# Linear Strands Supported on Regular CW Complexes

Keller VandeBogert

###### Abstract.

In this paper, we study ideals $I$ whose linear strand can be supported on a
regular CW complex. We provide a sufficient condition for the linear strand of
an arbitrary subideal of $I$ to remain supported on an easily described
subcomplex. In particular, we prove that a certain class of rainbow monomial
ideals always have linear strand supported on a regular CW complex, including
any initial ideal of the ideal of maximal minors of a generic matrix. We also
provide a sufficient condition for these ideals to have linear resolution,
which is also an equivalence under mild assumptions. We then employ a result
of Almousa, Fløystad, and Lohne to apply these results to polarizations of
Artinian monomial ideals. We conclude with further questions relating to
cellularity of certain classes of squarefree monomial ideals and the
relationship between initial ideals of maximal minors and algebra structures
on certain resolutions.

##  1\. Introduction

A common theme in the study of monomial ideals is the association of some kind
of combinatorial object $P$ that can recover the ideal of interest $I$. It is
then often the case that properties desirable from a combinatorial perspective
for $P$ translate into properties that are desirable from an algebraic
perspective for $I$. This kind of correspondence has been used to great effect
in, for instance, Stanley-Reisner theory, where any squarefree monomial ideal
has a bijective correspondence with a simplicial complex. This correspondence
yields nontrivial consequences, such as Hochster’s formula or Reisner’s
criterion for Cohen-Macaulayness.

From a homological perspective, the object that best captures the behavior of
a monomial ideal $I$ is a free resolution. Thus, a similar line of thinking
suggests that the association of a free resolution $F_{\bullet}$ of $I$ to
some combinatorial object $P$ will yield a dictionary for translating
properties of $P$ into properties of $F_{\bullet}$. This was first explored by
Bayer, Peeva, and Sturmfels [3], where free resolutions that could be
supported on simplicial complexes were studied. This means that the
differential of $F_{\bullet}$ and the natural differential on the simplicial
complex induce each other, and there is a bijective correspondence between
basis elements in a given homological degree and faces of a given dimension.
Examples of well-known complexes supported on simplicial complexes include the
Taylor resolution and the Scarf complex. However, simplicial complexes turn
out to be too restrictive in general to support more than the most well-
behaved complexes.

This leads to the idea of considering _cellular or CW resolutions_ , that is,
resolutions supported on so-called cell/regular CW complexes, introduced by
Bayer and Sturmfels in [4]. Cellular and CW resolutions have turned out to
occupy a “goldilocks” zone of generality; they are general enough to support
minimal free resolutions of many large classes of monomial ideals, including
the well-known Eliahou-Kervaire resolution (see [24]), but not so general as
to be combinatorially incomprehensible. There are many classes of ideals whose
minimal free resolutions can be supported on cell or regular CW complexes;
see, for instance [10], [2], [18], [13], [14], [15], or [11].

It turns out that not _all_ monomial ideals have cellular resolutions; indeed,
there are monomial ideals whose minimal free resolution cannot even be
supported on a CW-complex (this is due to Velasco in [36]). Reiner and Welker
(see [31]) have also constructed a relatively simple monomial ideal with
linear resolution, but such that no change of basis has differentials with
coefficients equal to $0$ or $\pm 1$. Minimal free resolutions of arbitrary
monomial ideals are not completely devoid of structure, however; a result of
Clark and Tchernev (see [9]) shows that every monomial ideal has minimal free
resolution that can be supported on a so-called hcw-poset.

In [26], a specific cell complex called the _complex of boxes_ was used to
support what is now called the _box polarization_ of squarefree strongly
stable ideals. In general, polarizations are a method of associating to any
monomial ideal $I$ a squarefree monomial $\widetilde{I}$ in a larger
polynomial ring that is homologically indistinguishable. There are _many_
different ways to polarize a monomial ideal; indeed, Almousa, Fløystad, and
Lohne in [1] describe all such polarizations for powers of the graded maximal
ideal in terms of spanning trees of certain maximal “down triangles”
associated to the graph of linear syzygies. Moreover, they also show that the
study of polarizations of an Artinian monomial ideals is equivalent via
Alexander duality to the study of ideals generated by _rainbow monomials_ with
linear resolution. Rainbow monomial ideals also appear as defining ideals for
sets of points in multiprojective spaces, where each color class corresponds
to a distinct copy of $\mathbb{P}^{n}$; see the work by Favacchio, Guardo, and
Migliore [16] or [17] for rainbow monomials from this perspective.

In this paper, we use these connections as motivation for studying a proper
subclass of all equigenerated rainbow monomial ideals, which we call _rainbow
determinantal facet ideals_ (or rainbow DFIs). These ideals are inspired by
_determinantal facet ideals_ (DFIs), a generalization of binomial edge ideals
(see [12], [27], [21], and [23]); the main difference is that rainbow DFIs
depend on both a term order $<$ _and_ a parametrizing simplicial complex
$\Delta$. We prove that the linear strand of _any_ rainbow DFI is supported on
a regular CW complex; in particular, any rainbow DFI with linear resolution
(hence Alexander dual to a polarization) has a CW resolution. Moreover, we
formulate a sufficient condition for any rainbow DFI to have linear
resolution; if the Alexander dual $\Delta^{\vee}$ of the associated simplicial
complex has nonmaximal overlap, this becomes an equivalence.

The techniques involved in proving these statements are of independent
interest and involve the formulation of more general statements on the
structure of the minimal free resolution of any initial ideal of the ideal of
maximal minors, the construction of linear strands of arbitrary modules, and
CW-ness of complexes whose associated poset is CW. Moreover, we examine
conditions guaranteeing that a sequence that is regular on $R/I$ is also
regular on $R/J$ for any fixed $J\subset I$; this is used for computing the
graded Betti numbers of rainbow DFIs (and is later used to give a
characterization of Cohen-Macaulayness).

The paper is organized as follows. In Section 2, we first introduce some
necessary background and notation relating to (rainbow) DFIs, the Eagon-
Northcott complex, and arbitrary initial ideals of ideals of maximal minors.
We perform a more detailed study of so-called _sparse_ Eagon-Northcott
complexes, including an explicit description of the differentials and the
multidegrees appearing in each homological degree.

In Section 3, we consider the construction of linear strands in general. We
first prove, using the machinery of _iterated trimming complexes_ (introduced
in [35]), that the linear strand of any subideal with $G(J)\subseteq G(I)$ is
obtained by taking an appropriate subcomplex of the linear strand of $I$. We
then apply this result to the case that the linear strands in question are
supported on cellular or CW complexes; we introduce the notion of _support
chains_ centered at a vertex $v$ of a regular CW complex to construct a
morphism of complexes. Taking the kernel of this morphism of complexes will
yield the linear strand upon deleting of the generator corresponding to the
vertex $v$. In the case that the CW complex is _linearly connected_ , the
resulting linear strand is obtained by simply restricting to the subcomplex
induced by deleting $v$.

In Section 4, we study conditions guaranteeing that certain posets associated
to a complex $F_{\bullet}$ give rise to a regular CW complex supporting
$F_{\bullet}$. Using results of Björner, CW-ness can be checked by showing
that the associated poset is CW. We use this result to prove that any initial
ideal of the ideal of maximal minors has CW resolution. Even more generally,
we combine this with the results of Section 3 to prove that the linear strand
of every rainbow DFI is supported on a regular CW complex.

In Section 5, we study conditions on the simplicial complex $\Delta$
associated to a rainbow DFI implying linearity of the minimal free resolution.
This condition is formulated in terms of the existence of an ordering on the
elements of $\Delta^{\vee}$ such that they form a so-called _free sequence_ ;
moreover, if elements of $\Delta^{\vee}$ have nonmaximal overlap, then every
ordering yields a free-sequence and this condition is also an equivalence. In
this latter situation, we also compute the Betti numbers explicitly; we
already know by the result of Section 4 that these ideals have CW resolution.

Finally, in Section 6, we apply our results to polarizations of Artinian
monomial ideals. Taking Alexander duals, all of the conditions guaranteeing
linearity of Section 5 directly translate to the property of being the
polarization of an Artinian monomial ideal. Moreover, assuming that all
elements of $\Delta^{\vee}$ have nonmaximal overlap, we show that rainbow DFIs
are Alexander dual to a polarization of a power of the maximal ideal if and
only if $|\Delta^{\vee}|=0$. We conclude with a variety of questions: we first
ask about whether or not the admission of an order for which the elements of
$\Delta^{\vee}$ form a free sequence is a necessary condition for the
associated rainbow DFI to have linear resolution. We also ask about the
existence of term orders for which a pre-selected set of generators are free
vertices of the regular CW complex supporting the resolution of the initial
ideal of maximal minors. Lastly, we pose a curious potential connection
between the selection of a multigraded algebra structure on the minimal
resolution of the ideal of all squarefree monomials, and an arbitrary initial
ideal of an ideal of maximal minors.

##  2\. Background and Sparse Eagon-Northcott Complexes

In this section, we introduce some notation related to (rainbow) determinantal
facet ideals that will be in play for the rest of the paper. We then delve
into so-called _sparse_ Eagon-Northcott complexes, which arise by homogenizing
the classical Eagon-Northcott complex with respect to a variable $t$ and term
order $<$, then setting $t=0$. It turns out that one can say _precisely_ what
the differentials and multidegrees must look like for any given term order.
These complexes will be used in later sections for proving cellularity/CW-ness
and establishing criteria for having linear resolution.

Throughout this paper, all complexes will be assumed to be nontrivial only in
nonnegative homological degrees.

######  Notation 2.1.

Let $R=k[x_{ij}\mid 1\leqslant i\leqslant n,\ 1\leqslant j\leqslant m]$ be a
polynomial ring over an arbitrary field $k$. Let $M$ be an $n\times m$ matrix
of variables in $R$ where $n\leqslant m$. For indices
$\textbf{a}=\\{a_{1},\ldots,a_{r}\\}$ and
$\textbf{b}=\\{b_{1},\ldots,b_{r}\\}$ such that $1\leqslant
a_{1}<\ldots<a_{r}\leqslant n$ and $1\leqslant b_{1}<\cdots<b_{r}\leqslant m$,
set

$[\textbf{a}|\textbf{b}]=[a_{1},\ldots,a_{r}|b_{1},\ldots,b_{r}]=\det\left(\begin{array}[]{ccc}x_{a_{1},b_{1}}&\cdots&x_{a_{1},b_{r}}\\\
\vdots&\ddots&\vdots\\\ x_{a_{r},b_{1}}&\cdots&x_{a_{r},b_{r}}\\\
\end{array}\right)$

where $[\textbf{a}|\textbf{b}]=0$ if $r>n$. When $r=n$, use the simplified
notation $[\textbf{a}]$ = $[1,\ldots,n|\textbf{a}]$. The ideal generated by
the $r$-minors of $M$ is denoted $I_{r}(M)$.

