# Linear Strands of Initial Ideals of Determinantal Facet Ideals

Ayah Almousa  Cornell University<EMAIL_ADDRESS>http://math.cornell.edu/~aalmousa and  Keller VandeBogert  University of South
Carolina<EMAIL_ADDRESS>http://people.math.sc.edu/kellerlv

###### Abstract.

A determinantal facet ideal (DFI) is an ideal $J_{\Delta}$ generated by
maximal minors of a generic matrix parametrized by an associated simplicial
complex $\Delta$. In this paper, we construct an explicit linear strand for
the initial ideal with respect to any diagonal term order $<$ of an arbitrary
DFI. In particular, we show that if $\Delta$ has no _1-nonfaces_ , then the
Betti numbers of the linear strand of $J_{\Delta}$ and its initial ideal
coincide. We apply this result to prove a conjecture of Ene, Herzog, and Hibi
on Betti numbers of closed binomial edge ideals in the case that the
associated graph has at most $2$ maximal cliques. More generally, we show that
the linear strand of the initial ideal (with respect to $<$) of _any_ DFI is
supported on a polyhedral cell complex obtained as an induced subcomplex of
the _complex of boxes_ , introduced by Nagel and Reiner.

###### Key words and phrases:

determinantal facet ideal, binomial edge ideal, initial ideals, linear strand,
free resolutions

AA was partially supported by the NSF GRFP under Grant No. DGE-1650441.

##  1\. Introduction

Let $R$ denote the coordinate ring of a generic $n\times m$ matrix $M$, over
some field $k$ (with $n\leq m$). The ideal of maximal minors $I_{n}(M)$
possesses many surprising and desirable properties; for example, a result of
Sturmfels and Zelevinsky [16] shows that the natural generating set consisting
of all maximal minors forms of $M$ a reduced Gröbner basis of $I_{n}(M)$, for
_any_ term order $<$. Boocher goes one step further in [3] and shows that the
graded Betti numbers of $I_{n}(M)$ and any of its initial ideals must also
agree. As a consequence, the minimal free resolution of any initial ideal of
the ideal of maximal minors can be obtained by simply setting some of the
entries in the matrix representation of the standard Eagon-Northcott
differentials equal to $0$.

One direction for generalizing ideals of maximal minors is to imagine that the
column-sets of minors appearing in the generating set are parametrized by some
simplicial complex $\Delta$. Such ideals are called _determinantal facet
ideals_ (DFIs), and have been studied in multiple contexts by a wide variety
of authors (see [6], [8], [17]). In the case that $\Delta$ has dimension $1$
(that is, $\Delta$ is a graph), DFIs are better known as _binomial edge
ideals_ , and are even more well behaved than arbitrary DFIs (see [14], [7],
and [9]). DFIs themselves have also been generalized in a few different
directions - Mohammadi and Rauh [12] have allowed for the minors to be
parametrized by hypergraphs, and in [2], DFIs for nonmaximal minors are
studied.

In [5], Ene, Herzog, and Hibi conjecture that Betti numbers for a _closed_
binomial edge ideal agree with that of its initial ideal with respect to any
diagonal term order. This conjecture is known to be true in the case of Cohen-
Macaulay binomial edge ideals, but has remained elusive in generality. In this
paper, we give further evidence and a proof of this conjecture in the
$2$-clique case using techniques related to the computation of linear strands.

Given a homogeneous minimal complex of initial degree $d$, the linear strand
can be obtained by restricting the $i$th differential to basis elements of
degree $i+d$, for all $i\geq 1$. The first results relating to linear strands
of DFIs were given by Herzog, Kiani, and Madani in [8]; the main result was
the fact that the linear strand of any DFI can be obtained as a so-called
_generalized_ Eagon-Northcott complex. In this paper, we prove a similar
result showing that the linear of of the initial ideal of certain classes of
DFIs is obtained as a _generalized sparse_ Eagon-Northcott complex (see
Definition 4.3).

More precisely, in this paper we study linear strands of initial ideals of
DFIs with respect to a diagonal term order. We first construct an explicit
resolution of $\operatorname{in}_{<}I_{n}(M)$ as a _sparse_ Eagon-Northcott
complex (see 3.6). Using this, one can restrict to an appropriate subcomplex
parametrized by the clique complex of the simplicial complex associated to the
DFI $J_{\Delta}$. This subcomplex is not acyclic in general; however, if one
imposes sufficient conditions on $\Delta$, the homology of this subcomplex
will vanish in “small” degrees. This implies that the Betti numbers of certain
DFIs and their initial ideals with respect to a diagonal term order coincide
on the linear strand (see Theorem 4.13).

