---
title: Monomial Almost Complete Intersections
url: https://www.emergentmind.com/topics/monomial-almost-complete-intersections
type: topic
---

# Monomial Almost Complete Intersections

Searching arXiv for recent and foundational papers on monomial almost complete intersections.
Monomial almost complete intersections are monomial ideals that exceed the complete intersection condition by exactly one minimal generator, and they occupy a central position at the interface of combinatorial commutative algebra, homological algebra, Rees algebras, and Lefschetz theory. In a polynomial ring \(S=K[x_1,\ldots,x_n]\), the standard form is an ideal whose minimal number of generators is one more than its height, typically written as a complete intersection block together with one additional monomial. Across the recent literature, this class appears in several related but distinct settings: as a testing ground for Stanley depth and cleanliness phenomena [1311.7303, 1112.4956], as a tractable family for explicit Betti numbers and regularity formulas [2505.18788, 2606.08009], as a source of Artinian algebras with subtle Strong and Weak Lefschetz behavior [2507.18516, 2603.11491], and as a family whose Rees algebras exhibit almost Cohen–Macaulay behavior [1902.03068, 1405.0531]. The subject is therefore not a single theorem but a network of structural results in which the monomial hypothesis permits exact formulas, combinatorial classifications, and constructive homological arguments.

## 1. Definition and basic forms

A monomial ideal \(I\subset S=K[x_1,\ldots,x_n]\) is an almost complete intersection if
\[
\mu(I)=\operatorname{ht}(I)+1,
\]
where \(\mu(I)\) is the cardinality of the minimal monomial generating set \(G(I)\) [1311.7303, 1902.03068]. In the standard monomial ACI presentation used repeatedly in the literature, one writes
\[
I=(u_1,\ldots,u_q,v),
\]
where \((u_1,\ldots,u_q)\) is a monomial complete intersection and \(q=\operatorname{ht}(I)\) [2505.18788, 2606.08009]. For monomials, the complete intersection condition is equivalent to pairwise disjoint supports of the generators \(u_1,\ldots,u_q\) [2505.18788].

In the Artinian case over a standard graded ring \(R=K[x_1,\ldots,x_n]\), monomial ACIs are often written as
\[
I=(x_1^{a_1},\ldots,x_n^{a_n},m),
\]
where \(m\notin (x_1^{a_1},\ldots,x_n^{a_n})\); then the number of minimal generators equals codimension \(+1\) [2507.18516]. A recurrent special case is
\[
I=(x_1^{a_1},\ldots,x_n^{a_n},x_1^{b_1}\cdots x_n^{b_n}),
\]
with \(0\le b_i<a_i\) and at least two \(b_i\) nonzero in the finite-colength setting [1902.03068, 1405.0531]. In three variables this becomes
\[
I=(x^{d_1},y^{d_2},z^{d_3},x^{a_1}y^{a_2}z^{a_3}),
\]
the standard form for Weak Lefschetz investigations [2603.11491]. In the ternary uniform case of Rees-algebra theory, one frequently studies
\[
I=(x^a,y^a,z^a,(xyz)^b)
\]
with \(0<b<a\) [1405.0531]. Another related but distinct family, sometimes also called an almost complete intersection in recent Gröbner-basis work, is
\[
I=(x_1^{m_1},\ldots,x_n^{m_n},L^k),
\]
where \(L\) is a general linear form; here the extra generator is not monomial, but the background complete intersection remains monomial [2506.24028].

A useful structural dichotomy is the dominant/semidominant distinction. A monomial ideal is dominant if each generator possesses a variable whose exponent is strictly larger than the corresponding exponent in every other generator; a monomial ACI is either dominant or semidominant in the sense that the extra generator is controlled by the least common multiple of a minimal subset of the complete intersection block [2606.08009, 2505.18788]. This dichotomy governs both regularity formulas and the shape of minimal resolutions.