######  Definition 2.2.

For a simplicial complex $\Delta$ and an integer $i$, the $i$-th skeleton
$\Delta^{(i)}$ of $\Delta$ is the subcomplex of $\Delta$ whose faces are those
faces of $\Delta$ with dimension at most $i$. Let $\mathcal{S}$ denote the set
of simplices $\Gamma$ with vertices in $[m]$ with $\dim(\Gamma)\geqslant n-1$
and $\Gamma^{(n-1)}\subset\Delta$.

Let $\Gamma_{1},\dotsc,\Gamma_{c}$ be maximal elements in $\mathcal{S}$ with
respect to inclusion, and let $\Delta_{i}:=\Gamma^{(n-1)}_{i}$. Each
$\Gamma_{i}$ is called a _maximal clique_ , and any induced subcomplex of
$\Gamma_{i}$ is a _clique_. The simplicial complex $\Delta^{\textrm{clique}}$
whose facets are the maximal cliques of $\Delta$ is called the _clique
complex_ associated to $\Delta$. The decomposition
$\Delta=\Delta_{1}\cup\cdots\cup\Delta_{c}$ is called the _maximal clique
decomposition_ of $\Delta$.

######  Definition 2.3.

Let $\Delta$ be a pure $(n-1)$-dimensional simplicial complex on the vertex
set $[m]$. Let $R=k[x_{ij}\mid 1\leqslant i\leqslant n,\ 1\leqslant j\leqslant
m]$ be a polynomial ring over an arbitrary field $k$. Let $M$ be an $n\times
m$ matrix of variables in $R$. The _determinantal facet ideal_ (or _DFI_)
$J_{\Delta}\subseteq R$ associated to $\Delta$ is the ideal generated by
determinants of the form $[\textbf{a}]$ where a supports an $(n-1)$-facet of
$\Delta$; that is, the columns of $[\textbf{a}]$ correspond to the vertices of
some facet of $\Delta$.

Let $<$ be an arbitrary term order. The _rainbow DFI_
$\operatorname{rain}_{<}(J_{\Delta})$ associated to $\Delta$ is the monomial
ideal generated by all monomials of the form $\textrm{in}_{<}([\textbf{a}])$,
where a supports and $(n-1)$-facet of $\Delta$.

Observe that the maximal clique decomposition of the associated simplicial
complex $\Delta$ is telling us the largest submatrices of $M$ from which _all_
minors are taken. That is, $J_{\Delta}=J_{\Delta_{1}}+\cdots+J_{\Delta_{r}}$.

######  Notation 2.4.

Let $\Delta$ be a pure $(n-1)$-dimensional simplicial complex on the vertex
set $[m]$. The notation $\Delta^{\vee}$ will denote the Alexander dual of
$\Delta$; that is, $\Delta^{\vee}$ is the unique $(n-1)$-pure simplicial
complex with $\Delta\cup\Delta^{\vee}$ equal to all $n$-subsets of $[m]$.

Recall that the set of maximal minors forms a universal Gröbner basis for
$I_{n}(M)$ by a result of Sturmfels and Zelevinsky (see [33]). This
immediately yields the following:

######  Observation 2.5.

Let $<$ be an arbitrary term order. For any $(n-1)$-pure simplicial complex
$\Delta$,

$\operatorname{rain}_{<}(J_{\Delta})+\operatorname{rain}_{<}(J_{\Delta^{\vee}})=\textrm{in}_{<}I_{n}(M).$

Next we define the _linear strand_ , which is one of the central objects of
study in this paper. These will be dealt with more intensively in Section 3.

######  Definition 2.6.

Let $F_{\bullet}$ be a minimal graded $R$-free complex with $F_{0}$ having
initial degree $d$. Then the _linear strand_ of $F_{\bullet}$, denoted
$F_{\bullet}^{\operatorname{lin}}$, is the complex obtained by restricting
$d_{i}^{F}$ to $(F_{i})_{d+i}$ for each $i\geqslant 1$.

######  Remark 2.7.

Observe that the minimality assumption in Definition 2.6 ensures that the
linear strand is well defined. Choosing bases, the linear strand can be
obtained by restricting to the columns where only linear entries occur in the
matrix representation of each differential.

The following result, due to Boocher, shows that with respect to any term
order $<$, the ideal $\textrm{in}_{<}I_{n}(M)$ specializes to the ideal of all
squarefree monomials of degree $n$ in $m$ variables.

######  Theorem 2.8 ([7, Proof of Theorem 3.1]).

For any term order $<$, the sequence of variable differences

$\\{x_{11}-x_{21},\dots,x_{11}-x_{n1}\\}\cup\cdots\cup\\{x_{1m}-x_{2m},\dotsc,x_{1m}-x_{nm}\\}$

forms a regular sequence on $R/\textrm{in}_{<}I_{n}(M)$. In particular,

$\beta_{ij}(R/I_{n}(M))=\beta_{ij}(R/\textrm{in}_{<}I_{n}(M))\ \textrm{for
all}\ i,j.$

The following definition is the definition of the well-known Eagon-Northcot
complex. Recall that the Eagon-Northcott complex is a minimal free resolution
of the quotient ring defined by the ideal of maximal minors $I_{n}(M)$; it
will turn out to be the basis from which all resolutions of
$R/\textrm{in}_{<}I_{n}(M)$ can be built.

######  Definition 2.9 (Eagon-Northcott complex).

Let $\phi:F\to G$ be a homomorphism of free modules of ranks $n$ and $m$,
respectively, with $n\geqslant m$. Let $c_{\phi}$ be the image of $\phi$ under
the isomorphism $\operatorname{Hom}_{R}(F,G)\xrightarrow{\cong}F^{*}\otimes
G$. The _Eagon-Northcott complex_ is the complex

$0\to D_{m-n}(G^{*})\otimes\bigwedge^{m}F\to
D_{m-n-1}(G^{*})\otimes\bigwedge^{m-1}F\to\cdots\to
G^{*}\otimes\bigwedge^{n+1}F\to\bigwedge^{n}F\to\bigwedge^{n}G$

with differentials in homological degree $\geqslant 2$ induced by
multiplication by the element $c_{\phi}\in F^{*}\otimes G$, and the map
$\bigwedge^{g}F\to\bigwedge^{g}G$ is $\bigwedge^{g}\phi$.

######  Notation 2.10.

Let $E_{\bullet}$ denote the Eagon-Northcott complex of Definition 2.9. If $F$
has basis $f_{1},\dots,f_{m}$ and $G$ has basis $g_{1},\dots,g_{n}$, then
define

$g^{*(\alpha)}\otimes f_{I}:=g_{1}^{*(\alpha_{1})}\cdots
g_{n}^{*(\alpha_{n})}\otimes f_{i_{1}}\wedge\cdots\wedge f_{i_{n+\ell}},$

where $\alpha=(\alpha_{1},\dots,\alpha_{n})$ and
$I=(i_{1}<\cdots<i_{n+\ell})$. Observe that $E_{\bullet}$ inherits a
$\mathbb{Z}^{n}\times\mathbb{Z}^{m}$-grading by setting

$\operatorname{mdeg}(g^{*(\alpha)}\otimes
f_{I})=(1+\alpha_{1}\epsilon_{1}+\cdots+\alpha_{n}\epsilon_{n},\epsilon_{i_{1}}+\cdots\epsilon_{i_{n+\ell}}),$

where $\epsilon_{k}$ denotes the appropriately sized vector with $1$ in the
$i$th spot and $0$ elsewhere, and $1$ denotes a length $n$ vector of $1$s.

######  Corollary 2.11.

Let $F_{\bullet}$ denote a multigraded resolution of
$\textrm{in}_{<}I_{n}(M)$. Then for every multidegree $\alpha$,

$\beta_{\alpha}(R/\textrm{in}_{<}I_{n}(M))\leqslant 1.$

###### Proof.

By Theorem 2.8, a minimal free resolution of $R/\textrm{in}_{<}I_{n}(M)$ may
be obtained by setting some of the entries in the matrix representation of the
differentials of Definition 2.9 equal to $0$. With respect to the
$\mathbb{Z}^{n}\times\mathbb{Z}^{m}$-grading of the Eagon-Northcott complex
$E_{\bullet}$, one has

$\beta_{\alpha}(R/\textrm{in}_{<}I_{n}(M))\leqslant 1.$

Since any $\mathbb{Z}^{nm}$-graded minimal free resolution is also
$\mathbb{Z}^{n}\times\mathbb{Z}^{m}$-graded, the result follows. ∎

######  Definition 2.12.

Let $R=k[x_{ij}\mid 1\leqslant i\leqslant n,\ 1\leqslant j\leqslant m]$ be a
polynomial ring over an arbitrary field $k$, endowed with some term order $<$.
The _sparse Eagon-Northcott_ $E_{\bullet}^{<}$ complex is the complex obtained
by:

  1. (1)

homogenizing $E_{\bullet}$ with a new variable $t$, with respect to the term
order $<$, then

  2. (2)

setting $t=0$ in the resolution.

As observed by Boocher, the minimal free resolution of
$R/\textrm{in}_{<}I_{n}(M)$ may be obtained as the sparse Eagon-Northcott
complex $E_{\bullet}^{<}$.

######  Definition 2.13.

Let $F$ be a free $R$-module with some fixed basis $B\subset F$. The _support_
of an element $f=\sum_{b\in B}c_{b}b\in F$ with respect to the basis $B$ is
defined to be:

$\operatorname{Supp}_{B}(f):=\\{b\in B\mid c_{b}\neq 0\\},$

where $c_{b}\in R$. When the basis $B$ is understood, the notation
$\operatorname{Supp}(f)$ will be used instead.

When dealing with the sparse Eagon-Northcott complex $E_{\bullet}^{<}$, the
support of any element will be assumed to be with respect to the basis
elements of Notation 2.10.

######  Observation 2.14.

Let $<$ be any term order. Then the associated sparse Eagon-Northcott complex
can be given a $\mathbb{Z}^{nm}$-multigrading.

