Squarefree Principal Vector-Spread Borel Ideals

• The paper presents an explicit minimal primary decomposition for squarefree principal vector-spread Borel ideals by analyzing t-spread supports and tail facets.
• It employs a recursive Alexander-dual splitting strategy that proves sequential Cohen–Macaulayness through vertex decomposability and linear quotient techniques.
• A complete criterion is established for the equality of ordinary and symbolic powers, linking spread constraints to normal torsionfreeness.

Searching arXiv for the most relevant papers on squarefree principal vector-spread Borel ideals and closely related classes. Squarefree principal vector-spread Borel ideals are principal ${\bf t}$-spread strongly stable monomial ideals in which the spread vector ${\bf t}=(t_1,\dots,t_{d-1})$ has positive entries, so the generators are squarefree and satisfy position-dependent gap constraints. In the modern formulation, one works in $S=K[x_1,\dots,x_n]$, fixes ${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$, and starts from a ${\bf t}$-spread monomial $u=x_{j_1}\cdots x_{j_d}$ with $j_{k+1}-j_k\ge t_k$; the associated principal ideal $B_{\bf t}(u)$ is the smallest ${\bf t}$-spread strongly stable ideal containing $u$ (Crupi et al., 9 Jul 2025). This squarefree vector-spread framework specializes both ordinary strongly stable theory when ${\bf t}=(t_1,\dots,t_{d-1})$0 and squarefree strongly stable theory when ${\bf t}=(t_1,\dots,t_{d-1})$1 (Crupi et al., 2022, Crupi et al., 2023). Recent work develops a dedicated theory for the squarefree principal case, including minimal primary decomposition, sequential Cohen–Macaulayness, and a complete classification of when symbolic and ordinary powers coincide (Crupi et al., 9 Jul 2025).

1. Definition and ambient framework

The ambient ring is the standard graded polynomial ring

${\bf t}=(t_1,\dots,t_{d-1})$2

A monomial

${\bf t}=(t_1,\dots,t_{d-1})$3

is called ${\bf t}=(t_1,\dots,t_{d-1})$4-spread, for ${\bf t}=(t_1,\dots,t_{d-1})$5, if

${\bf t}=(t_1,\dots,t_{d-1})$6

When ${\bf t}=(t_1,\dots,t_{d-1})$7 for all ${\bf t}=(t_1,\dots,t_{d-1})$8, every ${\bf t}=(t_1,\dots,t_{d-1})$9-spread monomial is squarefree (Crupi et al., 9 Jul 2025, Crupi et al., 2023).

A monomial ideal $S=K[x_1,\dots,x_n]$0 is a $S=K[x_1,\dots,x_n]$1-spread monomial ideal if every minimal generator is $S=K[x_1,\dots,x_n]$2-spread. It is $S=K[x_1,\dots,x_n]$3-spread strongly stable if for every $S=K[x_1,\dots,x_n]$4-spread monomial $S=K[x_1,\dots,x_n]$5, and any $S=K[x_1,\dots,x_n]$6 such that $S=K[x_1,\dots,x_n]$7 and $S=K[x_1,\dots,x_n]$8 is again $S=K[x_1,\dots,x_n]$9-spread, one has

${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$0

Given ${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$1-spread monomials ${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$2, the notation

${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$3

denotes the smallest ${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$4-spread strongly stable ideal containing them; when ${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$5, ${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$6 is a principal ${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$7-spread Borel ideal (Crupi et al., 9 Jul 2025).

The general vector-spread theory was established for ${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$8-spread strongly stable ideals before the squarefree principal case was isolated. In that broader theory, ${\bf t}\in \mathbb Z_{\ge 1}^{d-1}$9 recovers classical strongly stable ideals, while ${\bf t}$0 recovers squarefree strongly stable ideals (Crupi et al., 2022). A plausible summary is that squarefree principal vector-spread Borel ideals are the principal objects at the intersection of principal Borel closure, squarefreeness, and position-dependent spread constraints.

