Monomial Almost Complete Intersections

• Monomial almost complete intersections are monomial ideals generated by a complete intersection block plus one extra generator, forming a bridge between complete intersections and more complex ideal structures.
• Analyses employ combinatorial and homological techniques to derive explicit formulas for Betti numbers, regularity, multiplicity, and Rees algebra properties, distinguishing dominant and semidominant cases.
• These ideals serve as a testing ground for Stanley depth, Lefschetz properties, and Cohen–Macaulay conditions, offering practical insights for both computational methods and theoretical advances in algebra.

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 (Bandari et al., 2013, Cimpoeas, 2011), as a tractable family for explicit Betti numbers and regularity formulas (Mafi et al., 24 May 2025, Mafi et al., 6 Jun 2026), as a source of Artinian algebras with subtle Strong and Weak Lefschetz behavior (Chase et al., 24 Jul 2025, Booth et al., 12 Mar 2026), and as a family whose Rees algebras exhibit almost Cohen–Macaulay behavior (Lin et al., 2019, Simis et al., 2014). 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 $G(I)$ (Bandari et al., 2013, Lin et al., 2019). 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 $q=\operatorname{ht}(I)$ (Mafi et al., 24 May 2025, Mafi et al., 6 Jun 2026). For monomials, the complete intersection condition is equivalent to pairwise disjoint supports of the generators $u_1,\ldots,u_q$ (Mafi et al., 24 May 2025).

In the Artinian case over a standard graded ring $R=K[x_1,\ldots,x_n]$, monomial ACIs are often written as

$I\subset S=K[x_1,\ldots,x_n]$0

where $I\subset S=K[x_1,\ldots,x_n]$1; then the number of minimal generators equals codimension $I\subset S=K[x_1,\ldots,x_n]$2 (Chase et al., 24 Jul 2025). A recurrent special case is

$I\subset S=K[x_1,\ldots,x_n]$3

with $I\subset S=K[x_1,\ldots,x_n]$4 and at least two $I\subset S=K[x_1,\ldots,x_n]$5 nonzero in the finite-colength setting (Lin et al., 2019, Simis et al., 2014). In three variables this becomes

$I\subset S=K[x_1,\ldots,x_n]$6

the standard form for Weak Lefschetz investigations (Booth et al., 12 Mar 2026). In the ternary uniform case of Rees-algebra theory, one frequently studies

$I\subset S=K[x_1,\ldots,x_n]$7

with $I\subset S=K[x_1,\ldots,x_n]$8 (Simis et al., 2014). Another related but distinct family, sometimes also called an almost complete intersection in recent Gröbner-basis work, is

$I\subset S=K[x_1,\ldots,x_n]$9

where $\mu(I)=\operatorname{ht}(I)+1,$0 is a general linear form; here the extra generator is not monomial, but the background complete intersection remains monomial (Kling et al., 30 Jun 2025).

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 (Mafi et al., 6 Jun 2026, Mafi et al., 24 May 2025). 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 $\mu(I)=\operatorname{ht}(I)+1,$1, built from pairwise coprime square-free monomials $\mu(I)=\operatorname{ht}(I)+1,$2 (Bandari et al., 2013). 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

$\mu(I)=\operatorname{ht}(I)+1,$3

with $\mu(I)=\operatorname{ht}(I)+1,$4 a complete intersection, define $\mu(I)=\operatorname{ht}(I)+1,$5 to be the smallest integer such that $\mu(I)=\operatorname{ht}(I)+1,$6 divides the least common multiple of some $\mu(I)=\operatorname{ht}(I)+1,$7 of the $\mu(I)=\operatorname{ht}(I)+1,$8. Then the combinatorics of subsets

$\mu(I)=\operatorname{ht}(I)+1,$9

controls the cancellation pattern in the Taylor complex and yields closed Betti formulas (Mafi et al., 24 May 2025). In the complete intersection setting, $\mu(I)$0 consists of supersets of a fixed minimal subset of size $\mu(I)$1, so $\mu(I)$2 for $\mu(I)$3 (Mafi et al., 24 May 2025).