As an application, we prove the previously mentioned conjecture of Ene,
Herzog, and Hibi in the case that the associated graph $G$ has at most $2$
maximal cliques. Moreover, we pose more generally the conjecture that for any
lcm-closed DFI $J_{\Delta}$, the Betti numbers of $J_{\Delta}$ and its initial
ideal with respect to any diagonal term order coincide (see Conjecture 4.16).
Finally, we consider the linear strands of initial ideals (with respect to any
diagonal term order) of arbitrary DFIs. In this case, we find that the linear
strand in general is _always_ supported on a polyhedral cell complex. This
implies that the Betti numbers of a general linear strand can be obtained by
looking at certain induced subcomplexes of the so-called _complex of boxes_ of
Nagel and Reiner (see [13]). We conclude with examples of this construction
and remarks on further applications.

The paper is organized as follows. In Section 2, we introduce the notation and
terminology that will be in play for the rest of the paper. This includes the
definition of a DFI (see 2.3), linear strands, and the Eagon-Northcott
complex. We include the aforementioned result of Boocher on the Betti numbers
of initial ideals of ideals of maximal minors, and use this to deduce that the
multigraded Betti numbers of any initial ideal of the ideal of maximal minors
are either $0$ or $1$. In Section 3 we introduce a _sparse_ Eagon-Northcott
complex to be used for resolving the initial ideal of the ideal of maximal
minors with respect to any diagonal term order. As corollaries, we obtain
minimal free resolutions of the ideal of squarefree monomials of a given
degree and the box polarization for powers of the graded maximal ideal (and
hence by specialization, any power of the graded maximal ideal).

In Section 4, we consider subcomplexes of the sparse Eagon-Northcott complex
with basis elements parametrized by a simplicial complex $\Delta$. This
subcomplex in general is _not_ acyclic, but it turns out that combinatorial
properties of $\Delta$ will allow us to deduce exactly when homology is
nontrivial in linear degrees. This combinatorial condition is encoded in the
existence of so-called $i$-_nonfaces_ , a generalization of minimal nonfaces
(see Definition 4.5). We use this to show that the Betti numbers along the
linear strand of a DFI $J_{\Delta}$ and its initial ideal with respect to a
diagonal term order coincide if and only if the clique complex associated to
$\Delta$ has no $1$-nonfaces of cardinality $n+1$. We apply this result to
give a proof of the conjecture of Ene, Herzog, and Hibi for graphs having at
most $2$ maximal cliques (see Corollary 4.21).

In Section 5, we consider linear strands supported on cellular complexes. In
particular, we show that the linear strand of the initial ideal (with respect
to a diagonal term order) of _any_ DFI is supported on a polyhedral cellular
complex which can be obtained as the induced subcomplex of the _complex of
boxes_ introduced by Nagel and Reiner in [13] (see Theorem 5.11). This implies
that the multigraded Betti numbers of the linear strand of any such ideal are
$0$ or $1$ and moreover allows us to count the Betti numbers in the linear
strand as the $f$-vector of the associated polyhedral cell complex.

##  2\. Background

In this section we introduce some necessary background to be used for the rest
of the paper. To start, we discuss determinantal facet ideals and establish
some notation related to these ideals and the simplicial complexes that
parametrize them. We then give the definition of the _linear strand_ of a
minimal homogeneous complex, which will be used extensively in later sections.
Finally, we conclude with some results on ideals generated by maximal minors,
including the definition of the classical _Eagon-Northcott_ complex. We
conclude with an observation stating that the multigraded Betti numbers of the
minimal free resolution of the initial ideal of the ideal of maximal minors
with respect to any term order has multigraded Betti numbers that are either
$0$ or $1$.