2. Principal generation and combinatorial support data

For a principal ideal

${\bf t}$1

the minimal generators are obtained from ${\bf t}$2 by admissible lowering moves that preserve the ${\bf t}$3-spread condition. The paper on principal vector-spread Borel ideals repeatedly uses the degree-${\bf t}$4 characterization that a monomial

${\bf t}$5

belongs to ${\bf t}$6 exactly when the coordinates satisfy ${\bf t}$7 for all ${\bf t}$8, together with the ${\bf t}$9-spread inequalities (Crupi et al., 9 Jul 2025). In the earlier general theory, principal ideals are likewise understood as closure under all decreasing moves $u=x_{j_1}\cdots x_{j_d}$0 that preserve $u=x_{j_1}\cdots x_{j_d}$1-spreadness (Crupi et al., 2022).

A central invariant in the squarefree principal theory is the $u=x_{j_1}\cdots x_{j_d}$2-spread support. If

$u=x_{j_1}\cdots x_{j_d}$3

is $u=x_{j_1}\cdots x_{j_d}$4-spread, then

$u=x_{j_1}\cdots x_{j_d}$5

This differs from the ordinary support $u=x_{j_1}\cdots x_{j_d}$6: it records the forbidden intervals induced by the spread constraints rather than only the chosen indices (Crupi et al., 9 Jul 2025). In the special case $u=x_{j_1}\cdots x_{j_d}$7, this collapses to the familiar squarefree strongly stable support pattern; in the general vector-spread resolution theory, the related set

$u=x_{j_1}\cdots x_{j_d}$8

appears in the description of colon ideals and Betti numbers (Crupi et al., 2022).

The squarefree principal paper also introduces a second family of subsets, denoted $u=x_{j_1}\cdots x_{j_d}$9, of the form

$j_{k+1}-j_k\ge t_k$0

where $j_{k+1}-j_k\ge t_k$1 is a generator of a shorter principal vector-spread Borel ideal. These sets encode the non-generator facets of the Stanley–Reisner complex and are essential in the primary decomposition (Crupi et al., 9 Jul 2025). This suggests that the correct combinatorial object for the squarefree principal class is not merely the set of generators, but the pair consisting of $j_{k+1}-j_k\ge t_k$2-spread supports and tail-augmented truncated supports.

3. Minimal primary decomposition and associated simplicial complex

The main structural theorem of the 2025 paper gives a complete minimal primary decomposition for squarefree principal vector-spread Borel ideals. For

$j_{k+1}-j_k\ge t_k$3

one has

$j_{k+1}-j_k\ge t_k$4

where $j_{k+1}-j_k\ge t_k$5 and $j_{k+1}-j_k\ge t_k$6 is the family of truncated-support-plus-tail subsets described above (Crupi et al., 9 Jul 2025). Since the ideals are squarefree, these are prime components, so the formula is simultaneously a minimal primary and minimal prime decomposition.

This decomposition is best understood through the Stanley–Reisner complex $j_{k+1}-j_k\ge t_k$7 defined by $j_{k+1}-j_k\ge t_k$8. The facets of $j_{k+1}-j_k\ge t_k$9 are exactly the sets in

$B_{\bf t}(u)$0

so there are two facet types: generator facets coming from full $B_{\bf t}(u)$1-spread supports, and tail facets coming from truncated generators extended by an interval $B_{\bf t}(u)$2 (Crupi et al., 9 Jul 2025). This facet classification is the decisive combinatorial input for both homological and symbolic-power results.

An immediate corollary is the height formula

$B_{\bf t}(u)$3

for $B_{\bf t}(u)$4 (Crupi et al., 9 Jul 2025). This matches the broader Borel-type pattern that height is governed by the smallest index appearing among principal generators; for $B_{\bf t}(u)$5-spread strongly stable ideals more generally, the height formula

$B_{\bf t}(u)$6

was already known (Crupi et al., 2022).

The decomposition theorem may be viewed as the vector-spread analogue of several earlier squarefree-principal results. For squarefree principal Borel ideals in the ordinary $B_{\bf t}(u)$7-case, associated primes and stable sets can also be described explicitly from the principal generator and its localizations (Aslam, 2013). For $B_{\bf t}(u)$8-spread principal Borel ideals, the facet structure of the associated simplicial complex had already been described in terms of interval unions of length $B_{\bf t}(u)$9 (Andrei et al., 2018). The vector-spread result generalizes that scalar-${\bf t}$0 interval-block picture by replacing uniform blocks with the nonuniform ${\bf t}$1-spread support.