This leads to the explicit total Betti numbers

$\mu(I)$4

and

$\mu(I)$5

(Mafi et al., 24 May 2025). In the special case $\mu(I)$6,

$\mu(I)$7

for $\mu(I)$8 (Mafi et al., 24 May 2025). 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 $\mu(I)$9 and $G(I)$0 for which $G(I)$1 (Mafi et al., 24 May 2025).

Regularity also admits explicit formulas. For a dominant monomial ideal $G(I)$2 with $G(I)$3, if

$G(I)$4

then

$G(I)$5

(Mafi et al., 6 Jun 2026). For a monomial ACI $G(I)$6 with complete intersection part $G(I)$7, the semidominant formula is

$G(I)$8

where $G(I)$9 is determined by minimal divisibility of $I=(u_1,\ldots,u_q,v),$0 by lcms of generators from the complete intersection part (Mafi et al., 6 Jun 2026). Since

$I=(u_1,\ldots,u_q,v),$1

the correction term is read directly from gcd-data with the extra generator (Mafi et al., 6 Jun 2026). In the dominant ACI case one has

$I=(u_1,\ldots,u_q,v),$2

for the generators that meet the support of $I=(u_1,\ldots,u_q,v),$3 (Mafi et al., 6 Jun 2026).

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 $I=(u_1,\ldots,u_q,v),$4-graded module $I=(u_1,\ldots,u_q,v),$5, a Stanley decomposition is a direct sum

$I=(u_1,\ldots,u_q,v),$6

and the Stanley depth is the maximum, over all such decompositions, of the minimum $I=(u_1,\ldots,u_q,v),$7 (Bandari et al., 2013). Stanley’s conjecture asserts

$I=(u_1,\ldots,u_q,v),$8

For monomial ACI ideals $I=(u_1,\ldots,u_q,v),$9, one of the central results is that $(u_1,\ldots,u_q)$0 is pretty clean (Bandari et al., 2013). By the implications developed by Herzog–Popescu and Herzog–Vladoiu–Zheng, pretty cleanness yields

$(u_1,\ldots,u_q)$1

(Bandari et al., 2013). The proof proceeds by polarization: if $(u_1,\ldots,u_q)$2 is the polarization of $(u_1,\ldots,u_q)$3, then $(u_1,\ldots,u_q)$4 is ACI if and only if $(u_1,\ldots,u_q)$5 is square-free ACI, and a theorem of Soleyman Jahan relates pretty cleanness of $(u_1,\ldots,u_q)$6 to cleanness of the polarized quotient (Bandari et al., 2013). 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 (Bandari et al., 2013).

A parallel earlier result proves Stanley’s conjecture for both $(u_1,\ldots,u_q)$7 and $(u_1,\ldots,u_q)$8 when $(u_1,\ldots,u_q)$9 is a monomial almost complete intersection (Cimpoeas, 2011). That argument uses the numerical relation

$q=\operatorname{ht}(I)$0

together with lower bounds such as

$q=\operatorname{ht}(I)$1

and a decomposition with respect to a variable that appears in many generators (Cimpoeas, 2011). A related manuscript also proves that if $q=\operatorname{ht}(I)$2 is almost complete intersection, or generated by a filter-regular sequence or a $q=\operatorname{ht}(I)$3-sequence, then $q=\operatorname{ht}(I)$4 is pretty clean, sequentially Cohen–Macaulay, and satisfies both Stanley’s and $q=\operatorname{ht}(I)$5-regularity conjectures (Bandari et al., 2011).

