Multicomplex Configurations in Algebra

• Multicomplex configurations are specialized projective varieties built from polarizing Artinian monomial ideals, ensuring controlled Gröbner bases and combinatorial invariants.
• The construction uses finite divisor-closed sets of c-monomials with polarization to transform ideals, followed by substituting homogeneous forms to form configuration ideals.
• These ideals are Cohen–Macaulay, preserve graded Betti numbers, and lie in the Gorenstein liaison class of a complete intersection, with explicit primary decompositions.

Searching arXiv for the specified paper and closely related work on multicomplex structures and configurations. Multicomplex configurations are a class of projective varieties constructed via specialization of the polarizations of Artinian monomial ideals. In the formulation of "Multicomplex Configurations: a case study in Gorenstein Liaison" (Klein et al., 14 Jul 2025), the basic input is a finite divisor-closed set of bounded monomials, together with ordered families of homogeneous forms used to substitute for the polarization variables. Under suitable $x_i$-initial hypotheses and a distinctness condition on the induced complete intersections, the resulting ideals admit Gröbner bases with prescribed initial ideals, have irredundant primary decompositions by linear-space ideals, are Cohen–Macaulay, preserve graded Betti numbers, and lie in the Gorenstein liaison class of a complete intersection.

1. Combinatorial input: $c$-multicomplexes and Artinian monomial ideals

The theory is developed over an infinite field $K$, with polynomial rings

$$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$

and with a fixed monomial order $<$ on $S$ refining $x_1>x_2>\cdots>x_m$ (Klein et al., 14 Jul 2025).

For $c=(c_1,\ldots,c_n)\in\mathbb N^n$, a $c$-monomial in $R$ is a monomial $c$0 with $c$1. A finite set $c$2 of $c$3-monomials is a $c$4-multicomplex if whenever $c$5 and $c$6, then $c$7. The associated monomial ideal is

$c$8

Because $c$9 is closed under divisors, $K$0 is an Artinian, in fact zero-dimensional, $K$1-algebra whose standard monomials are exactly those in $K$2 (Klein et al., 14 Jul 2025).

Polarization provides the squarefree model behind the construction. One introduces new variables $K$3 and replaces each $K$4 by the product $K$5. This produces a squarefree monomial ideal $K$6 in

$K$7

The multicomplex configuration is then obtained by specializing these polarization variables to homogeneous forms in a larger polynomial ring.

2. Configuration ideals by specialization of polarization

Fix a $K$8-multicomplex $K$9. For each $$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$0, choose an ordered collection of homogeneous forms

$$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$1

all of the same degree, and set $$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$2. If $$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$3 denotes the minimal set of monomial generators of $$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$4, the multicomplex configuration ideal determined by $$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$5 is

$$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$6

Equivalently, one first forms the polarization $$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$7, then applies the $$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$8-algebra map $$R = K[x_1,\ldots,x_n] \subseteq S = K[x_1,\ldots,x_m], \qquad m \ge n,$$9 to obtain $<$0 as the image of $<$1 (Klein et al., 14 Jul 2025).

A basic example takes

$<$2

so that

$<$3

In $<$4, choose

$<$5

Then

$<$6

For this choice, $<$7 is four points in $<$8, each cut out by one of the pairwise-distinct ideals

$<$9

(Klein et al., 14 Jul 2025).

3. Gröbner bases, initial ideals, and primary decomposition

The central Gröbner-theoretic hypotheses are that each $S$0 is $S$1-initial, meaning that for every $S$2,

$S$3

and that the complete intersection ideals

$S$4

for $S$5 with $S$6 are all distinct. Under these assumptions, Theorem 3.1 gives three structural conclusions (Klein et al., 14 Jul 2025).

First,

$S$7

Second, the collection

$S$8

is a Gröbner basis of $S$9. Third,

$x_1>x_2>\cdots>x_m$0

is the irredundant primary decomposition.

The proof proceeds through a chain of inclusions and numerical equalities. For each minimal generator $x_1>x_2>\cdots>x_m$1,

$x_1>x_2>\cdots>x_m$2

so $x_1>x_2>\cdots>x_m$3. If $x_1>x_2>\cdots>x_m$4, then for each $x_1>x_2>\cdots>x_m$5 at least one coordinate satisfies $x_1>x_2>\cdots>x_m$6, so $x_1>x_2>\cdots>x_m$7 divides $x_1>x_2>\cdots>x_m$8; consequently every $x_1>x_2>\cdots>x_m$9 lies in

$c=(c_1,\ldots,c_n)\in\mathbb N^n$0

and hence $c=(c_1,\ldots,c_n)\in\mathbb N^n$1. The argument then uses the fact that both chains of inclusions

$c=(c_1,\ldots,c_n)\in\mathbb N^n$2

preserve codimension $c=(c_1,\ldots,c_n)\in\mathbb N^n$3 and degree $c=(c_1,\ldots,c_n)\in\mathbb N^n$4, which forces equality throughout (Klein et al., 14 Jul 2025).

These equalities have immediate geometric consequences. Since $c=(c_1,\ldots,c_n)\in\mathbb N^n$5 is Cohen–Macaulay of codimension $c=(c_1,\ldots,c_n)\in\mathbb N^n$6, the ideal $c=(c_1,\ldots,c_n)\in\mathbb N^n$7 is also Cohen–Macaulay. Moreover, each primary component has degree $c=(c_1,\ldots,c_n)\in\mathbb N^n$8, so

$c=(c_1,\ldots,c_n)\in\mathbb N^n$9

Thus the specialization preserves both a combinatorial counting invariant and a rigid linear-space decomposition.