4. Homological structure and sequential Cohen–Macaulayness

A central theorem states that every squarefree principal vector-spread Borel ideal is sequentially Cohen–Macaulay (Crupi et al., 9 Jul 2025). The proof is squarefree and Alexander-dual in character. For a squarefree monomial ideal ${\bf t}$2, sequential Cohen–Macaulayness is equivalent to componentwise linearity of the Alexander dual ${\bf t}$3. The paper proves that ${\bf t}$4 is vertex splittable, hence has linear quotients, hence is componentwise linear, which yields the result.

The key recursive decomposition is

${\bf t}$5

where ${\bf t}$6 and ${\bf t}$7 (Crupi et al., 9 Jul 2025). This is a vertex-splitting formula on the Alexander dual, obtained by decomposing the Stanley–Reisner complex according to whether a face contains the vertex ${\bf t}$8. A corollary is that the simplicial complex ${\bf t}$9 with $u$0 is vertex decomposable (Crupi et al., 9 Jul 2025).

This homological behavior sits naturally within the broader vector-spread literature. Earlier work proved that every $u$1-spread strongly stable ideal has linear quotients, hence is componentwise linear, with explicit colon ideals

$u$2

when generators are ordered in pure lexicographic order (Crupi et al., 2022). The same paper derived the graded Betti numbers

$u$3

and classified the Cohen–Macaulay $u$4-spread strongly stable ideals as precisely the $u$5-spread Veronese ideals in the equigenerated case (Crupi et al., 2022).

For squarefree principal vector-spread Borel ideals, this implies that the dedicated 2025 sequential Cohen–Macaulay theorem is consistent with a more general homological picture, but it is stronger in a different direction: it exploits the principal squarefree structure to obtain an explicit Alexander-dual recursion and primary decomposition unavailable in the general nonprincipal setting.

5. Ordinary powers, symbolic powers, and normal torsionfreeness

The strongest arithmetic classification in the squarefree principal theory concerns equality of ordinary and symbolic powers. For a squarefree monomial ideal

$u$6

the symbolic powers are

$u$7

In general $u$8, and equality for all $u$9 is equivalent to normal torsionfreeness (Crupi et al., 9 Jul 2025).

For

${\bf t}=(t_1,\dots,t_{d-1})$00

the classification theorem states that the following are equivalent: ${\bf t}=(t_1,\dots,t_{d-1})$01

${\bf t}=(t_1,\dots,t_{d-1})$02

${\bf t}=(t_1,\dots,t_{d-1})$03

Thus the ordinary and symbolic powers coincide for all ${\bf t}=(t_1,\dots,t_{d-1})$04 exactly when every coordinate of the principal generator sits no further to the right than the cumulative spread bound permits (Crupi et al., 9 Jul 2025).

Condition (c) is tight. Since ${\bf t}=(t_1,\dots,t_{d-1})$05-spreadness already imposes

${\bf t}=(t_1,\dots,t_{d-1})$06

the classification says that the generator must lie in the narrow range

${\bf t}=(t_1,\dots,t_{d-1})$07

for each ${\bf t}=(t_1,\dots,t_{d-1})$08. The proof of necessity constructs an explicit monomial in ${\bf t}=(t_1,\dots,t_{d-1})$09 whenever one of these inequalities fails, while sufficiency is obtained by combining the primary decomposition with a criterion of Sayedsadeghi–Nasernejad–Qureshi and a linear-relation-graph analysis (Crupi et al., 9 Jul 2025).