The Cohen–Macaulay question behaves differently. For ACIs, $q=\operatorname{ht}(I)$6 is Cohen–Macaulay if and only if $q=\operatorname{ht}(I)$7 is non-dominant (Mafi et al., 24 May 2025). Equivalently, for an ACI $q=\operatorname{ht}(I)$8, the following are equivalent: $q=\operatorname{ht}(I)$9 is Cohen–Macaulay, $u_1,\ldots,u_q$0 is unmixed, and $u_1,\ldots,u_q$1 is clean (Mafi et al., 24 May 2025). In contrast, for a dominant monomial ideal, $u_1,\ldots,u_q$2 is Cohen–Macaulay if and only if $u_1,\ldots,u_q$3 is a complete intersection (Mafi et al., 24 May 2025). 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

$u_1,\ldots,u_q$4

one has $u_1,\ldots,u_q$5 and $u_1,\ldots,u_q$6, so $u_1,\ldots,u_q$7 is a monomial ACI; $u_1,\ldots,u_q$8 is not clean, but it is pretty clean, and therefore

$u_1,\ldots,u_q$9

(Bandari et al., 2013). By contrast, for the square-free ACI

$R=K[x_1,\ldots,x_n]$0

the quotient is clean, and

$R=K[x_1,\ldots,x_n]$1

(Bandari et al., 2013).

4. Multiplicity, resolutions, and integral closure

The monomial structure also permits explicit formulas for multiplicity. If

$R=K[x_1,\ldots,x_n]$2

is a monomial ACI such that $R=K[x_1,\ldots,x_n]$3 is a complete intersection and $R=K[x_1,\ldots,x_n]$4, then

$R=K[x_1,\ldots,x_n]$5

(Alesandroni, 2019). 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 $R=K[x_1,\ldots,x_n]$6,

$R=K[x_1,\ldots,x_n]$7

while for a complete intersection,

$R=K[x_1,\ldots,x_n]$8

(Alesandroni, 2019).

For example, with

$R=K[x_1,\ldots,x_n]$9

the first product is $I\subset S=K[x_1,\ldots,x_n]$00, while the corrected factors have degrees $I\subset S=K[x_1,\ldots,x_n]$01, so the second product is $I\subset S=K[x_1,\ldots,x_n]$02, giving

$I\subset S=K[x_1,\ldots,x_n]$03

(Alesandroni, 2019).

Minimal free resolutions are equally explicit in the ACI setting. The formulas for $I\subset S=K[x_1,\ldots,x_n]$04 described above enable a direct construction of the minimal resolution from the Scarf subcomplex of the Taylor complex (Mafi et al., 24 May 2025). 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 (Mafi et al., 24 May 2025). 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\subset S=K[x_1,\ldots,x_n]$05 is dominant or almost complete intersection, then

$I\subset S=K[x_1,\ldots,x_n]$06

where $I\subset S=K[x_1,\ldots,x_n]$07 denotes the integral closure (Mafi et al., 6 Jun 2026). This gives a positive answer to the Küronya–Pintye conjecture for these two classes (Mafi et al., 6 Jun 2026). The proof for dominant ideals uses bounds on lcm-degrees of generators of $I\subset S=K[x_1,\ldots,x_n]$08, together with Lyubeznik-resolution arguments; for ACIs, a structural hypothesis on how generators of $I\subset S=K[x_1,\ldots,x_n]$09 sit relative to the complete intersection part allows the same conclusion (Mafi et al., 6 Jun 2026). The scope is explicit: the inequality is not claimed for all monomial ideals, and counterexamples exist in general dimension $I\subset S=K[x_1,\ldots,x_n]$10 (Mafi et al., 6 Jun 2026).

5. Rees algebras and almost Cohen–Macaulay blowup algebras

Another major branch of the subject concerns the Rees algebra

$I\subset S=K[x_1,\ldots,x_n]$11

and its defining ideal. For Artinian monomial ACIs

$I\subset S=K[x_1,\ldots,x_n]$12

with $I\subset S=K[x_1,\ldots,x_n]$13, the Rees algebra is almost Cohen–Macaulay, meaning

$I\subset S=K[x_1,\ldots,x_n]$14

(Lin et al., 2019). This confirms a conjecture of Vasconcelos for all Artinian almost complete intersection monomial ideals (Lin et al., 2019).

