition 3.10.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.12: fix n at least 4 and let J be a grade-3 homogeneous ideal of a polynomial ring whose minimal graded free resolution has the shifts displayed in the paper as (3.4). Then J is homogeneously licci, yet no sequence of sequentially bounded homogeneous links takes J to a complete intersection. The proof is a Betti-table analysis of all possible double links: each sequentially bounded double link of J is shown to be another ideal of type (star), or of the related types (star-star) and (diamond), and all three types have CM type 3 and at most one minimal Koszul relation. Since a direct link to a complete intersection would require enough minimal Koszul relations, none of these ideals can be directly linked to a complete intersection, so the chain can never terminate. The same machinery shows that the previously known non-minimally licci ideals and their minimal links are SBL, which isolates the new class as the genuine obstruction.

Load-bearing premise

The load-bearing premise is the Betti-table classification of sequential double links: in particular, that an ideal with Betti table (3.2) has at most one minimal Koszul relation (the one between the degree n and n+2 generators), and that an ideal with resolution (3.1) contains a complete intersection of type (2, n+1, 2n+4); both are asserted without proof, and if either enumeration is wrong, a sequentially bounded link sequence might exist.

Editorial extensions

If this is right

Question 1.1 is settled in the negative: homogeneously licci does not imply sequentially bounded licci, even for grade-3 ideals in polynomial rings.
The SBL property is not a Hilbert-function invariant: in three variables each counterexample shares its Hilbert function with a zero-dimensional licci monomial ideal (Theorem 5.4).
The Betti table, not the Hilbert function, carries the obstruction: the counterexamples are defined by the resolution (3.4), and every sequentially bounded double link lands in one of the three Betti classes.
The mechanism gives a sufficient condition for failure of SBL: a homogeneously licci ideal of CM type 3 with at most one minimal Koszul relation cannot be sequentially bounded licci, because no direct link to a complete intersection is possible.

Reading between the lines

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

The paper leaves open whether ideals of type (star-star) are themselves SBL; if any is, that would sharpen exactly which portion of the Betti table, rather than the whole table, forces the obstruction.
The restriction n at least 4 arises because smaller n would let the double-link chain reach a complete intersection of type (1, n-1, 2n+3); testing the same construction for n = 3 with a modified type could reveal whether the phenomenon is unique to large shifts.
The Hilbert-function matching between a non-SBL ideal and a monomial licci ideal suggests that any search for Eisenbud-Green-Harris counterexamples should focus on graded Betti numbers and minimal Koszul relations rather than on Hilbert functions.
A computer algebra search over the n = 4 case could produce a fully explicit certificate of the linkage chain, independent of the paper's case analysis.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I'll give it to you straight. The paper does something worthwhile: it constructs a class of grade 3 homogeneous ideals with prescribed Betti tables that are homogeneously licci but not sequentially bounded licci, answering Chong's question in the negative. The companion result—same Hilbert function as a licci monomial ideal in three variables—is a nice addition and gives evidence on Huneke–Ulrich's monomial question.

The main technique is to use Ferrand's mapping cone and the λ-invariant (minimal Koszul relations) to control what can happen in a sequentially bounded double link. The Betti table analysis is involved and, as far as I can tell, mostly careful. The appendices do real work, and the author is transparent about the limits of what is shown.

Now the soft spots, in proportion.

First, the reader's stress-test is on target. Proposition 3.10 begins with the claim that any ideal satisfying (♦) has λ(L) ≤ 1, and says the proof is omitted. This is load-bearing: the whole argument that no direct link of L is a complete intersection, and hence the non-SBL conclusion, depends on it. If λ(L) were 2, a direct link to a CI would not be excluded. The author says 'one can show' but doesn't, anywhere in the manuscript. This is fixable, and probably true, but a referee needs to see the proof.

Second, the construction of the class (⋆) in Section 3.1 asserts that an ideal with resolution (3.1) contains a complete intersection of type (2, n+1, 2n+4). This is not proved at that point. Section 5 shows it for one explicit ideal, but the general statement is needed. If existence fails for some ideals, the class (⋆) could be smaller than advertised.

Third, the proof of Theorem 3.12.ii after Remark 3.14 assumes N=3, with a remark that larger N reduces to this case by Artinian reduction. That reduction is not shown, and the theorem is stated for all N≥3. The gap is probably patchable, but it should be explicit.

There are also many 'easily checked' Betti-table computations. They're probably right, but they make verification slow.

Who is this for? Commutative algebraists working in liaison theory, especially those interested in the hierarchy of licci properties and the EGH conjecture. They'll get a genuine new construction and a useful set of tools.

My recommendation: send it to a serious referee, but a conditional one. The central idea is sound and the answer to Chong's question is likely correct. The missing proof of Proposition 3.10 must be supplied, and the N=3 reduction and CI existence need to be addressed. This is serious work that deserves to be completed, not a desk reject.