## 2. Combinatorial and homological structure

Several papers exploit the fact that monomial ACIs admit unusually explicit combinatorial descriptions. For square-free monomial ACIs, Kimura–Terai–Yoshida provide a classification into six types, denoted \(I_1,\ldots,I_6\), built from pairwise coprime square-free monomials \(A_i,B_j\) [1311.7303]. This classification is used to prove cleanness and to analyze Alexander duals, forest type, and linear quotients.

A complementary structural description appears in the Betti-number theory of semidominant ACIs. If
\[
I=(u_1,\ldots,u_q,v)
\]
with \((u_1,\ldots,u_q)\) a complete intersection, define \(s\) to be the smallest integer such that \(v\) divides the least common multiple of some \(s\) of the \(u_i\). Then the combinatorics of subsets
\[
L_i=\{\tau\subseteq \{1,\ldots,q\}: |\tau|=i,\; v\mid \operatorname{lcm}(u_j:j\in \tau)\}
\]
controls the cancellation pattern in the Taylor complex and yields closed Betti formulas [2505.18788]. In the complete intersection setting, \(L_i\) consists of supersets of a fixed minimal subset of size \(s\), so \(|L_i|=\binom{q-s}{i-s}\) for \(i\ge s\) [2505.18788].

This leads to the explicit total Betti numbers
\[
\beta_i(R/I)=\binom{q+1}{i}\quad \text{for } i=0,\ldots,s-1,
\]
and
\[
\beta_i(R/I)=\binom{q+1}{i}-\binom{q-s+1}{i-s}\quad \text{for } i=s,\ldots,q
\]
[2505.18788]. In the special case \(s=2\),
\[
\beta_i(R/I)=\binom{q+1}{i}-\binom{q}{i-2}
\]
for \(i\ge 2\) [2505.18788]. The same work shows that dominant ideals have minimal Taylor resolutions, while semidominant ACIs are resolved by a Scarf complex obtained by deleting precisely those Taylor faces \(\sigma\) and \(\sigma\cup\{v\}\) for which \(v\mid \operatorname{lcm}(u_\sigma)\) [2505.18788].

Regularity also admits explicit formulas. For a dominant monomial ideal \(I\) with \(G(I)=\{u_1,\ldots,u_m\}\), if
\[
a_i=\max\{\deg_{x_i}(u): u\in G(I)\},
\]
then
\[
\operatorname{reg}(R/I)=\deg(\operatorname{lcm}(u_1,\ldots,u_m))-m,
\qquad
\operatorname{reg}(I)=\sum_{i=1}^n a_i-m+1
\]
[2606.08009]. For a monomial ACI \(I=(u_1,\ldots,u_q,u_{q+1})\) with complete intersection part \(I'=(u_1,\ldots,u_q)\), the semidominant formula is
\[
\operatorname{reg}(I)=\operatorname{reg}(I')-\min\left\{\deg\!\left(\frac{u_j}{\gcd(u_j,u_{q+1})}\right)\mid j=1,\ldots,s\right\},
\]
where \(s\) is determined by minimal divisibility of \(u_{q+1}\) by lcms of generators from the complete intersection part [2606.08009]. Since
\[
\operatorname{reg}(I')=\sum_{i=1}^q \deg(u_i)-q+1,
\]
the correction term is read directly from gcd-data with the extra generator [2606.08009]. In the dominant ACI case one has
\[
\operatorname{reg}(I)=\sum_{l=1}^{q+1}\deg(u_l)-q-\sum_{i=1}^{h}\deg(\gcd(u_i,u_{q+1}))
\]
for the generators that meet the support of \(u_{q+1}\) [2606.08009].

These results show that monomial ACIs interpolate between complete intersections, where all homological data are product-like, and general monomial ideals, where no uniform closed formulas are expected.

