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Restrictions on the Betti tables of licci ideals

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Pith's one-line read The paper proves that for many homogeneous licci ideals the minimal number of generators is at most the largest shift in the last module of the minimal free resolution, and conjectures the bound always holds.

desk verdict Solid new conjectures and partial proofs for Betti-table bounds on licci ideals; the main theorems lean on two cited bounds that a referee should check. read the letter →

arxiv 2608.06019 v1 pith:5CAYJ7RH submitted 2026-08-06 math.AC

Craig Huneke , Claudia Polini , Bernd Ulrich This is my paper · ORCID
classification math.AC MSC 14C2013A3014M1014J17
keywords licciidealslinkageBettitablesminimalfreeresolutionnumberofgeneratorsdeviationregularityBoij-Söderbergtheory
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

Licci ideals—ideals obtained from complete intersections by repeated linkage—form a central class in commutative algebra, but their Betti tables are not fully understood. This paper introduces several conjectures comparing the number of minimal generators of a homogeneous licci ideal to the largest shift appearing in the last module of its minimal free resolution. The main conjecture states that $\mu(I) \le T_c(S/I)$ for a licci ideal of codimension $c$, equivalently that the deviation of $I$ is at most its regularity; the paper proves this in a large number of cases, including codimension two Cohen–Macaulay ideals, codimension three Gorenstein ideals, licci ideals with nearly pure resolutions, equigenerated Gorenstein monomial ideals, and ideals containing a maximal regular sequence of quadrics. A section on sums of links shows that the Gorenstein case would imply strong evidence for the general conjecture. If true, these restrictions would confirm the earlier expectation that square-free monomial licci ideals are of linear type with Cohen–Macaulay Rees algebras.

What carries the argument

The argument is carried by the interaction between two numerical bounds on the shifts of the minimal free resolution of a licci ideal $I$ of codimension $c$. The first, cited from the structure of linkage, gives $T_c(S/I) > (c-1)t_1(S/I)$; the second, cited from the regularity of Tor, gives $T_c(S/I) \le T_k(S/I)+(c-k)T_1(S/I)$ for each $k$. Together these force the inequalities $t_k > (k-1)T_1$ needed in the pure-diagram argument. The Boij–Söderberg decomposition expresses the Betti table of $S/I$ as a positive rational combination of pure diagrams bounded between the minimal and maximal shift sequences; reducing the generator bound to a pointwise inequality on pure diagrams yields $\mu(I) \le (c-1)t_1+1$, which the licci shift bound then upgrades to $\mu(I) \le T_c(S/I)$. In low codimension, Hilbert–Burch and Buchsbaum–Eisenbud supply the structural facts, and the sum-of-links construction $L=I+J$ (always Gorenstein of codimension $c+1$) connects the deviations of $I$, its link $J$, and $L$ through the identity $\Delta(I)+\Delta(J)-\Delta(L)=T_c(S/I)-t_1(S/I)-c$.

What would settle it

Find a licci ideal whose minimal free resolution has last module concentrated in degrees strictly below the number of minimal generators; the conjecture would be false. This can be checked by computing the Betti table of any licci ideal, for instance by linking a complete intersection, and comparing $\mu(I)$ with $T_c(S/I)$.

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Extended reading notes

Core claim

The central assertion of the paper is Conjecture 1.2: for every homogeneous licci ideal $I$ of codimension $c$ in a polynomial ring over a field, the minimal number of generators $\mu(I)$ is at most $T_c(S/I)$, the maximal degree occurring in the last syzygy module of $S/I$. This is equivalent—after cutting down by a regular sequence of linear forms—to the local statement that $m^{d(I)} \not\subset I$ for an $m$-primary licci ideal, where $d(I)=\mu(I)-c$ is the deviation. The authors prove this conjecture for several families: in codimension two for all Cohen–Macaulay ideals via Hilbert–Burch; in codimension three for Gorenstein ideals via the Buchsbaum–Eisenbud structure theorem together with the fact that non-hypersurface Golod Gorenstein rings cannot occur; for licci ideals with nearly pure resolutions and for equigenerated Gorenstein monomial ideals via the Boij–Söderberg decomposition combined with two bounds on Betti shifts; and for ideals containing a maximal regular sequence of quadrics by induction using links and sums of links. They also prove that if the conjecture holds for the Gorenstein ideal obtained as the sum of links of an ideal and its link, then it holds for at least one of the two linked ideals.

Load-bearing premise

The main theorems lean on two cited inequalities about the degrees appearing in the minimal free resolution of licci ideals; if either of those inequalities fails for some licci ideal, the proofs in this paper would not carry through, although the conjectures might still be true.