Referee Report

3 major / 4 minor

Summary. The paper studies the sequentially bounded licci (SBL) property for homogeneous grade-3 ideals in polynomial rings. It first shows that the non-minimally licci ideals of Huneke–Migliore–Nagel–Ulrich, and their minimal links, are SBL. It then defines a class of ideals with a fixed Betti table, condition (⋆), and claims in Theorem 3.12 that every such ideal is homogeneously licci but not sequentially bounded licci, giving a negative answer to Chong's Question 1.1. For the three-variable case, Theorem 5.4 constructs, for each ideal satisfying (⋆), a zero-dimensional licci monomial ideal with the same Hilbert function. The proof is a long case analysis based on Ferrand's mapping cone, minimal Koszul relations, and grade-jump lemmas.

Significance. If the main theorem is correct, it provides the first infinite family separating 'homogeneously licci' from 'sequentially bounded licci,' a distinction that is directly relevant to the Eisenbud–Green–Harris conjecture through Chong's Theorem 1.3. The paper's strengths include a systematic use of Ferrand's mapping cone, explicit Betti-table invariants, and a concrete monomial-ideal construction in Section 5 that shows the SBL property is not detected by the Hilbert function. The overall strategy is coherent and does not rely on circular definitions or fitted parameters.

major comments (3)

[Proposition 3.10] The proof opens with the assertion that every ideal L with Betti table (3.2) satisfies λ(L) ≤ 1, the only possible minimal Koszul relation being between the degree n and n+2 generators, and it explicitly states that the proof is omitted. This assertion is load-bearing: Claim 1 uses it to rule out direct links of L to a complete intersection, Proposition 3.15 and the lemmas in Section 3 assume the analogous λ ≤ 1 for classes (⋆) and (⋆⋆), and the final paragraph of Theorem 3.12.ii concludes that a sequentially bounded double link J′ satisfies λ(J′) ≤ 1 and therefore cannot be directly linked to a complete intersection. If λ(L) were 2, or if the unique minimal Koszul relation had a different pair of degrees, the case analysis would not exclude a sequentially bounded sequence ending at a complete intersection. The manuscript must supply the missing proof, or a reference containing it, before the non-SBL conclusion is established.
[Remark 3.14 and Theorem 3.12] Theorem 3.12 is stated for all N ≥ 3, but from Remark 3.14 onward the proof assumes N = 3. The remark says that for N > 3 one may 'extend to an infinite base field and study the Artinian reduction of the ideals via linear forms,' but no argument is given that the SBL property, the classes (⋆), (⋆⋆), (♦), the existence of the relevant homogeneous complete intersections, or the Betti-table invariants are preserved under this reduction. As written, the proof of Proposition 3.15 and the later propositions covers only the three-variable case. The reduction must be made precise, or the statement of Theorem 3.12 must be restricted to N = 3.
[Section 3.1, paragraph after (3.1)] The text asserts, without proof, that an ideal I with resolution (3.1) contains a complete intersection D of type (2, n+1, 2n+4). This is not an immediate consequence of the displayed Betti table or of the cited grade-jump lemmas, which determine only the minimal type (2, n, 2n+4). The only explicit verification in the paper is for the concrete three-variable ideal in Section 5. Since the construction of the class (⋆) as direct links of the ideals of Theorem 3.4 depends on the existence of D, the paper should either prove this existence in general or clarify that the construction is only carried out for the specific ideals for which D is exhibited.

minor comments (4)

[Throughout] There are several typographical errors, including 'Deﬁntion' in Definition 2.7, 'the the main result' in the Introduction, 'Propositon' before Proposition 5.3, and 'impossibe' in Proposition B.4. These should be corrected.
[Theorem 3.12.i] In the proof of part (i), 'set I′ = C : I' should read 'set I′ = C : J', since C is a complete intersection inside J.
[Proposition 3.10 and Lemma 3.16] Several Betti-table computations are summarized as 'easily checked' or justified by phrases such as 'there are not enough ghost terms.' Given the case-based nature of the proof, these computations should be expanded or placed in an appendix so that the reader can verify them without redoing the entire Ferrand mapping cone analysis.
[Section 3.1] The existence of a complete intersection D of type (2, n+1, 2n+4) inside an arbitrary ideal with resolution (3.1) is asserted rather than proved; even if this is not needed for the main theorem's internal logic, the construction of the family should be unambiguous about which existence statements are being used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the main theorem is a structural Betti-table argument built on external mapping-cone results; the cited gaps are omitted proofs, not circularities.

full rationale

The derivation chain is self-contained with respect to the target claim. The class (⋆) is defined by a Betti table, namely the table of resolution (3.4), and the non-SBL conclusion is obtained by classifying sequentially bounded double links into the classes (⋆), (⋆⋆), and (♦), then showing none of these can be directly linked to a complete intersection while respecting the sequentially bounded degree constraint. No parameter is fitted from data, and the definition of (⋆) does not presuppose that J is not SBL. The main external inputs are Ferrand's mapping cone (cited to [25]) and the non-minimally licci ideals of Huneke–Migliore–Nagel–Ulrich (cited to [18]); these are independent supporting results, not self-citations that carry the conclusion. The paper does openly flag two load-bearing assertions whose proofs are omitted: in Proposition 3.10 it says 'although the proof is omitted, one can show this must be a minimal Koszul relation' when asserting λ(L) ≤ 1 for ideals with Betti table (3.2), and in Section 3.1 it asserts without proof that an ideal with resolution (3.1) contains a complete intersection of type (2, n+1, 2n+4), an existence claim later validated only for one explicit ideal in Theorem 5.4. These are gaps in justification and potential correctness risks, but they are not circular: the conclusions are not equivalent by construction to the hypotheses, and the omitted arguments are intended to derive structural facts from the Betti tables rather than assume the target property. Accordingly, the circularity score is 0.