## 3. Cleanliness, Stanley depth, and Cohen–Macaulay properties

One of the foundational themes of the subject is that monomial ACIs satisfy strong forms of Stanley’s conjecture. For a finitely generated \(\mathbb{Z}^n\)-graded module \(M\), a Stanley decomposition is a direct sum
\[
M=\bigoplus_{i=1}^r u_iK[Z_i],
\]
and the Stanley depth is the maximum, over all such decompositions, of the minimum \(|Z_i|\) [1311.7303]. Stanley’s conjecture asserts
\[
\operatorname{sdepth}(M)\ge \operatorname{depth}(M).
\]

For monomial ACI ideals \(I\subset S\), one of the central results is that \(S/I\) is pretty clean [1311.7303]. By the implications developed by Herzog–Popescu and Herzog–Vladoiu–Zheng, pretty cleanness yields
\[
\operatorname{sdepth}(S/I)\ge \operatorname{depth}(S/I),
\qquad
\operatorname{sdepth}(S/I)=\operatorname{depth}(S/I)
\]
[1311.7303]. The proof proceeds by polarization: if \(I^P\) is the polarization of \(I\), then \(I\) is ACI if and only if \(I^P\) is square-free ACI, and a theorem of Soleyman Jahan relates pretty cleanness of \(S/I\) to cleanness of the polarized quotient [1311.7303]. The square-free case is then analyzed via the Kimura–Terai–Yoshida classification and a mix of forest-type arguments and Alexander-dual linear quotients [1311.7303].

A parallel earlier result proves Stanley’s conjecture for both \(S/I\) and \(I\) when \(I\) is a monomial almost complete intersection [1112.4956]. That argument uses the numerical relation
\[
\mu(I)=s+1,\qquad s=n-\operatorname{depth}(S/I),
\]
together with lower bounds such as
\[
\operatorname{sdepth}(S/I)\ge n-\mu(I),
\qquad
\operatorname{sdepth}(I)\ge n-\lfloor \mu(I)/2\rfloor
\]
and a decomposition with respect to a variable that appears in many generators [1112.4956]. A related manuscript also proves that if \(I\) is almost complete intersection, or generated by a filter-regular sequence or a \(d\)-sequence, then \(S/I\) is pretty clean, sequentially Cohen–Macaulay, and satisfies both Stanley’s and \(h\)-regularity conjectures [1112.5159].

The Cohen–Macaulay question behaves differently. For ACIs, \(R/I\) is Cohen–Macaulay if and only if \(I\) is non-dominant [2505.18788]. Equivalently, for an ACI \(I\), the following are equivalent: \(R/I\) is Cohen–Macaulay, \(I\) is unmixed, and \(R/I\) is clean [2505.18788]. In contrast, for a dominant monomial ideal, \(R/I\) is Cohen–Macaulay if and only if \(I\) is a complete intersection [2505.18788]. This distinction corrects a common oversimplification: monomial ACIs often satisfy Stanley-type inequalities, but they are not generally Cohen–Macaulay.

Concrete examples illustrate the difference between clean and pretty clean. In
\[
S=K[x,y],\qquad I=(x^2,xy),
\]
one has \(\mu(I)=2\) and \(\operatorname{ht}(I)=1\), so \(I\) is a monomial ACI; \(S/I\) is not clean, but it is pretty clean, and therefore
\[
\operatorname{depth}(S/I)=\operatorname{sdepth}(S/I)=0
\]
[1311.7303]. By contrast, for the square-free ACI
\[
I=(x_1x_2,x_1x_3,x_2x_3)\subset K[x_1,x_2,x_3],
\]
the quotient is clean, and
\[
\operatorname{depth}(S/I)=\operatorname{sdepth}(S/I)=1
\]
[1311.7303].

## 4. Multiplicity, resolutions, and integral closure

The monomial structure also permits explicit formulas for multiplicity. If
\[
M=(m_1,\ldots,m_q,m)
\]
is a monomial ACI such that \(M_1=(m_1,\ldots,m_q)\) is a complete intersection and \(\operatorname{codim}(S/M)=q\), then
\[
e(S/M)=\prod_{i=1}^{q}\deg(m_i)-\prod_{i=1}^{q}\deg\!\left(\frac{m_i}{\gcd(m_i,m)}\right)
\]
[1901.03291]. This expresses the multiplicity as the multiplicity of the complete intersection block minus a correction term measuring the degrees lost after removing the common factors with the extra generator. The formula is compatible with the extremes already known in the same work: in codimension \(1\),
\[
e(S/M)=\deg(\gcd(m_1,\ldots,m_q)),
\]
while for a complete intersection,
\[
e(S/M)=\prod_{i=1}^{q}\deg(m_i)
\]
[1901.03291].

