REVIEW 1 major objections 4 minor 22 references
Principal vector-spread Borel ideals
T0 review · 1 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The paper proves that for a squarefree principal vector-spread Borel ideal, symbolic powers equal ordinary powers exactly when the generator's indices satisfy $j_i \le t_1+\cdots+t_i$ for all $i$, through an explicit primary decomposition…
desk verdict Solid structural results with a genuine, fixable gap in the main classification proof. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The t-spread support of a monomial $v = x_{i_1}\cdots x_{i_\ell}$ is the union of intervals $[i_s, i_s + t_s - 1]$; these sets are the building blocks for the facets of the Stanley-Reisner complex whose ideal is $B_t(u)$. The load-bearing mechanism is the complete description of these facets (equation (1.1)): every facet is either the t-spread support of a minimal generator or a set $\operatorname{supp}_t(x_{\ell_1}\cdots x_{\ell_{s-1}}x_{j_s}) \cup [j_s+1,n]$. From that description flow the minimal primary decomposition, the vertex-splitting decomposition of the Alexander dual ideal in Lemma 2.3, and the enumeration of associated primes used throughout. Here vertex splitting means a recursive construction of a monomial ideal as $x_i I_1 + I_2$ with $I_2 \subseteq I_1$, a property that guarantees the linear-resolution behavior needed for the sequential Cohen-Macaulay conclusion. The proof machinery for the classification also includes monomial localization and the a-restriction criterion, which reduce the symbolic-power equality to a smaller ideal after removing one index of the generator.
What would settle it
Enumerate the facets of the Stanley-Reisner complex for an example such as $t=(2,2,1)$, $u=x_3x_6x_{10}x_{13}$ in $K[x_1,\ldots,x_{13}]$, and compare the list with the description in Theorem 1.2; any missing or spurious facet would falsify the primary decomposition. To test Theorem 3.1 directly, compute the ordinary square and the second symbolic power of an ideal satisfying $j_i \le \sum_{s=1}^i t_s$ for all $i$, for instance $t=(3,2,1)$, $u=x_4x_9x_{13}x_{15}$ in $K[x_1,\ldots,x_{15}]$; if they differ, the classification collapses.
Extended reading notes
Core claim
The central claim is Theorem 3.1: for a squarefree principal vector-spread Borel ideal $I = B_t(u)$, the properties 'normally torsionfree', '$I^{(k)} = I^k$ for all $k \ge 1$', and the inequalities $j_i \le \sum_{s=1}^i t_s$ for $i = 1,\ldots,d-1$ are equivalent. The proof rests on Theorem 1.2, which describes the minimal primary decomposition as the intersection of primes $P_{[n]\setminus G}$ over two explicitly listed families of subsets, and on Theorem 2.1, which establishes sequential Cohen-Macaulayness. To prove Theorem 2.1 the authors show the Alexander dual ideal $I^\vee$ — the squarefree monomial ideal generated by the facet complements of the Stanley-Reisner complex — admits the vertex-splitting decomposition $x_1 B'_t(u)^{\vee} + B'_{t'}(u/x_{j_1})^{\vee}$, allowing induction on degree and number of variables. To prove Theorem 3.1 they combine a monomial-localization reduction with the a-restriction criterion for symbolic powers. The paper claims these results hold for every vector $t_1,\ldots,t_{d-1} \ge 1$, extending previously known cases such as $d = 2$.
Load-bearing premise
The proof that the list of sets in Theorem 1.2 exhausts the facets of the Stanley-Reisner complex of $B_t(u)$ is the load-bearing step; if some facet were missed, the primary decomposition, the Alexander dual computation, and the final classification would not follow.
Editorial extensions
If this is right
- Deciding normal torsion-freeness for this class requires only checking finitely many integer inequalities on the generator's indices; no primary decomposition is needed in practice.
- Every squarefree principal vector-spread Borel ideal is sequentially Cohen-Macaulay, and its Stanley-Reisner complex is vertex decomposable, so the class inherits the homological and combinatorial consequences of those properties.