###### Proof.

Assign the multidegrees inductively. In homological degree $1$, $f_{I}$ has
multidegree $\textrm{in}_{<}([I])$; for basis elements in higher homological
degree, the multidegree is assigned as the lcm of the multidegrees appearing
in the support of the image under the differential. ∎

######  Notation 2.15.

Let $<$ be a term order. Every basis element of the sparse Eagon-Northcott
complex $E_{\bullet}^{<}$ can be assigned a _unique_ multidegree

$g^{*(\alpha)}\otimes
f_{I}\xleftrightarrow{}\prod_{i=1}^{n}\Big{(}x_{ib_{|\alpha_{\leqslant
i-1}|+i}}\cdots x_{ib_{|\alpha_{\leqslant i}|+i}}\Big{)},$

where $I=\\{b_{1},\dots,b_{n+\ell}\\}$. For ease of notation, the above
multidegree will be written

$\prod_{i=1}^{n}\Big{(}x_{ib_{|\alpha_{\leqslant i-1}|+i}}\cdots
x_{ib_{|\alpha_{\leqslant i}|+i}}\Big{)}=x_{1\textbf{b}_{1}}\cdots
x_{n\textbf{b}_{n}},$

where $\textbf{b}_{i}=(b_{|\alpha_{\leqslant
i-1}|+i},\dots,b_{|\alpha_{\leqslant i}|+i})$.

######  Theorem 2.16.

Let $<$ be any term order. Then the complex $E_{\bullet}^{<}$ has
differentials of the form

$g^{*(\alpha)}\otimes
f_{I}\mapsto\sum_{\begin{subarray}{c}i\in\operatorname{Supp}(\alpha)\\\
j\in\textbf{b}_{i}\\\ \end{subarray}}\operatorname{sgn}(j\in
I)x_{ij}g^{*(\alpha-\epsilon_{i})}\otimes f_{I\backslash j},$

where $\operatorname{mdeg}(g^{*(\alpha)}\otimes
f_{I})=x_{1\textbf{b}_{1}}\cdots x_{n\textbf{b}_{n}}$, and

$f_{I}\mapsto\textrm{in}_{<}([I]).$

###### Proof.

This follows by induction on the homological degree, where the base case for
$\ell=2$ is clear. Let $\ell>2$; since $(E_{\bullet}^{<},d_{\bullet})$ is a
complex, by definition $d_{\ell-1}\circ d_{\ell}=0$. Assume for sake of
contradiction that $\operatorname{Supp}(d_{\ell}(g^{*(\alpha)}\otimes f_{I})$
does not contain $g^{*(\alpha-\epsilon_{i})}\otimes f_{I\backslash j}$ for
some $i\in\operatorname{Supp}(\alpha)$, $j\in\textbf{b}_{i}$. Let $r$ be any
integer and $s\in\textbf{b}_{r}$ such that $g^{*(\alpha-\epsilon_{r})}\otimes
f_{I\backslash s}\in\operatorname{Supp}(d_{\ell}(g^{*(\alpha)}\otimes
f_{I}))$; by the inductive hypothesis,
$g^{*(\alpha-\epsilon_{r}-\epsilon_{i})}\otimes f_{I\backslash
s,j}\in\operatorname{Supp}(g^{*(\alpha-\epsilon_{r})}\otimes f_{I\backslash
s})$. However, if $g^{*(\alpha-\epsilon_{i})}\otimes f_{I\backslash
j}\notin\operatorname{Supp}(d_{\ell}(g^{*(\alpha)}\otimes f_{I}))$, then the
term $x_{ij}x_{rs}g^{*(\alpha-\epsilon_{r}-\epsilon_{i})}\otimes
f_{I\backslash s,j}$ does not cancel with any other term in the image
$d_{\ell-1}\circ d_{\ell}(g^{*(\alpha)}\otimes f_{I})$, contradicting the fact
that $E_{\bullet}^{<}$ is a complex. ∎

Theorem 2.16 immediately yields an explicit description of the multidegrees
appearing in each homological degree. This result will be essential later for
showing that the associated poset forms a meet semilattice.

######  Corollary 2.17.

The multidegrees appearing in homological degree $\ell>0$ of the complex
$E_{\bullet}^{<}$ may be constructed by choosing all degree $n+\ell-1$
monomials $x_{1\textbf{b}_{1}}\cdots x_{n\textbf{b}_{n}}$ with the property
that

$x_{1b_{i_{1}}}x_{2b_{i_{2}}}\cdots x_{nb_{i_{n}}}\in
G(\textrm{in}_{<}I_{n}(M))$

for all $b_{i_{j}}\in\textbf{b}_{j}$, $j=1,\dots,n$.

###### Proof.

This follows by induction on the homological degree combined with the
differential of Theorem 2.16. ∎

To conclude this section, we record the following observations which will be
used to prove that bounded intervals in the poset associated to the sparse
Eagon-Northcott complexes are in fact CL-shellable.

######  Observation 2.18.

Let $<$ be any term order. Let $1\leqslant a_{1},\dots,a_{n}\leqslant m$ and
set $\mu:=x_{1a_{1}}x_{2a_{2}}\cdots x_{na_{n}}$. Assume that $b$ and $c$ are
integers such that

$\frac{\mu x_{ib}}{x_{ia_{i}}}\quad\textrm{and}\quad\frac{\mu
x_{jb}}{x_{ja_{j}}}\in G(\textrm{in}_{<}I_{n}(M)),\ \textrm{where}\ i\neq j.$

Then $\frac{\mu x_{ib}x_{jc}}{x_{ia_{i}}x_{ja_{j}}}\in
G(\textrm{in}_{<}I_{n}(M))$.

###### Proof.

Let $L=\\{a_{1},\dots,\widehat{a_{i}},b,\dots,a_{j},c,\dots,a_{n}\\}$ (it is
not assumed that these are increasing), and consider the standard Eagon-
Northcott relation

$\sum_{\ell\in L}\operatorname{sgn}(\ell)x_{j\ell}[L\backslash\ell]=0.$

Taking lead terms in the above yields a relation on the initial ideal. Observe
that after taking lead terms, the term $x_{jc}\cdot\frac{\mu
x_{ib}}{x_{ia_{i}}}$ appears in the above sum. The only other term that could
possibly cancel with this term is the term appearing with coefficient
$x_{ja_{j}}$, which must be $\frac{\mu x_{ib}x_{jc}}{x_{ia_{i}}x_{ja_{j}}}$. ∎

######  Observation 2.19.

Let $<$ be any term order. Let $1\leqslant a_{1},\dots,a_{n}\leqslant m$ and
$1\leqslant b_{1},\dots,b_{n}\leqslant m$, and set

$\mu=x_{1a_{1}}\cdots x_{na_{n}},\qquad\nu=x_{1b_{1}}\cdots x_{nb_{n}}.$

Assume that $\textrm{lcm}(\mu,\nu)$ is a valid multidegree as in Corollary
2.17. Then there exists $1\leqslant i\leqslant n$ such that
$\frac{x_{ia_{i}}\nu}{x_{ib_{i}}}\in G(\textrm{in}_{<}I_{n}(M))$ _and_
$\frac{x_{ia_{i}}\nu}{x_{ib_{i}}}<\nu$.

###### Proof.

Recall that any term order $<$ on $R$ may be obtained as the ordering
corresponding to a weight vector $\omega\in\mathbb{Z}^{nm}$ (see, for
instance, [32, Proposition 1.11]). Observe that if $\textrm{lcm}(\mu,\nu)$ is
a valid multidegree as in Corollary 2.17, then for all $1\leqslant i\leqslant
n$, one has $\frac{x_{ia_{i}}\nu}{x_{ib_{i}}}\in G(\textrm{in}_{<}I_{n}(M))$.
Thus it suffices to show that $i$ may be chosen such that
$\frac{x_{ia_{i}}\nu}{x_{ib_{i}}}<\nu$.

In terms of the weight vector $\omega$, the assumption $\mu<\nu$ implies
$e(\mu)\cdot\omega<e(\nu)\cdot\omega$ (where $e(\cdot)$ denotes the
corresponding exponent vector); this means proving the statement is equivalent
to proving the following:

If $p_{1}+\cdots+p_{\ell}<0$ for some integers $p_{i}\in\mathbb{Z}$, then
$p_{t}<0$ for some $1\leqslant t\leqslant\ell$.

But this is trivially true, whence the statement follows. ∎

##  3\. On the Construction of Linear Strands

In this section, we consider the construction of linear strands. We first
formulate a more general statement on constructing the linear strand of
certain classes of ideals with the machinery of _iterated trimming complexes_.
With this more general result, we consider ideals whose linear strand is
supported on a regular CW or cell complex. After developing a fair amount of
machinery, this section culminates in Theorems 3.19 and 3.20. These results
show that ideals whose linear strand is supported on a _linearly connected_
cell/CW complex have the property that every subideal is supported on a
cell/CW complex that is simple to describe.

We begin this section with a general result on linear strands originally due
to Herzog, Kiani, and Madani; intuitively, this says that a linear complex
arises as a linear strand if and only if the homology is concentrated in
sufficienly large degrees.

######  Theorem 3.1 ([22], Theorem 1.1).

Let $R$ be a standard graded polynomial ring over a field $k$. Let
$G_{\bullet}$ be a finite linear complex of free $R$-modules with initial
degree $n$. Then the following are equivalent:

  1. (1)

The complex $G_{\bullet}$ is the linear strand of a finitely generated
$R$-module with initial degree $n$.

  2. (2)

The homology $H_{i}(G_{\bullet})_{i+n+j}=0$ for all $i>0$ and $j=0,\ 1$.

We will temporarily adopt the following general setup for the next two
results. In this setup, we are choosing specific generators of a given ideal
$I$ that we wish to trim off, then “rescale” by some new ideal. Theorem 3.3
shows how a (generally nonminimal) resolution can be computed for this new
ideal based on the free resolutions of all other ideals involved.

######  Setup 3.2.

Let $R$ be a standard graded polynomial ring over a field $k$. Let $I\subseteq
R$ be a homogeneous ideal and $(F_{\bullet},d_{\bullet})$ denote a homogeneous
free resolution of $R/I$.

Write $F_{1}=F_{1}^{\prime}\oplus\Big{(}\bigoplus_{i=1}^{t}Re_{0}^{i}\Big{)}$,
where, for each $i=1,\dotsc,t$, $e^{i}_{0}$ generates a free direct summand of
$F_{1}$. Using the isomorphism

$\operatorname{Hom}_{R}(F_{2},F_{1})=\operatorname{Hom}_{R}(F_{2},F_{1}^{\prime})\oplus\Big{(}\bigoplus_{i=1}^{t}\operatorname{Hom}_{R}(F_{2},Re^{i}_{0})\Big{)}$

write $d_{2}=d_{2}^{\prime}+d_{0}^{1}+\cdots+d^{t}_{0}$, where
$d_{2}^{\prime}\in\operatorname{Hom}_{R}(F_{2},F_{1}^{\prime})$ and
$d^{i}_{0}\in\operatorname{Hom}_{R}(F_{2},Re^{i}_{0})$.