For example, with
\[
M=(x^4,y^3,z^2,xy^2)\subset k[x,y,z],
\]
the first product is \(4\cdot 3\cdot 2=24\), while the corrected factors have degrees \(3,1,2\), so the second product is \(6\), giving
\[
e(S/M)=18
\]
[1901.03291].

Minimal free resolutions are equally explicit in the ACI setting. The formulas for \(\beta_i(R/I)\) described above enable a direct construction of the minimal resolution from the Scarf subcomplex of the Taylor complex [2505.18788]. In the dominant case, the full Taylor resolution is minimal; in the semidominant ACI case, the deleted faces are exactly those corresponding to divisibility of the extra generator by subset lcms [2505.18788]. A plausible implication is that monomial ACIs form one of the largest natural classes for which both Betti numbers and resolutions remain uniformly controllable by lcm-combinatorics.

Recent work also addresses integral closure. If \(I\) is dominant or almost complete intersection, then
\[
\operatorname{reg}(\overline{I})\le \operatorname{reg}(I),
\]
where \(\overline{I}\) denotes the integral closure [2606.08009]. This gives a positive answer to the Küronya–Pintye conjecture for these two classes [2606.08009]. The proof for dominant ideals uses bounds on lcm-degrees of generators of \(\overline I\), together with Lyubeznik-resolution arguments; for ACIs, a structural hypothesis on how generators of \(\overline I\) sit relative to the complete intersection part allows the same conclusion [2606.08009]. The scope is explicit: the inequality is not claimed for all monomial ideals, and counterexamples exist in general dimension \(4\) [2606.08009].

## 5. Rees algebras and almost Cohen–Macaulay blowup algebras

Another major branch of the subject concerns the Rees algebra
\[
\mathcal R(I)=R[It]=\bigoplus_{m\ge 0} I^mt^m
\]
and its defining ideal. For Artinian monomial ACIs
\[
I=\langle T_1^{a_1},\ldots,T_m^{a_m},T_1^{b_1}\cdots T_m^{b_m}\rangle
\]
with \(0\le b_i<a_i\), the Rees algebra is almost Cohen–Macaulay, meaning
\[
\operatorname{depth}(\mathcal R(I))\ge \dim(\mathcal R(I))-1
\]
[1902.03068]. This confirms a conjecture of Vasconcelos for all Artinian almost complete intersection monomial ideals [1902.03068].

The proof is Gröbner-theoretic. Writing \(S=R[X_1,\ldots,X_m,W]\) and \(L=\ker(\varphi)\) for the defining ideal of \(\mathcal R(I)\), one constructs an infinite Gröbner basis from binomials of two types:
\[
T_0=\{P(X_i,X_j)\},
\qquad
T_1=\{P(W^c,X^c): c\in \mathbb N^m,\ |c|\ge 2\}
\]
[1902.03068]. Finite closed subsets yield finite Gröbner bases, and successive colon ideals of the initial ideals are shown to be extended from the coefficient ring \(R\). A depth induction via the Depth Lemma then proves
\[
\operatorname{depth}(S/L)\ge m=\dim(S/L)-1
\]
[1902.03068].