- The explicit primary decomposition gives a closed-form list of associated primes and the height of the ideal, which is the smallest index of the generator.
- The classification covers arbitrary spread vectors, subsuming the known one-dimensional and squarefree strongly stable cases.
- The induction pattern used in the proof — remove one index of the generator and pass to a smaller spread vector — provides a recursive certificate for the equality of powers.
Reading between the lines
- The contradiction argument in the proof targets the second symbolic power, so in this class the full equality of symbolic and ordinary powers may already be forced by the single equality of the ordinary square and the second symbolic power; a reader could test whether condition (c) is equivalent to that single equality.
- The vertex-splitting recursion used here is limited to squarefree ideals, but the same recursion suggests a route into the non-squarefree case that the paper leaves open, provided the t-spread support construction is replaced by a suitable non-squarefree analogue.
- The analytic spread analysis in Proposition 3.6 implies that under condition (c) the maximal ideal is never an associated prime of any power; examining the depths of the quotients $S/I^k$ might reveal stronger homological stabilization than the paper states.
- Because the class is parameterized by an arbitrary spread vector and a generator, it gives a two-parameter family of monomial ideals with a closed-form normal-torsion-freeness answer, which could serve as a test bed for more general criteria.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies squarefree principal vector-spread Borel ideals B_t(u), where t=(t_1,...,t_{d-1}) and u is a t-spread monomial of degree d. Its main results are: Theorem 1.2 gives an explicit minimal primary decomposition in terms of t-spread supports; Theorem 2.1 proves these ideals are sequentially Cohen-Macaulay by showing their Alexander duals are vertex splittable; Theorem 3.1 classifies when such ideals satisfy I^(k)=I^k for all k, by showing this is equivalent to the inequalities j_i ≤ t_1+...+t_i for all i<d. The paper also records corollaries on height, vertex decomposability, and non-finiteness of associated primes.
Significance. If Theorem 3.1 is correct, the paper provides a complete and elegant classification of normally torsionfree ideals in this class, and the explicit primary decomposition in Theorem 1.2 is independently valuable and likely to be useful for further work on vector-spread Borel ideals. The sequential Cohen-Macaulayness result in Theorem 2.1 is also a natural strengthening of known properties of Borel ideals. The proof of Theorem 3.1, however, contains a false intermediate assertion in the implication (b)=> (c), and that implication is the core of the classification. The other parts of the paper appear sound to me; in particular, I did not find a concrete gap in the facet-exhaustion argument of Theorem 1.2, although that proof is terse. The paper also reports computational verification of examples, but the main theorems do not rely on those computations.
major comments (1)
- [§3, proof of Theorem 3.1, (b)⇒(c), around Eq. (3.3)] The assertion 'Since p2 ≥ p1 + t1 it follows that p2 ∈ {j2,...,jd}' is false in general. For example, take d=3, t=(3,1), and u=x4x7x8 in K[x1,...,x8]. Then the proof's monomial is w=(x1x2x3x4)(x7x8), and v1=x1x4x7 is a generator of B_t(u): it is t-spread and satisfies the upper bounds p1=1≤4, p2=4≤7, p3=7≤8. It divides w, yet p2=4=t1+1 is not in {j2,j3}={7,8}. Thus the claim that p2 belongs to the right-hand factor is not valid, and the proof does not rule out w∈I^2 as written. Since this is the only step that excludes w from I^2, the implication (b)⇒(c) is not established by the given argument. The theorem may still be true—degree and support considerations rule out this particular example—but a corrected argument is required.
minor comments (4)
- [Lemma 2.3, text near Eq. (2.1)] The expression 'B′t′(u/j1)' should read 'B′t′(u/x_{j1})', consistent with the notation in Eq. (2.1).
- [Theorem 1.2, converse direction] The proof that every face G is contained in one of the listed facets is rather compressed; in the first case it is not immediately clear which generator v has t-spread support equal to the containing set. I verified that v = x_{ℓ1}⋯x_{ℓk}x_{j_{k+1}}⋯x_{jd} works, so this is a readability issue rather than a mathematical gap, but the generator should be written explicitly.