This result sharply contrasts with other Borel-type families. For principal ${\bf t}=(t_1,\dots,t_{d-1})$10-Borel ideals ${\bf t}=(t_1,\dots,t_{d-1})$11, all symbolic powers equal ordinary powers: ${\bf t}=(t_1,\dots,t_{d-1})$12 (Moreno et al., 2020). That phenomenon does not persist uniformly in squarefree principal vector-spread Borel ideals; instead, equality is exceptional and fully characterized by condition (c) (Crupi et al., 9 Jul 2025). In the constant-${\bf t}=(t_1,\dots,t_{d-1})$13 case, there is an analogous but differently formulated normal-torsionfreeness criterion for ${\bf t}=(t_1,\dots,t_{d-1})$14-spread principal Borel ideals: ${\bf t}=(t_1,\dots,t_{d-1})$15 is normally torsionfree iff ${\bf t}=(t_1,\dots,t_{d-1})$16, equivalently iff it is the edge ideal of a ${\bf t}=(t_1,\dots,t_{d-1})$17-uniform ${\bf t}=(t_1,\dots,t_{d-1})$18-partite hypergraph (Nasernejad et al., 2021). The vector-spread theorem is the nonuniform counterpart of that scalar-${\bf t}=(t_1,\dots,t_{d-1})$19 classification.

6. Related theories, special cases, and broader significance

Squarefree principal vector-spread Borel ideals sit at the intersection of several neighboring theories. When ${\bf t}=(t_1,\dots,t_{d-1})$20, they recover squarefree principal Borel ideals, for which persistence of associated primes, explicit stable sets, and Waldschmidt-constant bounds are known (Aslam, 2013, Moreno et al., 2021). When ${\bf t}=(t_1,\dots,t_{d-1})$21, they specialize to ${\bf t}=(t_1,\dots,t_{d-1})$22-spread principal Borel ideals, where sequential Cohen–Macaulayness, normal Cohen–Macaulay Rees algebras, strong persistence, and asymptotic depth of powers were established earlier (Andrei et al., 2018).

From the homological perspective, vector-spread Borel ideals in general have linear quotients, explicit Betti formulas, bigraded Poincaré series, a characterization of extremal Betti numbers, and a complete Cohen–Macaulay classification (Crupi et al., 2022). From the enumerative perspective, ${\bf t}=(t_1,\dots,t_{d-1})$23-spread strongly stable ideals admit a Macaulay–Kruskal–Katona theory: each such ideal has an ${\bf t}=(t_1,\dots,t_{d-1})$24-vector, there is a unique ${\bf t}=(t_1,\dots,t_{d-1})$25-spread lex ideal with the same ${\bf t}=(t_1,\dots,t_{d-1})$26-vector, and the ${\bf t}=(t_1,\dots,t_{d-1})$27-spread lex ideal maximizes Betti numbers among ideals with that ${\bf t}=(t_1,\dots,t_{d-1})$28-vector (Crupi et al., 2023). These general results apply to the principal squarefree class once principality is forgotten.

The squarefree principal class is also distinguished by its explicit primary decomposition and symbolic-power classification, which are not available in the same form for arbitrary vector-spread strongly stable ideals (Crupi et al., 9 Jul 2025). A plausible implication is that principality restores enough rigidity to make combinatorial prime descriptions possible. In this respect the theory parallels earlier work on principal ${\bf t}=(t_1,\dots,t_{d-1})$29-Borel ideals and principal ${\bf t}=(t_1,\dots,t_{d-1})$30-Borel ideals, where closure under a structured family of lowering moves likewise leads to unexpectedly explicit formulas for associated primes, analytic spread, or Rees algebras (Moreno et al., 2020, DiPasquale et al., 2020).

One stated limitation is that the 2025 theory is squarefree: the authors explicitly leave open whether the sequential Cohen–Macaulay result extends to the non-squarefree setting (Crupi et al., 9 Jul 2025). This restriction is structural rather than cosmetic, since the proofs rely on Stanley–Reisner duality, vertex splittability, and squarefree prime decompositions. The squarefree condition is therefore intrinsic to the current state of the theory.

In summary, squarefree principal vector-spread Borel ideals form a rigid principal subclass of ${\bf t}=(t_1,\dots,t_{d-1})$31-spread strongly stable ideals, characterized by a nonuniform spread vector and a single Borel generator. Their modern theory consists of three tightly connected ingredients: an explicit minimal prime decomposition via ${\bf t}=(t_1,\dots,t_{d-1})$32-spread supports and tail facets, a recursive Alexander-dual splitting that yields sequential Cohen–Macaulayness, and a complete criterion

${\bf t}=(t_1,\dots,t_{d-1})$33

for equality of symbolic and ordinary powers (Crupi et al., 9 Jul 2025). These results place the class among the most tractable squarefree spread-constrained Borel families presently known.