For each $i=1,\dotsc,t$, let $\mathfrak{a}_{i}$ denote any homogeneous ideal
with

$d^{i}_{0}(F_{2})\subseteq\mathfrak{a}_{i}e^{i}_{0},$

and $(G^{i}_{\bullet},m^{i}_{\bullet})$ be a homogeneous free resolution of
$R/\mathfrak{a}_{i}$.

Use the notation
$K^{\prime}:=\operatorname{im}(d_{1}|_{F_{1}^{\prime}}:F_{1}^{\prime}\to R)$,
$K^{i}_{0}:=\operatorname{im}(d_{1}|_{Re^{i}_{0}}:Re^{i}_{0}\to R)$, and let
$J:=K^{\prime}+\mathfrak{a}_{1}\cdot K^{1}_{0}+\cdots+\mathfrak{a}_{t}\cdot
K_{0}^{t}$.

######  Theorem 3.3.

Adopt notation and hypotheses as in Setup 3.2. Then there is a morphism of
complexes

(3.1)
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{k+1}}$$\textstyle{F_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q_{k-1}}$$\scriptstyle{d_{k}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{3}}$$\textstyle{F_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{2}^{\prime}}$$\scriptstyle{Q_{1}}$$\textstyle{F_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigoplus
m^{i}_{k}}$$\textstyle{\bigoplus_{i=1}^{t}G^{i}_{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigoplus
m^{i}_{k-1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigoplus
m^{i}_{2}}$$\textstyle{\bigoplus_{i=1}^{t}G^{i}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\sum_{i=1}^{t}m^{i}_{1}(-)\cdot
d_{1}(e^{i}_{0})}$$\textstyle{R.}$

Moreover, the mapping cone of 3.1 is a free resolution of $R/J$.

######  Definition 3.4.

The _iterated trimming complex_ associated to the data of Setup 3.2 is the
complex of Theorem 3.3.

Restricting the diagram 3.1 to the linear strands of all complexes involved,
one immediately obtains the following:

######  Corollary 3.5.

Adopt notation and hypotheses as in Setup 3.2. Assume that the complexes
$F_{\bullet}$, $G_{\bullet}^{j}$ ($j=1,\dots,t$) are minimal and that $I$ has
initial degree $\ell$. For $i>1$, let

$C_{i}:=\operatorname{Ker}\Big{(}(Q_{i-1})_{i+\ell}:(F_{i})_{i+\ell}\to\bigoplus_{j=1}^{t}(G_{i}^{j})_{i+\ell}\Big{)}.$

Then the complex

$\cdots\xrightarrow{d_{k+1}}C_{k}\xrightarrow{d_{k}}\cdots\xrightarrow[]{d_{3}}C_{2}\xrightarrow[]{d_{2}^{\prime}}F_{1}^{\prime}\to
0$

is the linear strand of $J$.

Observe that if one chooses $\mathfrak{a}_{i}:=(K^{\prime}:K_{0}^{i})$ in
Setup 3.2, the the mapping cone of 3.1 will be a (necessarily nonminimal)
resolution of $R/K^{\prime}$. This fact is what will allow us to construct
linear strands for subideals of a given ideal $K$.

For the remainder of this section, we will transition into dealing with
cell/CW complexes supporting linear complexes. We will have to build up a
decent amount of machinery before constructing a morphism of complexes whose
kernel can recover the linear strand of certain subideals. In some of the
literature, the terms cell complex or CW complex are used interchangeably; we
will only use cellular to refer to _polyhedral cell complexes_ , and CW will
mean _regular CW-complex_. We begin with the definition of a polyhedral cell
complex; for more precise details, see [8, Chapter 6.2] or [25, Chapter 4].

######  Definition 3.6.

A _polyhedral cell complex_ $\mathcal{P}$ is a finite collection of convex
polytopes (called _cells_ or _faces_ of $\mathcal{P}$) in some Euclidean
space, satisfying the following two properties:

  * •

if $H$ is a polytope in $\mathcal{P}$, then every face of $H$ also lies in
$\mathcal{P}$, and

  * •

if $H_{i},H_{j}$ are both in $\mathcal{P}$, then $H_{i}\cap H_{j}$ is a face
of both $H_{i}$ and $H_{j}$.

Denote by $V(\mathcal{P})$ the set of vertices (or $0$-dimensional cells) of
$\mathcal{P}$. If $\mathcal{X}\subseteq V(\mathcal{P})$, the _induced
subcomplex_ of $\mathcal{P}$ on $\mathcal{X}$ is the subcomplex
$\\{F\in\mathcal{P}\mid V(F)\subseteq\mathcal{X}\\}$. The $f$-vector of a
$d$-dimensional polyhedral cell complex $\mathcal{P}$ is the vector
$(f_{0},f_{1},\dots,f_{d})$, where $f_{i}$ is the number of $i$-dimensional
cells of $\mathcal{P}$.

For the definition of a regular CW-complex, the reader is referred to [20].
Alternatively, one can define an object to be a regular CW complex if its
associated face poset os a CW poset, as in Definition 4.6.

Let $\mathcal{P}$ denote a cell/CW complex supporting a linear complex. Let
$v$ be a vertex of $\mathcal{P}$ and let $N_{v}$ denote the neighbors of $v$;
that is, all vertices connected to $v$ in the graph of linear syzygies ($v$ is
not considered a neighbor of itself). The incidence function associated to any
given cell/CW complex will be denoted by $\epsilon$.

Lemmas 3.7 and 3.8 will be essential for establishing the existence so-called
support chains centered at a vertex (see Definition 3.11). For the remainder
of this section, the results will be stated only for regular CW complexes with
the understanding that they hold for polyhedral cell complexes as well.

######  Lemma 3.7.

Let $\mathcal{P}$ be a CW complex supporting a linear complex. Then the
correspondence

$P\mapsto V(P)$

is injective.

###### Proof.

Assume $P$ and $Q$ are faces of $\mathcal{P}$ with the property that
$V(P)=V(Q)$. Observe that since the multidegree of the basis elements $e_{P}$
and $e_{Q}$ are completely determined by their vertex sets, $P$ and $Q$ must
have the same dimension. Since $\mathcal{P}$ is a CW complex, $P\cap Q$ is a
face of strictly smaller dimension such that $e_{P\cap Q}$ has the same
multidegree. Since $\mathcal{P}$ is linear, no such basis element can exist. ∎

######  Lemma 3.8.

Let $\mathcal{P}$ be a CW complex supporting a linear complex. Then for any
vertex $p\in V(P)$, there exists a unique codimension $1$ face $Q\subset P$
with $p\notin V(Q)$.

###### Proof.

Observe first that for any codimension $1$ face $Q\subset P$,
$|V(Q)|=|V(P)|-1$. This follows from the fact that if there were at least $2$
vertices not contained in $V(Q)$, then the induced CW complex differential
would have degree at least $2$. Thus, if any $Q$ as in the statement were to
exist, it must be unique by Lemma 3.8.

It remains to show that such a $Q$ exists. One may assume without loss of
generality that there exists some codimension $2$ face $R$ with $p\notin
V(R)$, since otherwise every face of $P$ must contain $p$. Since $\mathcal{P}$
is a CW complex, $R$ may be written as the intersection of two codimension $1$
faces of $P$. This implies that one of these faces _cannot_ contain $p$,
whence the result. ∎

######  Notation 3.9.

Let $R$ be a standard graded polynomial ring over a field $k$ endowed with
some term order $<$. Let $v\in\mathcal{P}$ be a vertex of some CW complex
supporting a linear complex. For every $v^{\prime}\in N_{v}$, define

$x_{v^{\prime}}:=\frac{m_{v,v^{\prime}}}{m_{v}}.$

There is an induced order on elements of $N_{v}$ defined by

$v_{1}<v_{2}\iff x_{v_{1}}<x_{v_{2}},\quad v_{1},v_{2}\in N_{v}.$

######  Lemma 3.10.

Let $\mathcal{P}$ be a CW complex supporting a linear complex and let $v$ be
any vertex of $\mathcal{P}$. Let $P$ be an $n$-dimensional face of
$\mathcal{P}$ with $V(P)\cap N_{v}=\\{v_{1}<\cdots<v_{n}\\}$. Then there
exists a _unique_ chain

$P=P_{n}\supset P_{n-1}\supset\cdots\supset P_{1}\supset P_{0}=v$

such that:

  1. (1)

$P_{i}$ is a codimension $1$ face of $P_{i+1}$ for each $i=0,\dots,n-1$, and

  2. (2)

$P_{i}\cap N_{v}=\\{v_{n-i}<\cdots<v_{n}\\}$.

###### Proof.

This is just an iteration of Lemmas 3.7 and 3.8. ∎

######  Definition 3.11.

Let $P$ be an $n$-dimensional face of $\mathcal{P}$ with $V(P)\cap
N_{v}=\\{v_{1}<\cdots<v_{n}\\}$. Then the _support chain centered at $v$_
associated to $P$ is the unique chain of subfaces of Lemma 3.10.

Given a face $P$ as above, define

$c(P):=\epsilon(P_{n},P_{n-1})\cdot\epsilon(P_{n-1},P_{n-2})\cdots\epsilon(P_{0},P_{1}),$

where

$P=P_{n}\supset P_{n-1}\supset\cdots\supset P_{1}\supset P_{0}=v$

is the associated support chain centered at $v$.

######  Lemma 3.12.

Let $P$ be an $n$-dimensional face of $\mathcal{P}$ with $V(P)\cap
N_{v}=\\{v_{1}<\cdots<v_{n}\\}$. Let $Q$ be the unique codimension $1$ face of
$P$ such that $v_{i}\notin V(Q)$ for some $1\leqslant i\leqslant n$. Then,

$\epsilon(Q,P)c(Q)=(-1)^{i+1}c(P).$

###### Proof.

Let

$P=P_{n}\supset P_{n-1}\supset\cdots\supset P_{1}\supset
P_{0}=v\quad\textrm{and}$ $Q=Q_{n-1}\supset Q_{n-2}\supset\cdots\supset
Q_{1}\supset Q_{0}=v$

be the support chains centered at $v$ associated to $P$ and $Q$, respectively.
Observe that $P_{n-i+1}=Q_{n-i+1}$; moreover, since $P_{n-i+1}$ is a
codimension $2$ face of $Q_{n-i+3}$, $Q_{n-i+2}$ and $P_{n-i+2}$ must be the
unique codimension $1$ faces of $Q_{n-i+3}$ containing $P_{n-i+1}$. By
definition of the incidence function,

$\epsilon(Q_{n-i+2},Q_{n-i+1})\epsilon(P_{n-i+1},Q_{n-i+2})=-\epsilon(P_{n-i+2},Q_{n-i+1})\epsilon(P_{n-i+1},P_{n-i+2}).$

Repeating this argument $i$ more times, the result follows. ∎

With the existence of support chains established, there are a few more steps
required for defining the relevant morphism of complexes. The most important
of these steps is Lemma 3.15, which will imply that the morphism of Lemma 3.16
is well-defined.