- [§3, proof of Theorem 3.1, (c)⇒(a)] In the definition of ℓ, the condition 'jp − jp−1 = tp−1' is used for p=1, but j0 is never defined. The range of p should be adjusted or j0 should be declared, to avoid ambiguity.
- [Throughout] There are several minor typographical slips, such as 'witht-spread Borel generators' in the Introduction; these should be corrected in a final proofread.
Circularity Check
No significant circularity: the classification is derived from definitions and independent external theorems; the same-author citations are auxiliary tools, not fitted inputs or self-justifying premises.
full rationale
The paper's main results (Theorems 1.2, 2.1, 3.1) are established by direct arguments from the definition of t-spread strongly stable ideals, Stanley-Reisner theory, Alexander duality, vertex splitting, and standard results in monomial ideal theory. Theorem 1.2 explicitly computes the facets of the Stanley-Reisner complex and derives the minimal primary decomposition; no parameter is fitted and no target property is assumed. Theorem 2.1 follows from Lemma 2.3, which proves the Alexander dual is vertex splittable by induction using the facet description, and then from the independent theorem on componentwise linearity. Theorem 3.1 reduces (a) and (b) via the standard equivalence with normal torsion-freeness, and its proof of (c) implies (a) uses Theorem 3.2 from [21] plus an induction that re-applies the theorem only to restrictions; these are genuine reductions. The citations to the authors' earlier work ([3], [4], [12], [17]) are used as tools or as points of comparison, not as the source of the target equivalence. In particular, [4] is invoked in Corollary 3.5 only to obtain vertex splittability and thereby the Cohen-Macaulayness of the Rees algebra via [19]; this is an auxiliary external input and does not define the classified condition. No step of the form 'fitted parameter renamed as prediction' or 'defined in terms of the target' occurs. A separate proof gap in the (b)⇒(c) direction has been noted by a reader: the case p2 = t1+1 is not covered by the claim 'Since p2 ≥ p1 + t1 it follows that p2 ∈ {j2,...,jd}'. That is a correctness issue in the write-up, not a circularity, so it does not affect the circularity score.
Assumptions & free parameters
assumptions (6)
- standard math Stanley-Reisner correspondence: for a squarefree monomial ideal I, there is a unique simplicial complex Delta with I = I_Delta, and the minimal primary decomposition is I = intersection_{F in F(Delta)} P_{[n]\F}.
- standard math A squarefree ideal I is sequentially Cohen-Macaulay if and only if its Alexander dual I^∨ is componentwise linear.
- standard math Vertex splittable ideals have linear quotients and are componentwise linear.
- domain assumption For a vertex splittable ideal (or vertex decomposable complex), the Rees algebra is Cohen-Macaulay and associated primes ascend.
- standard math The analytic spread of a monomial ideal with linear relations is computed by its linear relation graph as ℓ(I) = r - s + 1.
- standard math Criterion for normal torsionfreeness via a squarefree monomial v in P \ P^2 for all P in Ass(I) and restrictions I^{<=1-e_i} being normally torsionfree.
Cite this review
Pith. "Pith review of Principal vector-spread Borel ideals." pith.science (2026). https://pith.science/paper/DRG3SOFE
@misc{pith2026250707022,
author = {Pith},
title = {Pith review of: Principal vector-spread Borel ideals},
year = {2026},
howpublished = {\url{https://pith.science/paper/DRG3SOFE}},
note = {Machine review of arXiv:2507.07022}
}
read the original abstract
We study the class of squarefree principal vector-spread Borel ideals. We compute the minimal primary decomposition of these ideals and thereby we prove that they are sequentially Cohen-Macaulay. As the final conclusion of our results, we completely classify the ideals in our class having the property that their ordinary and symbolic powers coincide.
Reference graph
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Reviewed August 6, 2026 · model on record in the stance chip above.
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