The proof is Gröbner-theoretic. Writing $I\subset S=K[x_1,\ldots,x_n]$15 and $I\subset S=K[x_1,\ldots,x_n]$16 for the defining ideal of $I\subset S=K[x_1,\ldots,x_n]$17, one constructs an infinite Gröbner basis from binomials of two types: $I\subset S=K[x_1,\ldots,x_n]$18 (Lin et al., 2019). Finite closed subsets yield finite Gröbner bases, and successive colon ideals of the initial ideals are shown to be extended from the coefficient ring $I\subset S=K[x_1,\ldots,x_n]$19. A depth induction via the Depth Lemma then proves

$I\subset S=K[x_1,\ldots,x_n]$20

(Lin et al., 2019).

In low dimension, the structure can be sharpened through Sylvester forms. For binary monomial ACIs

$I\subset S=K[x_1,\ldots,x_n]$21

the Rees ideal is generated by the initial syzygies together with iterated Sylvester forms determined by the Euclidean algorithm for $I\subset S=K[x_1,\ldots,x_n]$22 (Simis et al., 2014). The resulting Rees algebra is almost Cohen–Macaulay, and it is Cohen–Macaulay if and only if $I\subset S=K[x_1,\ldots,x_n]$23 (Simis et al., 2014). In the ternary uniform case

$I\subset S=K[x_1,\ldots,x_n]$24

the Rees ideal is generated by six syzygies together with four Sylvester forms $I\subset S=K[x_1,\ldots,x_n]$25 (or $I\subset S=K[x_1,\ldots,x_n]$26 when $I\subset S=K[x_1,\ldots,x_n]$27), and the Rees algebra is again almost Cohen–Macaulay (Simis et al., 2014).

The ternary Sylvester forms are explicit: $I\subset S=K[x_1,\ldots,x_n]$28 and

$I\subset S=K[x_1,\ldots,x_n]$29

or

$I\subset S=K[x_1,\ldots,x_n]$30

(Simis et al., 2014). Mapping-cone arguments applied to explicit colon ideals then yield a free resolution of length at most $I\subset S=K[x_1,\ldots,x_n]$31, proving the almost Cohen–Macaulay property (Simis et al., 2014).

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 $I\subset S=K[x_1,\ldots,x_n]$32, 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 (Chase et al., 24 Jul 2025).

A major recent theorem classifies the SLP for monomial ACIs whose non-pure-power generator has support in two variables. If

$I\subset S=K[x_1,\ldots,x_n]$33

over a field of characteristic $I\subset S=K[x_1,\ldots,x_n]$34, then after relabeling so that $I\subset S=K[x_1,\ldots,x_n]$35, $I\subset S=K[x_1,\ldots,x_n]$36 has the SLP if and only if one of four conditions holds: $I\subset S=K[x_1,\ldots,x_n]$37; or $I\subset S=K[x_1,\ldots,x_n]$38 and $I\subset S=K[x_1,\ldots,x_n]$39; or the two-variable factor

$I\subset S=K[x_1,\ldots,x_n]$40

is almost centered; or, equivalently, the explicit inequalities

$I\subset S=K[x_1,\ldots,x_n]$41

and one of

$I\subset S=K[x_1,\ldots,x_n]$42

hold (Chase et al., 24 Jul 2025). The two-variable Hilbert series of $I\subset S=K[x_1,\ldots,x_n]$43 is unimodal; it is symmetric if and only if

$I\subset S=K[x_1,\ldots,x_n]$44

in the notation $I\subset S=K[x_1,\ldots,x_n]$45, and its socle degree is

$I\subset S=K[x_1,\ldots,x_n]$46

(Chase et al., 24 Jul 2025). If the Hilbert series of a monomial ACI is symmetric, then the algebra always has the SLP in characteristic $I\subset S=K[x_1,\ldots,x_n]$47 (Chase et al., 24 Jul 2025).