######  Observation 3.13.

Let $\mathcal{P}$ be a CW complex supporting the linear strand of some ideal
$(I,g)$, where $g\in R$. Then the linear strand of $(I:g)$ is obtained as the
linear strand of the ideal $(x_{v^{\prime}}\mid v^{\prime}\in N_{v})$.

######  Setup 3.14.

Let $R$ be a standard graded polynomial ring over a field endowed with some
term order $<$. Let $\mathcal{P}$ be a CW complex supporting a linear complex
$F_{\bullet}$ and suppose that $v\in\mathcal{P}$ is some vertex. Let
$K_{\bullet}$ denote the Koszul complex resolving $R/(x_{v^{\prime}}\mid
v^{\prime}\in N_{v})$.

######  Lemma 3.15.

Let $\mathcal{P}$ be a CW complex supporting a linear complex. Let $P$ be an
$n$-dimensional face of $\mathcal{P}$ supported on $v$. Then, $|V(P)\cap
N_{v}|=n$.

In the following proof, we will say that $P$ is _maximally supported on $v$_
if $|V(P)\cap N_{v}|=n$, as in the statement of the Lemma.

###### Proof.

The proof follows by induction on the dimension of any given face of
$\mathcal{P}$. If $P$ is a $1$-dimensional face containing $v$, then by
definition $P$ contains a neighbor of $v$. Let $P$ be an $i$-dimensional face
and assume by induction that all faces of dimension $\leqslant i-1$ containing
$v$ are maximally supported on $v$. By Lemma 3.8, there exists a codimension
$1$ face containing $v$ that is also not maximally supported on $v$, which is
a contradiction. ∎

Lemma 3.16 defines the morphism of complexes whose kernel can be used to
compute the linear strand of certain subideals, as guaranteed by Corollary
3.5.

######  Lemma 3.16.

Adopt notation and hypotheses as in Setup 3.14. Define

$\displaystyle Q_{n-1}:F_{n}$ $\displaystyle\to K_{n-1}$ $\displaystyle e_{P}$
$\displaystyle\mapsto\begin{cases}c(P)e_{V(P)\cap N_{v}}&\textrm{if}\ v\in
V(P)\\\ 0&\textrm{otherwise}.\\\ \end{cases}$

Then for all $i\geqslant 2$, the following diagram commutes:

(3.1)
$\textstyle{F_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q_{i}}$$\textstyle{F_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q_{i-1}}$$\textstyle{K_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{i-1}}$

###### Proof.

Observe first that each $Q_{i}$ is well defined by Lemma 3.15. Going clockwise
around the diagram 3.1, one obtains:

$\displaystyle e_{P}$ $\displaystyle\mapsto\sum_{Q\subset
P}\epsilon(Q,P)m_{P}/m_{Q}e_{Q}$ $\displaystyle\mapsto\sum_{|V(Q)\cap
N_{v}|=n-1}\epsilon(Q,P)c(Q)m_{P}/m_{Q}e_{V(Q)\cap N_{v}}.$

Going counterclockwise around 3.1,

$\displaystyle e_{P}$ $\displaystyle\mapsto c(P)e_{V(P)\cap N_{v}}$
$\displaystyle\mapsto\sum_{i=1}^{n}(-1)^{i+1}c(P)x_{v_{j_{i}}}e_{V(P)\cap
N_{v}\backslash v_{j_{i}}}.$

The result then follows from Lemma 3.12. ∎

In the case of polyhedral cell complexes, it turns out that the kernels of the
maps $Q_{i}$ as above may _not_ yield a complex supported on a polyhedral cell
complex in general. The following Definition will end up being a property
sufficient for guaranteeing that the resulting kernel can be easily described
as an induced subcomplex.

######  Definition 3.17.

A CW complex $\mathcal{P}$ is _linearly connected with respect to $v$_ if for
all $v_{i},v_{j}\in N_{v}$, $(v_{i},v_{j})$ is an edge of $\mathcal{P}$
whenever $m_{v_{i}}$ and $m_{v_{j}}$ have a linear syzygy.

######  Lemma 3.18.

Let $\mathcal{P}$ be a CW complex supporting the linear strand of some module.
If $\mathcal{P}$ is linearly connected with respect to $v$, then
$x_{v_{i}}\neq x_{v_{j}}$ for all distinct $v_{i},\ v_{j}\in N_{v}$.

###### Proof.

Let $v_{1},v_{2}\in N_{v}$ be such that $x_{v_{1}}=x_{v_{2}}$ and suppose for
sake of contradiction that $\\{v_{1},v_{2}\\}$ is a face of $\mathcal{P}$. One
may assume that the differentials in the presenting matrix have the following
form:

$\displaystyle e_{v,v_{1}}$
$\displaystyle\mapsto\frac{m_{v,v_{1}}}{m_{v}}e_{v}-\frac{m_{v,v_{1}}}{m_{v_{1}}}e_{v_{1}},$
$\displaystyle e_{v,v_{2}}$
$\displaystyle\mapsto\frac{m_{v,v_{2}}}{m_{v}}e_{v}-\frac{m_{v,v_{2}}}{m_{v_{2}}}e_{v_{2}},$
$\displaystyle e_{v_{1},v_{2}}$
$\displaystyle\mapsto\frac{m_{v_{1},v_{2}}}{m_{v_{1}}}e_{v_{1}}-\frac{m_{v_{1},v_{2}}}{m_{v_{2}}}e_{v_{2}}.$

Since $x_{v_{1}}=x_{v_{2}}$, one has
$m_{v,v_{1}}=m_{v,v_{2}}=m_{v_{1},v_{2}}$. It is then clear that
$e_{v,v_{1}}-e_{v,v_{2}}+e_{v_{1},v_{2}}$ is a cycle that cannot possibly be a
boundary. By Theorem 3.1, no such element can exist. ∎

######  Theorem 3.19.

Let $\mathcal{P}$ be a CW complex supporting the linear strand of some
monomial ideal $I$. Assume that $\mathcal{P}$ is linearly connected with
respect to $v$. Then the CW complex induced by removing all faces supported on
$v$ is the linear strand of the monomial ideal with minimal generating set
$G(I)\backslash m_{v}$.

###### Proof.

Let $J$ denote the ideal with minimal generating set $G(I)\backslash m_{v}$.
By Corollary 3.5, the linear strand of $J$ is obtained by restricting to the
kernel of each $Q_{i}$ map as in Lemma 3.16. Moreover, for every $V\subset
N_{v}$, there is _at most_ one face with support containing $V$; if there were
more than one such face, then the intersection of these faces would also be
supported on $V$, a contradiction to Lemma 3.15. Thus, by Lemma 3.18, the only
possible basis elements contained in the kernel are those basis elements
corresponding to faces that do not contain $v$. ∎

######  Theorem 3.20.

Let $\mathcal{P}$ be a CW complex supporting the linear strand of some
monomial ideal $I$, and assume that every $1$-dimensional face of
$\mathcal{P}$ has a unique multidegree. Then the linear strand of any monomial
ideal $J$ with $G(J)\subseteq G(I)$ is supported on the CW complex induced by
restricting to the vertices corresponding to generators of $J$.

In particular, if $J$ has linear minimal free resolution, then $J$ has a CW
minimal free resolution.

###### Proof.

The assumption that every $1$-dimensional face has a unique multidegree
implies that for any vertex $v\in\mathcal{P}$, every pair of distinct vertices
$v_{1},v_{2}\in N_{v}$ has the property that $x_{v_{1}}\neq x_{v_{2}}$. The
proof then proceeds identically as in the proof of Theorem 3.19. ∎

##  4\. CW Posets and Cellularity of Sparse Eagon-Northcott Complexes

In this section, we consider the question of when a particular complex may be
supported on a CW complex. We first recall the construction of so-called
_frames_ of monomial ideals, as described by Peeva and Velasco in [30]. We
then show that the poset $\operatorname{po}(E_{\bullet}^{<})$ associated to
the sparse Eagon-Northcott complex is a CW poset, whence Corollary 4.11
immediately establishes that $E_{\bullet}^{<}$ is CW for _any_ term order $<$.
Combining this with the results of Section 3, we obtain the previously
mentioned fact that every rainbow DFI has linear strand supported on a CW
complex (see Corollary 4.13).

######  Definition 4.1.

An $r$-frame $U_{\bullet}$ is a finite complex of finite $k$-vector spaces
with differential $\partial$ and a fixed basis satisfying

  1. (1)

$U_{0}=k$,

  2. (2)

$U_{1}=k^{r}$,

  3. (3)

$\partial(w_{j})=1$ for each basis vector $w_{j}\in U_{1}$.

Recall that for any choice of $r$ monomials $m_{1},\dots,m_{r}$, an $r$-frame
$U_{\bullet}$ may be homogenized to a (not necessarily acyclic) complex
$F_{\bullet}$ with $H_{0}(F_{\bullet})=R/(m_{1},\dots,m_{r})$. This is
accomplished inductively by assigning a multidegree to a basis element $e_{P}$
by taking the lcm of the multidegrees of the basis elements appearing in the
support of $\partial(e_{P})$. Once the multigrading has been assigned, the
differential may be computed as follows:

$d(e_{P})=\sum_{e_{Q}\in\operatorname{Supp}(\partial(e_{P}))}a_{P,Q}\frac{\operatorname{mdeg}(e_{P})}{\operatorname{mdeg}(e_{Q})}e_{Q},$

where the coefficients $a_{P,Q}$ are determined by the associated $r$-frame.
Complete details are found in Construction $55.2$ of [29].

######  Theorem 4.2 ([30, Theorem 4.14]).

Let $R$ be a polynomial ring over a field and $I\subseteq R$ a monomial ideal.
Then the $I$-homogenization of any frame of the multigraded minimal free
resolution $F_{\bullet}$ of $R/I$ is $F_{\bullet}$.

######  Corollary 4.3.

Let $R$ be a polynomial ring over a field and $I\subseteq R$ a monomial ideal.
Then the $I$-homogenization of any frame of the multigraded linear strand
$F_{\bullet}^{\operatorname{lin}}$ of $R/I$ is
$F_{\bullet}^{\operatorname{lin}}$.

###### Proof.

Viewing $F_{\bullet}^{\operatorname{lin}}$ as a subcomplex of the minimal free
resolution, this follows immediately from Theorem 4.2. ∎

The following Definition is a method of associating any resolution (with a
fixed basis in each homological degree) to a poset whose elements correspond
to the basis elements. There is a natural way to try to associate any such
poset to a face lattice of some geometric object, but in general this will not
yield a cell or CW complex that is actually realizable in some Euclidean
space.