######  Notation 2.1.

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

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

where $[\mathbf{a}|\mathbf{b}]=0$ if $r>n$. When $r=n$, use the simplified
notation $[\mathbf{a}]$ = $[1,\ldots,n|\mathbf{a}]$. 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 $\dim(\Gamma)\geq n-1$ 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 $S=k[x_{ij}\mid 1\leq i\leq n,\ 1\leq j\leq m]$ be a polynomial
ring over an arbitrary field $k$. Let $M$ be an $n\times m$ matrix of
variables in $S$. The _determinantal facet ideal_ (or _DFI_)
$J_{\Delta}\subseteq S$ associated to $\Delta$ is the ideal generated by
determinants of the form $[\mathbf{a}]$ where $\mathbf{a}$ supports an
$(n-1)$-facet of $\Delta$; that is, the columns of $[\mathbf{a}]$ correspond
to the vertices of some facet of $\Delta$.

######  Notation 2.4.

Let $\Delta$ be a pure $(n-1)$-dimensional simplicial complex on the vertex
set $[m]$ with maximal clique decomposition
$\Delta=\Delta_{1}\cup\cdots\cup\Delta_{c}$. The notation $J_{\Delta_{i}}$
will be used to denote the DFI associated to the simplicial complex whose
facets come from all $\mathbf{a}\in\Delta_{i}$ with $|\mathbf{a}|=r$.

######  Definition 2.5.

Let $\Delta$ be a pure $(n-1)$-dimensional simplicial complex on $m$ vertices
with maximal clique decomposition $\Delta=\bigcup_{i=1}^{c}\Delta_{i}$. The
DFI $J_{\Delta}$ is _lcm-closed_ if the following condition holds:

  1. (*)

For all $[\mathbf{a}]\in J_{\Delta_{i}}$, $[\mathbf{a}^{\prime}]\in
J_{\Delta_{j}}$ with $(a_{k},b_{k})=(a_{k}^{\prime},b_{k}^{\prime})$ for some
$1\leq i\leq r$ and $[\mathbf{a}],\ [\mathbf{a}^{\prime}]\notin
J_{\Delta_{i}\cap\Delta_{j}}$, there exists $[\mathbf{c}]\in
J_{\Delta_{i}\cap\Delta_{j}}$ such that $\operatorname{in}([\mathbf{c}])$
divides
$\operatorname{lcm}\big{(}\operatorname{in}([\mathbf{a}]),\operatorname{in}([\mathbf{a}^{\prime}])\big{)}$.

In [2], it is shown that the standard minimal generating set of an lcm-closed
DFI forms a reduced Gröbner basis; conjecturally, we believe that Definition
2.5 is equivalent to being a Gröbner basis for DFIs. The following definition
introduces the main theme of the current paper.

######  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 $i\geq 1$.

######  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 $\operatorname{in}_{<}I_{n}(M)$ specializes to the ideal
of all squarefree monomials of degree $n$ in $m$ variables.

######  Theorem 2.8 ([3, 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 $R/\operatorname{in}_{<}I_{n}(M)$. In particular,

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

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

Let $\phi:F\to G$ be a homomorphism of free modules of ranks $n$ and $m$,
respectively, with $n\geq m$. 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 $\geq 2$ 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
$\operatorname{in}_{<}I_{n}(M)$. Then for every multidegree $\alpha$,

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

###### Proof.

By Theorem 2.8, a minimal free resolution of $R/\operatorname{in}_{<}I_{n}(M)$
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

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

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

##  3\. Sparse Eagon-Northcott Complexes

In this section, we construct an explicit example of a _sparse_ Eagon-
Northcott complex. The most complicated part of the construction ends up being
the definition of the differentials; as it turns out, ayclicity will follow
immediately from Theorem 2.8. As a consequence, we deduce that certain
specializations of this complex also yield minimal free resolutions of the
ideal generated by all squarefree monomials of a given degree and powers of
the graded maximal ideal in a polynomial ring. We begin this section with the
following setup:

######  Setup 3.1.

Let $R=k[x_{ij}\mid 1\leq i\leq n,1\leq j\leq m]$ and $M=(x_{ij})_{1\leq i\leq
n,1\leq j\leq m}$ denote a generic $n\times m$ matrix, where $n\leq m$. View
$M$ as a homomorphism $M:F\to G$ of free modules $F$ and $G$ of rank $m$ and
$n$, respectively.

Let $f_{i}$, $i=1,\dots,m$, $g_{j}$, $j=1,\dots,n$ denote the standard bases
with respect to which $M$ has the above matrix representation. Let $<$ denote
any diagonal term order on $R$ and $\operatorname{in}_{<}I_{n}(M)$ the initial
ideal with respect to $<$ of the ideal of maximal minors of $M$.