In low dimension, the structure can be sharpened through Sylvester forms. For binary monomial ACIs
\[
I=(x^d,y^d,x^by^{d-b}),\qquad \gcd(d,b)=1,
\]
the Rees ideal is generated by the initial syzygies together with iterated Sylvester forms determined by the Euclidean algorithm for \((d,b)\) [1405.0531]. The resulting Rees algebra is almost Cohen–Macaulay, and it is Cohen–Macaulay if and only if \(d=2\) [1405.0531]. In the ternary uniform case
\[
I=(x^a,y^a,z^a,(xyz)^b),
\]
the Rees ideal is generated by six syzygies together with four Sylvester forms \(H_1,H_2,H_3,E\) (or \(E'\) when \(a>3b\)), and the Rees algebra is again almost Cohen–Macaulay [1405.0531].

The ternary Sylvester forms are explicit:
\[
H_1=(xy)^{a-2b}w^2-z^{2b}tu,\quad
H_2=(xz)^{a-2b}w^2-y^{2b}tv,\quad
H_3=(yz)^{a-2b}w^2-x^{2b}uv,
\]
and
\[
E=w^3-(xyz)^{3b-a}tuv \quad \text{if } a\le 3b,
\]
or
\[
E'=(xyz)^{a-3b}w^3-tuv \quad \text{if } a>3b
\]
[1405.0531]. Mapping-cone arguments applied to explicit colon ideals then yield a free resolution of length at most \(4\), proving the almost Cohen–Macaulay property [1405.0531].

These results show that monomial ACIs are unusually well behaved among ideals that are not of linear type: their Rees ideals generally require higher equations, but those equations can still be organized explicitly.

## 6. Lefschetz properties and Artinian monomial ACIs

The Lefschetz theory of Artinian monomial ACIs has developed rapidly. For a standard graded Artinian algebra \(A=R/I\), the Weak Lefschetz Property (WLP) means that multiplication by a linear form has maximal rank in each degree, while the Strong Lefschetz Property (SLP) requires maximal rank for all powers of that form [2507.18516].

A major recent theorem classifies the SLP for monomial ACIs whose non-pure-power generator has support in two variables. If
\[
A=K[x_1,\ldots,x_n]/(x_1^{a_1},\ldots,x_n^{a_n},x_1^\alpha x_2^\beta)
\]
over a field of characteristic \(0\), then after relabeling so that \(a_1+\beta\le a_2+\alpha\), \(A\) has the SLP if and only if one of four conditions holds: \(n=2\); or \(n=3\) and \(a_3\le 2\); or the two-variable factor
\[
B=K[x_1,x_2]/(x_1^{a_1},x_2^{a_2},x_1^\alpha x_2^\beta)
\]
is almost centered; or, equivalently, the explicit inequalities
\[
a_2<a_1+\beta+2
\]
and one of
\[
a_1=\alpha+1,\qquad \beta=1,\qquad a_2\ge a_1+\beta-1
\]
hold [2507.18516]. The two-variable Hilbert series of \(B\) is unimodal; it is symmetric if and only if
\[
a+\beta=b
\]
in the notation \(B=K[x,y]/(x^a,y^b,x^\alpha y^\beta)\), and its socle degree is
\[
D=b+\alpha-2
\]
[2507.18516]. If the Hilbert series of a monomial ACI is symmetric, then the algebra always has the SLP in characteristic \(0\) [2507.18516].

The WLP in three-variable level monomial ACIs is subtler. For
\[
A=\mathbb F[x,y,z]/(x^{d_1},y^{d_2},z^{d_3},x^{a_1}y^{a_2}z^{a_3}),
\]
with \(\operatorname{char}(\mathbb F)=0\), levelness is equivalent to
\[
d_1-a_1=d_2-a_2=d_3-a_3=t
\]
[2603.11491]. The analysis reduces the WLP problem to the two-variable colon ideal
\[
(x^{d_1},y^{d_2}):(x+y)^{a_3},
\]
whose generators are described explicitly by binomial-coefficient formulas [2603.11491]. In the level case, WLP fails exactly when the determinant of a certain explicitly constructed \((a_1+a_2)\times (a_1+a_2)\) matrix vanishes; for fixed parity of \(t\), this gives a polynomial criterion
\[
P_1(t)=0 \quad \text{or} \quad P_2(t)=0
\]
depending on whether \(t\) is even or odd [2603.11491]. The same work proves new cases of the Migliore–Miró-Roig–Nagel conjecture near the boundary
\[
t=\frac{a_1+a_2+a_3}{3}+1,\qquad a_3=2(a_1+a_2)-3a,\qquad 0\le a\le 3
\]
[2603.11491].