######  Definition 4.4.

Let $F_{\bullet}$ be a complex with a fixed basis $B_{i}$ in each homological
degree $i$. The associated poset $\operatorname{po}(F_{\bullet})$ is defined
as follows: given basis elements $e\in F_{i}$, $f\in F_{i-1}$,

$f\leqslant e\iff f\in\operatorname{Supp}_{B_{i-1}}(d^{F}(f)).$

Extending $\leqslant$ transitively yields a partial order on basis elements
appearing in all homological degrees.

######  Example 4.5.

Let $\psi:F\to R$ be an $R$-module homomorphism, where $F$ is a free
$R$-module. The induced Koszul complex is the complex $K_{\bullet}$ with
$K_{i}=\bigwedge^{i}F$ and differential

$\displaystyle\bigwedge^{i}F$
$\displaystyle\xrightarrow{\textrm{comultiplication}}F\otimes\bigwedge^{i-1}F$
$\displaystyle\xrightarrow{\psi\otimes
1}R\otimes\bigwedge^{i-1}F=\bigwedge^{i-1}F.$

Let $F$ have basis $f_{1},\dots,f_{n}$. If one chooses the standard bases for
each exterior power $K_{i}$, the poset $\operatorname{po}(K_{\bullet})$ is
precisely the power set $2^{[n]}$ partially ordered by inclusion (ie, the
Boolean poset).

The following Definition is due to Björner [5].

######  Definition 4.6.

A poset $S$ is a _CW-poset_ if

  1. (a)

$S$ has a least element $0$,

  2. (b)

$S$ has more than $1$ element, and

  3. (c)

for all $x\in S\backslash\\{0\\}$, the open intercal $(0,x)$ is homeomorphic
to a sphere.

As previously mentioned, one can take the definition of a regular CW-complex
to be such that its associated face poset is CW, by the following Theorem:

######  Theorem 4.7 ([5, Proposition 3.1]).

A poset $S$ is a CW poset if and only if it is isomorphic to the face poset of
a regular CW-complex.

######  Proposition 4.8.

Let $S$ be a nontrivial poset with respect to some binary relation
$\leqslant$. If:

  1. (a)

$S$ has a least element, denoted $0$.

  2. (b)

Every interval $[x,y]$ of length two has cardinality $4$; that is, $S$ is
_thin_.

  3. (c)

Every interval $[0,x]$, $x\in S$, is finite and shellable.

Then, $S$ is a CW-poset.

As previously mentioned, choosing a face lattice defined by
$\operatorname{po}(F_{\bullet})$ does not always yield a CW complex. However,
the following lemmas will be used to prove that (with respect to the basis
represented by valid multidegrees), the poset associated to the sparse Eagon-
Northcott complexes are the face poset of a regular CW-complex. For the
definition of CL-shellable in the following result, see [6, Definition 2.2].

######  Lemma 4.9.

Adopt notation and hypotheses as in Theorem 2.16. Then every interval of the
form $[0,x]$ in $\operatorname{po}(E_{\bullet}^{<})$ is finite and CL-
shellable.

###### Proof.

Recall that a finite poset $S$ is semimodular if, for any two elements $u$ and
$v$ covering some $x\in S$, there exists a $z\in S$ covering $u$ and $v$. The
poset $S$ is totally semimodular if every interval is semimodular. We will
first prove that for any elements
$\mu,\nu\in\operatorname{po}(E_{\bullet}^{<})$ with $\nu\geqslant\mu$ and
$\operatorname{rank}\mu\geqslant 1$, the interval $[\mu,\nu]$ is totally
semimodular.

Let $\mu$ be any multidegree appearing in homological degree $\geqslant 1$.
The statement that $\mu$ is covered by two elements is equivalent to saying
that $x_{ib}\mu$ and $x_{jc}\mu$ are valid multidegrees in $E_{\bullet}^{<}$
for some integers $i,j,b,c$. If $i=j$, then Corollary 2.17 immediately yields
that $x_{ib}x_{jc}\mu$ is a valid multidegree. If $i\neq j$, then one simply
combines Observation 2.18 with Corollary 2.17 to conclude that $x_{ib}\mu$ and
$x_{jc}\mu$ are covered by some element of $[\mu,\nu]$.

Next, the proof will be complete by showing that every interval of the form
$[0,\nu]$ admits a recursive atom ordering; recall that CL-shellability is
equivalent to the existence of a recursive atom ordering by [6, Theorem 3.2].
More precisely:

Claim: The ordering induced by the term order $<$ (on the multidegrees of)
each atom of the interval $[0,\nu]$ is a recursive atom ordering.

The claim will follow upon proving the following conditions (these conditions
are a translation of the definition of a recursive atom ordering as given in
[6, Definition 3.1]):

  1. (1)

If $\mu_{1}<\mu_{2}$ are atoms of $\operatorname{po}(E_{\bullet}^{<})$ such
that $\textrm{lcm}(\mu_{1},\mu_{2})$ is a valid multidegree as in Corollary
2.17, then there exists a $\mu_{3}<\mu_{2}$ such that $\mu_{2}$ and $\mu_{3}$
are both covered by some element.

  2. (2)

For every atom $\mu\in[0,\nu]$, the interval $[\mu,\nu]$ admits a recursive
atom ordering for which any atom of $[\mu,\nu]$ that is also an atom of
$[\mu^{\prime},\nu]$ for some $\mu^{\prime}<\mu$ comes first in the order.

Proof of (1): This is just a retranslation of the statement of Observation
2.19, so there is nothing to prove here.

Proof of (2): As proved above, every interval of the form $[\mu,\nu]$ is
totally semimodular. By [6, Theorem 5.1], _every_ atom ordering of a totally
semimodular poset is a recursive atom ordering. Every atom of $[\mu,\nu]$ is
of the form $x_{ia}\mu$ for some variable $x_{ia}$, implying that $x_{ia}\mu$
is _always_ an atom of $[\mu^{\prime},\nu]$ for
$\mu^{\prime}=\frac{x_{ia}\mu}{x_{ib}}$, where $x_{ib}$ divides $\mu$. Order
all atoms of the form $x_{ia}\mu$ with $x_{ia}<x_{ib}$ with the term order
$<$, then simply choose all other atoms to come after these terms. By
construction this ordering satisfies $(2)$, and the result follows. ∎

######  Lemma 4.10.

Adopt notation and hypotheses as in Theorem 2.16. Then
$\operatorname{po}(E_{\bullet}^{<})$ is a CW poset.

###### Proof.

It is clear that $E_{\bullet}^{<}$ has a least element, represented by the
multidegree $1$. Moreover, $\operatorname{po}(E_{\bullet}^{<})$ is thin
because $E_{\bullet}^{<}$ is a linear resolution with unique multigraded Betti
numbers. Thus the only nontrivial property to check is that $[0,x]$ is
shellable for all $x\in\operatorname{po}(E_{\bullet}^{<})$.

By [6, Proposition 2.3], any CL-shellable poset is also shellable, whence the
result follows from Lemma 4.9. ∎

Combining the previous two results immediately yields the following:

######  Corollary 4.11.

Let $R=k[x_{ij}\mid 1\leqslant i\leqslant n,\ 1\leqslant j\leqslant m]$ be a
polynomial ring over an arbitrary field $k$ and $M$ be an $n\times m$ matrix
of variables in $R$ where $n\leqslant m$. Let $<$ denote any term order. Then
$R/\textrm{in}_{<}I_{n}(M)$ has minimal free resolution supported on a regular
CW complex.

###### Proof.

Observe that the frame associated to the sparse Eagon-Northcott complex has
coefficients $0$ or $\pm 1$. By Lemma 4.10, the frame may be supported on a
regular CW complex, and the homogenization of this frame is precisely
$E_{\bullet}^{<}$. ∎

######  Remark 4.12.

For the remainder of the paper, no distinction will be made between the poset
$\operatorname{po}(E_{\bullet}^{<})$ and the CW complex that it induces.

$x_{12}x_{24}$$x_{12}x_{23}$$x_{14}x_{23}$$x_{12}x_{21}$$x_{14}x_{21}$$x_{13}x_{21}$

$x_{13}x_{21}$$x_{13}x_{24}$$x_{12}x_{24}$$x_{13}x_{22}$$x_{11}x_{24}$$x_{11}x_{22}$

Figure 1. Two examples of cellular resolutions arising from initial ideals of
maximal minors of a $2\times 4$ matrix.

Using the results of Section 3, Corollary 4.11 immediately implies the much
stronger result that _every_ rainbow DFI has linear strand supported on a CW
complex.

######  Corollary 4.13.

Adopt notation and hypotheses as in Corollary 4.11. Let $\Delta$ be an
$(n-1)$-dimensional pure simplicial complex. Then the linear strand of
$\operatorname{rain}_{<}(J_{\Delta})$ is supported on a CW complex.

In particular, if $\operatorname{rain}_{<}(J_{\Delta})$ has linear resolution,
then $\operatorname{rain}_{<}(J_{\Delta})$ has a CW resolution.

###### Proof.

This is immediate by Corollary 4.11 combined with Theorem 3.20 and Corollary
2.11. ∎

##  5\. Preservation of Regularity for Subideals and Linearity

Corollary 4.13 establishes that any rainbow DFI with linear resolution must
have a CW resolution. This immediately begs the question of which rainbow DFIs
have linear resolution. In this section, we consider conditions that guarantee
linearity. We first study rainbow DFIs for which the sequence of variable
differences

$\\{x_{11}-x_{21},\dots,x_{11}-x_{n1}\\}\cup\cdots\cup\\{x_{1m}-x_{2m},\dotsc,x_{1m}-x_{nm}\\}$

forms a regular sequence on the associated quotient ring. This can be used to
show that certain ideals for which the squarefree complement (see Definition
5.1) has no linear relations must have CW resolution.

Next, using a result of Nagel and Reiner [26], we see that rainbow DFIs with
the property that the Alexander dual $\Delta^{\vee}$ admits an ordering for
which it is a _free sequence_ (see Definition 5.8) must have linear
resolution. Moreover, if the elements of the Alexander dual have nonmaximal
overlap, this condition is an equivalence; the Betti table for these ideals
can also be computed explicitly.

We begin with some definitions and notation related to monomial ideals and
their complementary ideals.