The WLP in three-variable level monomial ACIs is subtler. For

$I\subset S=K[x_1,\ldots,x_n]$48

with $I\subset S=K[x_1,\ldots,x_n]$49, levelness is equivalent to

$I\subset S=K[x_1,\ldots,x_n]$50

(Booth et al., 12 Mar 2026). The analysis reduces the WLP problem to the two-variable colon ideal

$I\subset S=K[x_1,\ldots,x_n]$51

whose generators are described explicitly by binomial-coefficient formulas (Booth et al., 12 Mar 2026). In the level case, WLP fails exactly when the determinant of a certain explicitly constructed $I\subset S=K[x_1,\ldots,x_n]$52 matrix vanishes; for fixed parity of $I\subset S=K[x_1,\ldots,x_n]$53, this gives a polynomial criterion

$I\subset S=K[x_1,\ldots,x_n]$54

depending on whether $I\subset S=K[x_1,\ldots,x_n]$55 is even or odd (Booth et al., 12 Mar 2026). The same work proves new cases of the Migliore–Miró-Roig–Nagel conjecture near the boundary

$I\subset S=K[x_1,\ldots,x_n]$56

(Booth et al., 12 Mar 2026).

A related but distinct direction concerns ideals of the form

$I\subset S=K[x_1,\ldots,x_n]$57

with $I\subset S=K[x_1,\ldots,x_n]$58 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 (Kling et al., 30 Jun 2025). This yields

$I\subset S=K[x_1,\ldots,x_n]$59

where $I\subset S=K[x_1,\ldots,x_n]$60, providing a new proof of the SLP for monomial complete intersections in characteristic $I\subset S=K[x_1,\ldots,x_n]$61 (Kling et al., 30 Jun 2025). The same enumeration connects Gröbner-basis degree counts to Catalan, Motzkin, and Riordan numbers (Kling et al., 30 Jun 2025).

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 $I\subset S=K[x_1,\ldots,x_n]$62 with system of parameters $I\subset S=K[x_1,\ldots,x_n]$63, one asks whether

$I\subset S=K[x_1,\ldots,x_n]$64

For $I\subset S=K[x_1,\ldots,x_n]$65, this statement is equivalent to the Monomial Conjecture, and hence valid; under additional small-multiplicity hypotheses such as $I\subset S=K[x_1,\ldots,x_n]$66, all positive Koszul homologies are annihilated by $I\subset S=K[x_1,\ldots,x_n]$67, and residual approximation complexes resolve the residue field (Tavanfar, 2017). 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; $I\subset S=K[x_1,\ldots,x_n]$68 is not clean	(Bandari et al., 2013)
Monomial ACIs are automatically Cohen–Macaulay	$I\subset S=K[x_1,\ldots,x_n]$69 is Cohen–Macaulay iff the ACI is non-dominant	(Mafi et al., 24 May 2025)
Rees algebras of monomial ACIs are always Cohen–Macaulay	The general result is almost Cohen–Macaulay, not necessarily Cohen–Macaulay	(Lin et al., 2019, Simis et al., 2014)

The scope of current theory is also uneven. Stanley-depth, cleanliness, and explicit homological invariants are well developed for monomial ACIs (Bandari et al., 2013, Mafi et al., 24 May 2025, Mafi et al., 6 Jun 2026). Rees-algebra structure is understood for Artinian monomial ACIs in general and with finer generators in binary and ternary low-dimensional families (Lin et al., 2019, Simis et al., 2014). 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 (Chase et al., 24 Jul 2025), and WLP in three-variable level cases still depends on determinant criteria rather than a fully closed classification (Booth et al., 12 Mar 2026). For non-monomial ACIs, most of these explicit formulas and combinatorial constructions are outside current scope (Bandari et al., 2013).

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.