A related but distinct direction concerns ideals of the form
\[
I=(x_1^{m_1},\ldots,x_n^{m_n},L^k)
\]
with \(L\) a general linear form. Although the extra generator is not monomial, the complete intersection part is monomial, and the Gröbner theory is highly explicit. The reduced Gröbner basis depends only on the variable ranking, and the initial ideal is generated by the pure powers together with a set of critical monomials defined via lattice-path reflection across a parameter-dependent red line [2506.24028]. This yields
\[
\operatorname{HS}(R/I;t)=\bigl[(1-t^k)\operatorname{HS}(R/A;t)\bigr],
\]
where \(A=(x_1^{m_1},\ldots,x_n^{m_n})\), providing a new proof of the SLP for monomial complete intersections in characteristic \(0\) [2506.24028]. The same enumeration connects Gröbner-basis degree counts to Catalan, Motzkin, and Riordan numbers [2506.24028].

These Lefschetz results underline an important distinction. Stanley-depth phenomena for monomial ACIs are robust and largely characteristic-free, whereas Lefschetz properties are sensitive to support, symmetry of Hilbert series, parity conditions, levelness, and characteristic.

## 7. Related directions, misconceptions, and scope

Monomial ACIs also arise in local algebra through annihilators of Koszul homology. In a Noetherian local almost complete intersection \(R\) with system of parameters \(\mathbf x\), one asks whether
\[
((\mathbf x):\mathfrak m)\subseteq 0:_R H_i(\mathbf x;R)\qquad \text{for all } i\ge 1.
\]
For \(H_1\), this statement is equivalent to the Monomial Conjecture, and hence valid; under additional small-multiplicity hypotheses such as \(\mathfrak m^2\subseteq (\mathbf x)\), all positive Koszul homologies are annihilated by \(\mathfrak m\), and residual approximation complexes resolve the residue field [1702.01111]. Although this work is not restricted to monomial ideals, it shows that ACIs remain a central testing ground for homological conjectures beyond the graded monomial setting.

Several misconceptions are corrected by the literature.

| Misconception | Correction | Source |
|---|---|---|
| Every monomial ACI quotient is clean | In general only pretty clean; \(K[x,y]/(x^2,xy)\) is not clean | [1311.7303] |
| Monomial ACIs are automatically Cohen–Macaulay | \(R/I\) is Cohen–Macaulay iff the ACI is non-dominant | [2505.18788] |
| Rees algebras of monomial ACIs are always Cohen–Macaulay | The general result is almost Cohen–Macaulay, not necessarily Cohen–Macaulay | [1902.03068, 1405.0531] |

The scope of current theory is also uneven. Stanley-depth, cleanliness, and explicit homological invariants are well developed for monomial ACIs [1311.7303, 2505.18788, 2606.08009]. Rees-algebra structure is understood for Artinian monomial ACIs in general and with finer generators in binary and ternary low-dimensional families [1902.03068, 1405.0531]. By contrast, SLP is completely classified only for certain Artinian classes, notably when the extra generator has support in two variables or when the Hilbert series is symmetric [2507.18516], and WLP in three-variable level cases still depends on determinant criteria rather than a fully closed classification [2603.11491]. For non-monomial ACIs, most of these explicit formulas and combinatorial constructions are outside current scope [1311.7303].

This suggests a coherent picture. Monomial ACIs are not merely “one generator away” from complete intersections in a numerical sense; they are among the rare non-complete-intersection families for which multiple deep invariants—Betti numbers, multiplicities, regularity, Stanley depth, Rees equations, and in important cases Lefschetz behavior—remain accessible through exact combinatorial or homological models.

Source: https://www.emergentmind.com/topics/monomial-almost-complete-intersections