Editorial extensions

If this is right

  • For every codimension-two Cohen–Macaulay ideal and every codimension-three Gorenstein ideal, the local form of the conjecture holds: $m^{d(I)} \not\subset I$, bounding the deviation by the regularity.
  • Licci ideals with nearly pure resolutions and equigenerated Gorenstein monomial licci ideals satisfy the stronger inequality $\mu(I) \le (c-1)t_1(S/I)+1$, which implies $\mu(I) \le T_c(S/I)$.
  • If the conjecture holds for Gorenstein licci ideals, then for any licci ideal $I$ with link $J$ and Gorenstein sum $L$, at least one of $I$ or $J$ satisfies the conjecture; moreover the identity relating the three deviations shows that the Gorenstein case controls the general case.
  • Every licci ideal containing a maximal regular sequence of quadrics has deviation at most its socle degree, so the conjecture holds for that whole family.
  • The tables assembled from the cited classifications of licci monomial and binomial edge ideals confirm the conjecture for every licci ideal in those lists, while the non-licci examples in the lists fail it.

Reading between the lines

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

  • The two shift bounds used here are not proved in the paper; a direct proof or counterexample for them would sharpen the whole program, since the Boij–Söderberg reduction in Theorem 3.1 works for any ideal satisfying $t_k > (k-1)T_1$.
  • The sum-of-links identity suggests an inductive strategy on codimension: proving the Gorenstein case might let one pass to arbitrary licci ideals by linking, reducing the conjecture to checking a $t_1$-to-deviation inequality.
  • Conjecture 1.2, if true, would imply the square-free monomial case Conjecture 1.1, so the linear-type and Cohen–Macaulay Rees algebra statements for those ideals would follow from a purely numerical bound on resolutions.
  • A systematic computer search over licci ideals of small codimension could directly test Conjecture 1.2 beyond the families proved here, since the Betti tables of linked ideals are computable.
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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

1 major / 4 minor

Summary. Let S be a polynomial ring over a field. For a homogeneous licci ideal I of codimension c, the paper introduces Conjecture 1.2, asserting that μ(I) ≤ T_c(S/I), where T_c(S/I) is the maximal shift in the last module of the minimal free resolution of S/I; it also gives the square-free monomial Conjecture 1.1 and the local m-primary Conjecture 1.3. The authors prove these conjectures in several cases: codimension-two Cohen-Macaulay ideals (Theorem 2.1), codimension-three Gorenstein ideals (Theorem 2.2), licci ideals with nearly pure resolutions (Theorem 3.2), equigenerated Gorenstein monomial licci ideals (Corollary 3.4), and licci ideals containing a maximal regular sequence of quadrics (Theorem 5.2). They also prove the stronger bound b_1 ≤ (c−1)t_1 + 1 in these cases. Section 4 relates the invariants Δ(I), Δ(J), and Δ(L) for an ideal, its link, and their sum, and shows that the Gorenstein case of the conjecture implies the conjecture for at least one of I or J.

Significance. If correct, these conjectures express a new numerical restriction on the Betti tables of licci ideals, complementing the known Cohen-Macaulay and strongly Cohen-Macaulay properties. The partial results cover a wide range of known licci classes, and the proofs use Boij–Söderberg decompositions, mapping-cone resolutions of links, and Golod/Gorenstein structure theory in a transparent way. The authors are explicit about what is conditional: Section 4 isolates the Gorenstein case as a hypothesis, and the tables in Section 6 provide concrete checks on classified families. The arguments are reproducible from the cited literature, and I found no internal inconsistency in the main derivations.

major comments (1)
  1. [§3, Theorems 3.1–3.3; §5, Theorem 5.2] The proofs of Theorems 3.1, 3.2, 3.3, and 5.2 depend on two external numerical inequalities that are cited but never stated precisely: the licci bound T_c(S/I) > (c−1)t_1(S/I) from [23, Corollary 5.13] and the bound T_c ≤ T_k + (c−k)T_1 from [10, Corollary 4.2]. These inequalities are load-bearing for deriving (c−1)t_1 + 1 ≤ T_c and for propagating the bound through intermediate homological degrees, so the manuscript should state their exact hypotheses and conclusions in the graded setting and confirm that they apply to every homogeneous licci ideal, not only to equigenerated or Gorenstein ideals. If either bound has an unstated hypothesis or is misstated, the deductions in these theorems do not survive; the conjectures themselves might still be true, but the present proofs would need repair.
minor comments (4)
  1. [Theorem 2.2] The phrase 'I is neither the unit ideal norm 2' appears to contain a typo; it should presumably read 'nor m^2'.
  2. [Theorem 5.2] In the final paragraph of the proof, 'K contains no linear forms' should read 'K′ contains no linear forms,' since K contains the linear forms x_{n−a−b+1},...,x_n by construction.
  3. [Theorem 3.3] The sentence 'Hence ... for all 0 ≤ k ≤ c−1' should either be restricted to 2 ≤ k ≤ c−1 or reconciled with the fixed integer k in the preceding sentence, since the displayed inequality was derived for 2 ≤ k ≤ c−1.
  4. [Introduction] The sentence 'using AI prepared a table' would benefit from a brief explanation of how the table was generated and rechecked, for reproducibility; the tables themselves are a useful supplement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proofs depend on cited prior theorems, not on the conjectures they aim to establish.