4. Gorenstein liaison and the inductive mechanism

A principal result is that $c$0 is in the Gorenstein liaison class of a complete intersection, or glicci. The proof is by double induction on $c$1 and on $c$2, and it uses geometric polarization, geometric vertex decomposition, and elementary G-biliaison (Klein et al., 14 Jul 2025).

For one-step geometric polarization, fix a variable $c$3 and a polynomial

$c$4

with $c$5 and no $c$6 divisible by $c$7. Introducing a new variable $c$8, define

$c$9

If $R$0 is a Gröbner basis of an ideal $R$1 all of whose leading terms come from the $R$2-part, then $R$3 is a Gröbner basis of the polarized ideal in $R$4, provided $R$5 remains a nonzerodivisor.

For geometric vertex decomposition, with $R$6 fixed and the term order $R$7-compatible, suppose $R$8 admits a Gröbner basis of the form

$R$9

with no lead term divisible by $c$00 except the $c$01. Then

$c$02

where

$c$03

is the geometric link and

$c$04

is the geometric deletion.

Elementary G-biliaison is formulated for unmixed Cohen–Macaulay ideals $c$05 of the same height, with $c$06 a Cohen–Macaulay $c$07 ideal such that

$c$08

and such that there is a graded isomorphism

$c$09

over $c$10. Such a step sits inside two direct G-links, hence preserves glicci.

The inductive proof for multicomplex configurations splits into two cases. If $c$11 divides every minimal generator of $c$12, one peels off the linear form $c$13 and reduces to the case $c$14. Otherwise, one applies one-step polarization in the $c$15-direction to the Gröbner basis $c$16. In this situation, $c$17 is regular, the geometric link is the configuration ideal for the smaller multicomplex

$c$18

in the ring with variable $c$19, and the geometric deletion is the configuration for

$c$20

Both $c$21 and $c$22 have strictly smaller size or support, so by induction their configuration ideals are glicci. The resulting elementary G-biliaison links $c$23 to these smaller ideals, and this yields the glicci property for $c$24 (Klein et al., 14 Jul 2025).

5. Invariants, Betti numbers, and the model example

The summary theorem packages the principal invariants of the construction. If $c$25 is a $c$26-multicomplex, $c$27 is a monomial order on $c$28 refining $c$29, and for each $c$30 one chooses an $c$31-initial set of linear forms

$c$32

such that the complete-intersection ideals

$c$33

are pairwise distinct, then $c$34 satisfies the following properties (Klein et al., 14 Jul 2025):

1. $c$35 is an irredundant intersection of linear-space ideals of codimension $c$36.
2. $c$37 is a Gröbner basis and $c$38.
3. $c$39, $c$40, and $c$41 is Cohen–Macaulay.
4. $c$42 is in the Gorenstein liaison class of a complete intersection.
5. The graded Betti numbers are preserved: $c$43

The Betti-number statement is attributed to the usual distraction arguments: because each $c$44 is $c$45-initial, the graded Betti numbers of $c$46 in $c$47 coincide with those of $c$48 in $c$49. This makes the construction a specialization that is geometrically nontrivial but homologically controlled.

The running example illustrates all of these statements concretely. For

$c$50

one has

$c$51

$c$52

and

$c$53

The ideal is Cohen–Macaulay, and one-step polarization in $c$54 introduces $c$55 and leads to a G-biliaison linking $c$56 to two smaller configurations, each of which is a complete intersection or known to be glicci; hence $c$57 is glicci (Klein et al., 14 Jul 2025).

6. Terminological scope and related uses of “multicomplex”

In this subject, “multicomplex configuration” refers specifically to the algebraic-geometric construction obtained by specializing polarizations of Artinian monomial ideals (Klein et al., 14 Jul 2025). The term “multicomplex,” however, has broader usage across several research areas.

In commutative algebra and algebraic structures, the purely complex-signature family $c$58 is exactly the classical multicomplex algebra $c$59 of Segre–Price, and more generally commutative analogues of Clifford algebras are either isomorphic to a multicomplex space or to a multi split-complex space (Sharma et al., 28 Apr 2025). In complex dynamics, multicomplex numbers support Multibrot and Julia theories: every multicomplex tridimensional principal slice of the Multibrot sets is equivalent to a tricomplex slice up to an affine transformation (Brouillette et al., 2018), while for filled-in Julia sets associated to $c$60 there are four principal 3D-slice classes when $c$61 is even and eight or nine when $c$62 is odd, depending on $c$63 and $c$64 (Charles et al., 2 May 2025). In homological topology, a first-quadrant multicomplex is a bigraded module

$c$65

with maps

$c$66

satisfying

$c$67

and the Morse–Bott multicomplex combines singular cubical chains and Morse chains in the same multicomplex (Hurtubise, 2012).

This suggests that the qualifier “configuration” is not a generic synonym for multicomplex structures, but a designation for a particular liaison-theoretic and Gröbner-theoretic specialization framework. Within that framework, the main significance of the theory is that it transfers combinatorial data from a $c$68-multicomplex to a projective configuration while retaining controlled initial ideals, explicit primary decompositions, Cohen–Macaulayness, glicci behavior, and the graded Betti numbers of the originating Artinian monomial ideal (Klein et al., 14 Jul 2025).