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

No new entities are postulated; the paper only introduces Betti-table-defined classes of ideals (properties diamond, star, star-star), which are not invented objects but families of ideals. The axioms listed are the external results and standard background the central claim relies on.

assumptions (5)

standard math Ferrand's mapping cone theorem and Peskine-Szpiro linkage basics
Used throughout Section 2 and in every link resolution calculation; accepted as standard commutative algebra background.
domain assumption Theorem 3.4 from [18]: existence of homogeneous grade 3 ideals that are homogeneously licci but not minimally licci
The base ideals I with resolution (3.1) are taken from this cited construction; the current paper does not reprove it.
domain assumption Lemma A.3 ([18], Remark 3.1): any complete intersection contained in a perfect ideal of grade n has type sum at least the maximal shift in the last Betti module
Used repeatedly to identify minimal types and rule out complete intersections; cited from previous literature.
domain assumption Migliore-Nagel theorem that grade 3 homogeneous Gorenstein ideals are minimally licci
Used in Proposition 3.5 to conclude that ideals with r(R/I)=2 and lambda(I)=1 are SBL.
standard math Macaulay inverse system correspondence and Hilbert function formula for links (Prop 5.3)
Used in Section 5 to prove the monomial companion theorem.

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Cite this review

Pith. "Pith review of Homogeneous Liaison and the Sequentially Bounded Licci Property." pith.science (2026). https://pith.science/paper/WWB2P56L

@misc{pith2026190800676,
  author       = {Pith},
  title        = {Pith review of: Homogeneous Liaison and the Sequentially Bounded Licci Property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WWB2P56L}},
  note         = {Machine review of arXiv:1908.00676}
}

abstract

In CI-Liaison, significant effort has been made to study ideals that are in the linkage class of a complete intersection, which are called licci ideals. In a polynomial ring, recently E. Chong defined a "sequentially bounded" condition on the degrees of the forms generating the regular sequences, and used this condition to find a large class of licci ideals satisfying the Eisenbud-Green-Harris Conjecture (among them, grade $3$ homogeneous Gorenstein ideals). He raised the question of whether all homogeneous licci ideals are sequentially bounded licci. In this paper we construct a class of examples that are homogeneously licci, but not sequentially bounded licci, thus answering his question in the negative. The structure of certain Betti tables plays a central role in our proof.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Brown, A structure theorem for a class of grade three perfect ideals , J

A.E. Brown, A structure theorem for a class of grade three perfect ideals , J. Algebra 105 (1987), 308-327

work page 1987

[2] [2]

Buchweitz, Contributions à la thèorie des singularités , Thesis, l’Université de Paris, 1981

R.-O. Buchweitz, Contributions à la thèorie des singularités , Thesis, l’Université de Paris, 1981

work page 1981

[3] [3]

Chong, Face vectors and Hilbert functions , Thesis, Cornell Univ., 2015

K.F.E. Chong, Face vectors and Hilbert functions , Thesis, Cornell Univ., 2015. HOMOGENEOUS LIAISON AND THE SEQUENTIALLY BOUNDED LICCI PRO PERTY 33

work page 2015

[4] [4]

Chong, An application of liaison theory to to Eisenbud-Green-Harris c onjecture, J

K.F.E. Chong, An application of liaison theory to to Eisenbud-Green-Harris c onjecture, J. Alge- bra 445 (2016), 221-231

work page 2016

[5] [5]

Eisenbud, M

D. Eisenbud, M. Green, and J. Harris, Cayley-Bacharach theorems and conjectures , Bull. Amer. Math. Soc. 33 (1996), 295-324

work page 1996

[6] [6]

Gaeta, Détermination de la chaine syzygétique des idéaux matriciels parfaits et son application à la posulation de leurs variétés algébriques associées , C.R

F. Gaeta, Détermination de la chaine syzygétique des idéaux matriciels parfaits et son application à la posulation de leurs variétés algébriques associées , C.R. Acad. Sci. Paris 234 (1954), 1833- 1835

work page 1954

[7] [7]

Geramita, T

A.V. Geramita, T. Harima, J.C. Migliore, and Y.S. Shin, The Hilbert function of a level algebra , Mem. Am. Math. Soc. 186 (2007) vi+139

work page 2007

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Hartshorne, Some examples of gorenstein liaison in codimension three , Collect

R. Hartshorne, Some examples of gorenstein liaison in codimension three , Collect. Math. 53 (2002), 21-48

work page 2002