######  Definition 5.1.

Let $R=k[x_{1},\dots,x_{n}]$ be a standard graded polynomial ring over a field
$k$. Let $K$ denote an equigenerated monomial ideal with generators in degree
$d$. Define

$G(K):=\textrm{unique minimal generating set of}\ K\ \textrm{consisting of
monic monomials.}$

Given a monomial ideal $K$, define the (squarefree) complementary ideal
$\overline{K}$ to be the ideal with minimal generating set:

$G(\overline{K})=\begin{cases}\\{\textrm{degree}\ d\ \textrm{squarefree
monomials}\\}\backslash G(K)&\textrm{if}\ K\ \textrm{squarefree},\\\
\\{\textrm{degree}\ d\ \textrm{monomials}\\}\backslash
G(K)&\textrm{otherwise.}\\\ \end{cases}$

######  Lemma 5.2 ([34]).

Let $K^{\prime}$ be a squarefree equigenerated momomial ideal of degree $n$
with $\overline{K^{\prime}}=(x^{I_{1}},\dots,x^{I_{r}})$ such that $|I_{i}\cap
I_{j}|<n-1$. Then $R/K^{\prime}$ has a linear minimal free resolution.

It is worth noting that [34] not only shows that ideals as in the above
statement have linear resolution, but also produces the minimal free
resolution explicitly. As previously mentioned, we wish to study when the
sequence of variable differences

$\\{x_{11}-x_{21},\dots,x_{11}-x_{n1}\\}\cup\cdots\cup\\{x_{1m}-x_{2m},\dotsc,x_{1m}-x_{nm}\\}$

forms a regular sequence on the quotient defined by a rain DFI. This problem
can be easily abstracted to the following question, which we call the
_preservation of regularity_ :

If $J\subseteq I$ and $\mathfrak{a}$ is regular on $R/I$, then when is
$\mathfrak{a}$ regular on $R/J$?

The initial answer to this question, although simple, turns out to be
surprisingly useful.

######  Lemma 5.3.

Let $R$ be a commutative ring and let $\mathfrak{a}$ be regular on $R/(J,g)$.
Then $\mathfrak{a}$ is regular on $R/J$ if and only if $\mathfrak{a}$ is
regular on $(J:g)$.

###### Proof.

Without loss of generality assume that $\mathfrak{a}=a$ has length $1$. The
proof follows immediately from the string of inclusions:

$\operatorname{Ass}(J:g)\subseteq\operatorname{Ass}(J)\subseteq\operatorname{Ass}(J:g)\cup\operatorname{Ass}(J,g).$

∎

If we assume furthermore that the ideal $I$ is a monomial ideal with a
convenient splitting, then Lemma 5.3 can be upgraded considerably.

######  Corollary 5.4.

Let $R$ be a polynomial ring over a field. Let $I$ be a linearly presented,
equigenerated monomial ideal of degree $d$. Assume that $I=J+K$, where $J$ and
$K$ are monomial ideals with $\beta_{1,d+1}(K)=0$. Let $\mathfrak{a}$ be
regular on $R/I$.

Then $\mathfrak{a}$ is regular on $R/J$ if and only if $\mathfrak{a}$ is
regular on $R/(J:g)$ for all $g\in G(K)$.

###### Proof.

Write $G(K)=\\{g_{1},\dots,g_{t}\\}$. Since $I$ is linearly presented and $K$
has no linear relations on any of its generators (this is the assumption
$\beta_{1,d+1}(K)=0$), one finds

$(J+(g_{1},\dots,g_{i}):g_{i+1})=(J:g_{i+1})\ \textrm{for all}\
i=0,\dots,t-1.$

The result then follows by iterating Lemma 5.3. ∎

The following Proposition is a standard exercise which we include here for
convenience.

######  Proposition 5.5.

Let $M$ be a finitely generated $R$-module. Assume $\mathfrak{a}$ is an
$M$-regular sequence and let $F_{\bullet}$ be any free resolution of $M$ over
$R$. Then $F_{\bullet}\otimes R/\mathfrak{a}$ is a free resolution of
$M/\mathfrak{a}M$ over $R/\mathfrak{a}$.

###### Proof.

By induction it is of no loss of generality to assume $\mathfrak{a}=a$ has
length $1$. There is a short exact sequence of complexes:

$0\to F_{\bullet}\xrightarrow{a}F_{\bullet}\to F_{\bullet}/aF_{\bullet}\to 0.$

The long exact sequence of homology implies
$H_{i}(F_{\bullet}/aF_{\bullet})=0$ for $i\geqslant 2$, and
$H_{1}(F_{\bullet}/aF_{\bullet})=0$ by the assumption that $a$ is regular on
$M$. ∎

######  Setup 5.6.

Let $R=k[x_{ij}\mid 1\leqslant i\leqslant n,\ 1\leqslant j\leqslant m]$ be a
polynomial ring over an arbitrary field $k$ and $M$ be an $n\times m$ matrix
of variables in $R$ where $n\leqslant m$. Let $\Delta$ be an
$(n-1)$-dimensional pure simplicial complex and assume that $<$ is an
arbitrary term order on $R$. Assume that $\Delta^{\vee}$ satisfies the
following:

  1. (*)

for all $\sigma,\ \tau\in\Delta^{\vee}$, $|\sigma\cap\tau|<n-1$.

######  Corollary 5.7.

Adopt notation and hypotheses as in Setup 5.6. Then the sequence of variable
differences

$\\{x_{11}-x_{21},\dots,x_{11}-x_{n1}\\}\cup\cdots\cup\\{x_{1m}-x_{2m},\dotsc,x_{1m}-x_{nm}\\}$

forms a regular sequence on $R/\operatorname{rain}_{<}(J_{\Delta})$ if and
only if $\operatorname{grade}(\operatorname{rain}_{<}(J_{\Delta}):g)=m-n$ for
all $g\in\operatorname{rain}_{<}(J_{\Delta^{\vee}})$.

###### Proof.

Observe that
$\operatorname{grade}(\operatorname{rain}_{<}(J_{\Delta}):g)\geqslant m-n$ for
all $g\in\operatorname{rain}_{<}(J_{\Delta^{\vee}})$ by Theorem 2.8. The grade
after specializing must be $m-n$, so the variable differences are regular if
and only if $\operatorname{grade}(\operatorname{rain}_{<}(J_{\Delta}):g)=m-n$
for all $g\in\operatorname{rain}_{<}(J_{\Delta^{\vee}})$. The result then
follows by Corollary 5.4. ∎

The next definition introduces the notion of a free sequence of vertices in a
CW complex. This definition will end up being the “correct” condition for
characterizing rainbow DFIs with linear resolution.

######  Definition 5.8.

Let $\mathcal{P}$ be a CW complex. A vertex is _free_ if it is contained in a
unique facet of $\mathcal{P}$. A _free sequence_ is a sequence of vertices
$\\{v_{1},\dots,v_{k}\\}\subseteq\mathcal{P}$ such that $v_{i}$ is a free
vertex of $\mathcal{P}\backslash\\{v_{1},\dots,v_{i-1}\\}$ for each
$1=1,\dots,k$.

######  Lemma 5.9 ([26]).

Let $C$ be a polytopal complex and $v$ a vertex in $C$ that lies in a unique
facet $P$, with $\dim P>0$. Then the subcomplex induced by deleting the vertex
$v$ is homotopy equivalent to $C$.

Given an $(n-1)$-dimensional pure simplicial complex $\Delta$, the facets of
$\Delta$ have a one-to-one correspondence with vertices of
$\operatorname{po}(E_{\bullet}^{<})$. We will tacitly employ this association
for the remainder of the paper.

######  Corollary 5.10.

Adopt notation and hypotheses as in Corollary 4.11. Let $\Delta$ be an
$(n-1)$-dimensional pure simplicial complex. If the facets of $\Delta^{\vee}$
admit an ordering for which they are a free sequence in
$\operatorname{po}(E_{\bullet}^{<})$, then
$\operatorname{rain}_{<}(J_{\Delta})$ has linear minimal free resolution
supported on a CW complex.

###### Proof.

The fact that the minimal free resolution is CW is Corollary 4.13. The
linearity follows from Lemma 5.9; since $\textrm{in}_{<}I_{n}(M)$ has linear
resolution, removing a free vertex at each step leaves the homology unchanged.
This implies that $\operatorname{rain}_{<}(J_{\Delta})$ has linear resolution.
∎

######  Theorem 5.11.

Adopt notation and hypotheses as in Setup 5.6. Then
$\operatorname{rain}_{<}(J_{\Delta})$ has a linear minimal free resolution if
and only if $\operatorname{grade}(\operatorname{rain}_{<}(J_{\Delta}):g)=m-n$
for all $g\in G(\operatorname{rain}_{<}(J_{\Delta^{\vee}}))$ (that is, each
element of $\Delta^{\vee}$ is a free vertex of
$\operatorname{po}(E_{\bullet}^{<})$).

###### Proof.

$\implies:$ Argue by contraposition. Recall that a free resolution of
$R/\operatorname{rain}_{<}(J_{\Delta})$ may be obtained by an iterated
trimming complex taking the form of a mapping cone of a morphism of complexes:

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{k+1}}$$\textstyle{F_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q_{k-1}}$$\scriptstyle{d_{k}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{3}}$$\textstyle{F_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{2}^{\prime}}$$\scriptstyle{Q_{1}}$$\textstyle{F_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigoplus
m^{i}_{k}}$$\textstyle{\bigoplus_{i=1}^{t}K^{i}_{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigoplus
m^{i}_{k-1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigoplus
m^{i}_{2}}$$\textstyle{\bigoplus_{i=1}^{t}K^{i}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\sum_{i=1}^{t}m^{i}_{1}(-)\cdot
d_{1}(e^{i}_{0})}$$\textstyle{R.}$

In the above, the complex on the top row is built from the minimal free
resolution of $R/\textrm{in}_{<}I_{n}(M)$, which has length $m-n+1$. The
bottom row is a direct sum of the minimal free resolutions of
$(\operatorname{rain}_{<}(J_{\Delta}):g)$ for each
$g\in\operatorname{rain}_{<}(J_{\Delta^{\vee}})$; for complete details on this
construction see [35]. The minimal free resolution will be linear if and only
if the comparison maps $Q_{i}$ are surjective in each homological degree. If
$\operatorname{grade}(\operatorname{rain}_{<}(J_{\Delta}):g)>m-n$ for some
$g\in\operatorname{rain}_{<}(J_{\Delta^{\vee}})$, however, then $Q_{m-n+1}$
will not be surjective because $F_{m-n+2}=0$.