full rationale

The derivation chain is self-contained relative to its cited external theorems. Conjectures 1.1–1.3 are introduced as open statements and are never assumed in their own proofs. Theorem 2.1 uses the Hilbert–Burch theorem, and Theorem 2.2 uses the Buchsbaum–Eisenbud structure theorem together with Lofwall's Golod criterion; neither argument invokes the conjectures. Theorem 3.1 proves the bound b_1 ≤ (c−1)t_1 + 1 from the Boij–Söderberg decomposition under the explicitly stated hypothesis t_k > (k−1)T_1, and then invokes [23, Corollary 5.13] only for the final comparison (c−1)t_1 + 1 ≤ T_c. Theorems 3.2 and 3.3 combine [23, Corollary 5.13] and [10, Corollary 4.2] with structural assumptions about nearly pure resolutions or Gorenstein duality; these are published theorems with stated hypotheses and are not restatements of Conjecture 1.2. Section 4 explicitly assumes Δ(L) ≥ 0, i.e., the Gorenstein case of the conjecture, and derives conditional implications for I and J; this is a reduction of the general case to a special case, not a circular proof. Theorem 5.2 is an induction using [23, Corollary 5.13(a)] and [24, Proposition 3], and no fitted parameter is renamed as a prediction. The tables in Section 6 verify known classifications of licci ideals against the conjecture; they are not used to derive the conjecture. The only notable feature is that two of the load-bearing inequalities come from the authors' earlier work, but those are independent published results and nothing in the text shows that they encode the target conjecture. The skeptic's concern about the hypotheses of [23, Corollary 5.13] and [10, Corollary 4.2] is a correctness or robustness contingency, not a circularity. Consequently, no circular step can be quoted from the paper, and the appropriate circularity score is 0.

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

The central claims rest on established theorems in commutative algebra, chiefly the numerical bounds from [23] and [10] and the Boij-Soderberg decomposition. The paper introduces no fitted constants, no new algebraic objects beyond the conjectures themselves, and no entities with independent evidence requirements.

assumptions (7)
  • domain assumption Boij-Soderberg decomposition of Betti tables of Cohen-Macaulay modules into positive rational combinations of pure diagrams
    Used in Theorem 3.1 to reduce the bound on b_1 to a bound on each pure diagram; a standard theorem from Eisenbud-Schreyer, reference [11].
  • domain assumption Numerical bound T_c(S/I) > (c-1) t_1(S/I) for licci ideals I of codimension c
    Load-bearing external result from Huneke-Ulrich, Corollary 5.13 of reference [23], used in Theorems 3.1, 3.2, 3.3 and 5.2 to produce the final inequality and to establish hypotheses.
  • domain assumption Inequality T_c <= T_k + (c-k) T_1 for the shifts of a licci ideal
    Cited to Eisenbud-Huneke-Ulrich, Corollary 4.2 of reference [10], used in Theorem 3.2 to propagate the licci bound from homological degree c to intermediate degrees.
  • domain assumption Buchsbaum-Eisenbud structure theorem for codimension three Gorenstein ideals
    Used in Theorem 2.2 to get I subset m^n when mu(I) = 2n+1, from the Pfaffian structure of the resolution.
  • domain assumption Avramov's classification of Golod Gorenstein rings with I subset m^2 as hypersurface rings
    Used in Theorem 2.2 to derive a contradiction in the codimension three Gorenstein case.
  • domain assumption Linkage theory results of Peskine-Szpiro and Ulrich, including that sums of linked ideals are Gorenstein and licci when the input is licci
    Used throughout Section 4 to relate the invariants Delta(I), Delta(J) and Delta(L).
  • standard math Krull's Altitude Theorem
    Used in the proof of Theorem 5.2 to bound the dimension of the image of (gamma) in K/mK.

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Pith. "Pith review of Restrictions on the Betti tables of licci ideals." pith.science (2026). https://pith.science/paper/5CAYJ7RH

@misc{pith2026260806019,
  author       = {Pith},
  title        = {Pith review of: Restrictions on the Betti tables of licci ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5CAYJ7RH}},
  note         = {Machine review of arXiv:2608.06019}
}
read the original abstract

We introduce several conjectures which mainly deal with restrictions on the Betti tables of licci ideals. We focus on a series of questions that compare the number of generators of homogeneous licci ideals in polynomial rings to the maximal last shift in their graded free resolution. We prove these conjectures in a large number of cases.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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