$\impliedby:$ The assumption
$\operatorname{grade}(\operatorname{rain}_{<}(J_{\Delta}):g)=m-n$ for all
$g\in\operatorname{rain}_{<}(J_{\Delta^{\vee}})$ implies that each
$g\in\Delta^{\vee}$ is a free vertex in $\operatorname{po}(E_{\bullet}^{<})$.
The conclusion follows from Corollary 5.10. ∎

######  Corollary 5.12.

Adopt notation and hypotheses as in Setup 5.6. Then
$R/\operatorname{rain}_{<}(J_{\Delta})$ has Betti table

| $0$ | $1$ | $\cdots$ | $\ell$ | $\cdots$ | $m-n+1$
---|---|---|---|---|---|---
$0$ | $1$ | $\cdot$ | $\cdots$ | $\cdot$ | $\cdots$ | $\cdot$
$\vdots$ | $\cdot$ | $\cdot$ | $\cdots$ | $\cdot$ | $\cdots$ | $\cdot$
$n-1$ | $\cdot$ | $\binom{m}{n}-r$ | $\cdots$ | $\binom{n+\ell-2}{\ell-1}\binom{m}{n+\ell-1}-r\binom{m-n}{\ell-1}$ | $\cdots$ | $\binom{m-1}{m-n}-r.$

###### Proof.

This follows from Corollary 5.7 and Proposition 5.5 combined with the Betti
tables given in [34, Corollary 5.12]. ∎

##  6\. Applications to Polarizations

In this section, we consider applications of the machinery and results
produced in the previous sections to polarizations of Artinian monomial
ideals. We recall the result of Almousa, Fløystad, and Lohne that shows
Alexander duality provides an equivalence between rainbow monomial ideals and
polarizations of Artinian monomial ideals. This immediately implies
Proposition 6.4 and Corollary 6.6. Furthermore, Corollary 6.6 shows that any
rainbow DFI $J_{\Delta}$ with $\Delta^{\vee}$ consisting only of free vertices
is a polarization of a power of the maximal ideal if and only if
$\operatorname{rain}_{<}J_{\Delta}=\textrm{in}_{<}I_{n}(M)$. We conclude with
a variety of questions related to the results of this paper.

We begin with the definition of an Alexander dual of a monomial ideal.

######  Definition 6.1.

Let $I$ be a monomial ideal. The Alexander dual $I^{\vee}$ is the ideal with
generators consisting of all monomials with nontrivial common divisor with
every generator of $I$.

######  Example 6.2.

Let
$I=\left({x}_{1}{x}_{2},{x}_{1}{x}_{3},{x}_{2}{x}_{3},{x}_{1}{x}_{4},{x}_{2}{x}_{4},{x}_{3}{x}_{4}\right)$.
Then one computes:

$I^{\vee}=\left({x}_{1}{x}_{2}{x}_{3},{x}_{1}{x}_{2}{x}_{4},{x}_{1}{x}_{3}{x}_{4},{x}_{2}{x}_{3}{x}_{4}\right).$

Next, we give the definition of a polarization. As previously mentioned, the
intuition behind polarization is to replace an ideal with a squarefree ideal
that is homologically indistinguishable. The existence of polarizations
translates the problem of minimal free resolutions of arbitrary monomial
ideals into the problem of minimal free resolutions of _squarefree_ monomial
ideals.

######  Definition 6.3.

Let $I$ be an Artinian monomial ideal in the polynomial ring
$S=k[x_{1},\ldots,x_{n}]$, such that for every index $i$, $x_{i}^{m_{i}}$ is a
minimal generator. Let $\check{X}_{i}=\\{x_{i1},x_{i2},\ldots,x_{im_{i}}\\}$
be a set of variables, and let
$\tilde{S}=k[\check{X}_{1},\ldots\check{X}_{n}]$ be the polynomial ring in the
union of these variables. An ideal $\tilde{I}\subset\tilde{S}$ is a
_polarization_ of $I$ if

$\displaystyle\sigma=\\{x_{11}-x_{12},x_{11}-x_{13},\ldots,x_{11}-x_{1m_{1}}\\}\cup\\{x_{21}-x_{22},\ldots,x_{21}-x_{2m_{2}}\\}\cup\ldots$
$\displaystyle\cup\\{x_{n1}-x_{n2},\ldots,x_{n1}-x_{n,m_{n}}\\}$

is a regular sequence in $\tilde{S}/\tilde{I}$ and
$\tilde{I}\otimes\tilde{S}/\sigma\cong I$.

######  Proposition 6.4 ([1]).

Let $J$ be an ideal generated by rainbow monomials with linear resolution
(with all variables in the ambient ring occurring in some generator of $J$).
Then $J^{\vee}$ is a polarization of an Artinian monomial ideal.

######  Corollary 6.5.

Let $\Delta$ be an $(n-1)$-dimensional simplicial complex such that
$\Delta^{\vee}$ admits an ordering for which it is a free sequence of
$\operatorname{po}(E_{\bullet}^{<})$. Then
$(\operatorname{rain}_{<}(J_{\Delta}))^{\vee}$ is the polarization of an
Artinian monomial ideal.

###### Proof.

Combine Proposition 6.4 with Corollary 5.10. ∎

######  Corollary 6.6.

Adopt notation and hypotheses as in Setup 5.6. Then:

  1. (1)

$(\operatorname{rain}_{<}(J_{\Delta}))^{\vee}$ is the polarization of an
Artinian monomial ideal if and only if
$\operatorname{grade}\big{(}\operatorname{rain}_{<}(J_{\Delta}):g\big{)}=m-n$
for all $g\in G(\operatorname{rain}_{<}(J_{\Delta^{\vee}}))$, and

  2. (2)

in the situation of $(1)$, $(\operatorname{rain}_{<}(J_{\Delta}))^{\vee}$ is a
polarization of a power of the graded maximal ideal if and only if
$|\Delta^{\vee}|=0$ (that is,
$\operatorname{rain}_{<}(J_{\Delta})=\textrm{in}_{<}I_{n}(M)$).

###### Proof.

The statement of $(1)$ is immediate from Theorem 5.11. To prove $(2)$, it
suffices to show that if $|\Delta^{\vee}|>0$, then the monomial ideal $K$ with
$G(\overline{K})=\\{x_{\textbf{a}}\mid\textbf{a}\in\Delta^{\vee}\\}$ is not
Cohen-Macaulay. That is, $K^{\vee}$ does not have linear resolution. Let
$x_{\textbf{a}}=x_{a_{1}}\cdots x_{a_{n}}\in G(\overline{K})$ and write
$[m]\backslash\textbf{a}=\\{b_{1}<\cdots<b_{m-n}\\}$. It is clear that
$x_{b_{1}}\cdots x_{b_{m-n}}\in K^{\vee}$. By Corollary 5.12, $R/K^{\vee}$ has
regularity $m-n+1$, whence $R/K^{\vee}$ does not have linear resolution. ∎

The following illustrates many of the results in this paper in a simple
example.

######  Example 6.7.

Let $<$ denote the standard diagonal term order, with $n=3$ and $m=5$. Assume
that $\Delta^{\vee}=\\{(1,2,3),(3,4,5)\\}$. The associated cellular resolution
is precisely the complex of boxes introduced by Nagel and Reiner. One can
verify:

$\operatorname{rain}_{<}(J_{\Delta}):x_{11}x_{22}x_{33}=(x_{34},x_{35}),\quad\operatorname{rain}_{<}(J_{\Delta}):x_{13}x_{24}x_{35}=(x_{11},x_{12});$

this shows that $(1,2,3)$ and $(3,4,5)$ are free vertices in the complex of
boxes. As guaranteed by Corollary 5.12,
$R/\operatorname{rain}_{<}(J_{\Delta})$ has Betti table

$\begin{matrix}&0&1&2&3\\\ \text{total:}&1&8&11&4\\\
\text{0:}&1&\text{.}&\text{.}&\text{.}\\\
\text{1:}&\text{.}&\text{.}&\text{.}&\text{.}\\\ \text{2:}&\text{.}&8&11&4\\\
\end{matrix}$

and one can verify using Macaulay2 [19] that
$(\operatorname{rain}_{<}(J_{\Delta}))^{\vee}$ is a polarization of

$(x_{1}^{2},x_{1}x_{2}^{2},x_{2}^{3},x_{1}x_{2}x_{3},x_{2}^{2}x_{3},x_{3}^{2}).$

######  Question 6.8.

Is Corollary 5.10 an equivalence? That is, if a rainbow DFI
$\operatorname{rain}_{<}(J_{\Delta})$ has linear resolution, then do the
elements of $\Delta^{\vee}$ admit an ordering for which they are a free
sequence in $\operatorname{po}(E_{\bullet}^{<})$?

Question 6.8, if answered in the affirmative, yields an interesting connection
between polarizations and free sequences on the CW complex
$\operatorname{po}(E_{\bullet}^{<})$. Moreover, using the combinatorics of the
initial ideals provided by Sturmfels and Zelevinsky [33], it may be possible
to explicitly enumerate all free sequences on
$\operatorname{po}(E_{\bullet}^{<})$, thus constructing all possible
polarizations with Alexander dual contained in $\textrm{in}_{<}I_{n}(M)$ for a
given term order.

######  Question 6.9.

Let $\Delta$ be an $(n-1)$-dimensional simplicial complex with the property
that $|\sigma\cap\tau|<n-1$ for all $\sigma,\tau\in\Delta^{\vee}$. Does there
exist a term order $<$ on $R$ such that every facet of $\Delta^{\vee}$
corresponds to a free vertex of $\operatorname{po}(E_{\bullet}^{<})$?

A positive answer to Question 6.9 would immediately yield that a large class
of equigenerated monomial ideals (considered in [34]) has CW resolution, since
one can employ Corollary 5.7 to specialize. More precisely, to the authors
knowledge, it is not known if every equigenerated squarefree monomial ideal
whose squarefree complement has no linear relations must have CW resolution.

######  Question 6.10.

Let $<$ be any term order. Then does the sparse Eagon-Northcott complex admit
the structure of an associative DG-algebra? If so, then how does this product
compare to the product on the squarefree Eliahou-Kervaire resolution
constructed by Peeva [28] upon specialization?

The first part of Question 6.10 is likely not as difficult to answer as the
second part. It would be interesting to see how multigraded DG-products on
$E_{\bullet}^{<}$ specialize to give DG-products on the resolution of all
squarefree monomial ideals of a given degree. It would be even _more_
interesting if these products were all distinct. If so, this would also yield
a fascinating correspondence between choices of initial ideals of
$\textrm{in}_{<}(M)$ and choices of algebra structures on the minimal free
resolution of all squarefree monomials of degree $n$ in $m$ variables. Up to
taking linear combinations, this may provide a method of parametrizing all
such multigraded products.

## Acknowledgements

Thanks to Ayah Almousa for very helpful discussions related to this work.
Thanks to Paolo Mantero for helpful comments and suggestions on an earlier
draft of this paper.

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