######  Notation 3.2.

Let $\alpha=(\alpha_{1},\dotsc,\alpha_{n})$. Define

$\alpha_{\leq i}:=(\alpha_{1},\dotsc,\alpha_{i}),$

where $\alpha_{\leq i}=\varnothing$ if $i\leq 0$ and $\alpha_{\leq i}=\alpha$
if $i\geq n$.

######  Definition 3.3.

Let $\alpha=(\alpha_{1},\dotsc,\alpha_{n})$ with $|\alpha|=\ell$ and
$I=(i_{1}<\cdots<i_{n+\ell})$. Define the indexing set

$\mathcal{I}_{<}(\alpha,I):=\\{(i,I_{i+j})\mid i\in\\{k\mid\alpha_{k}>0\\},\
|\alpha_{\leq i-1}|\leq j\leq|\alpha_{\leq i}|\\}$

######  Example 3.4.

One easily computes:

$\mathcal{I}_{<}((1,1,1),(1,2,3,4,5,6))=\\{(1,1),(1,2),(2,3),(2,4),(3,5),(3,6)\\}$
$\mathcal{I}_{<}((1,0,2),(1,2,3,4,5,6))=\\{(1,1),(1,2),(3,4),(3,5),(3,6)\\}$
$\mathcal{I}_{<}((2,1),(1,2,4,5,6))=\\{(1,1),(1,2),(1,4),(2,5),(2,6)\\}$

######  Remark 3.5.

Each basis element of the Eagon-Northcott complex also has a
$\mathbb{Z}^{nm}$-grading and can be viewed as a monomial in the ideal
$\operatorname{in}(J_{\Delta})$:

$g^{*(\alpha)}\otimes f_{\sigma}\leftrightarrow(x_{1\sigma_{1}}\dots
x_{1\sigma_{\alpha_{1}+1}})\cdot(x_{2\sigma_{\alpha_{1}+2}}\dots
x_{2\sigma_{\alpha_{1}+\alpha_{2}+2}})\cdots(x_{n\sigma_{n+i-\alpha_{n}-2}}\dots
x_{n\sigma_{n+i-1}})=:m_{\alpha,\sigma}.$

Observe then that $\mathcal{I}(\alpha,\sigma)$ chooses precisely those indices
for which $m_{\alpha,\sigma}/x_{rs}\in\operatorname{in}_{<}I_{n}(M)$ for all
$(r,s)\in\mathcal{I}(\alpha,\sigma)$.

Using the indexing set of Definition 3.3, we can define what will end up being
a complex as follows:

######  Definition 3.6.

Adopt notation and hypotheses as in Setup 3.1. Let $E_{\bullet}^{\prime}$
denote the sequence of module homomorphisms with

$E_{\ell}^{\prime}=\begin{cases}\bigwedge^{n}G\qquad\textrm{if}\ \ell=0\\\
D_{\ell}(G^{*})\otimes\bigwedge^{n+\ell}F\qquad\textrm{otherwise},\\\
\end{cases}$

and first differential $d^{\prime}_{1}:\bigwedge^{n}F\to\bigwedge^{n}G$
sending $f_{I}\mapsto\operatorname{in}_{<}(M(f_{I}))$. For $\ell\geq 2$,
$d^{\prime}_{\ell}:D_{\ell-1}(G^{*})\otimes\bigwedge^{n+\ell-1}F\to
D_{\ell-2}(G^{*})\otimes\bigwedge^{n+\ell-2}F$ is the map

$d_{\ell}(g^{*(\alpha)}\otimes
f_{I})=\sum_{\\{i\mid\alpha_{i}>0\\}}\sum_{(i,I_{j})\in\mathcal{I}_{<}(\alpha,I)}(-1)^{j+1}x_{iI_{j}}g^{*(\alpha-\epsilon_{i})}\otimes
f_{I\backslash I_{j}}.$

######  Proposition 3.7.

The sequence of homomorphisms $E^{\prime}_{\bullet}$ of Definition 3.6 forms a
complex.

###### Proof.

Observe first that the map $d^{\prime}_{1}:\bigwedge^{n}F\to\bigwedge^{n}G$
sends

$f_{I}\mapsto x_{1I_{1}}\cdots x_{nI_{n}}g_{[n]}.$

We first verify that $d^{\prime}_{1}\circ d^{\prime}_{2}=0$. Let
$g_{k}^{*}\otimes f_{I}\in G^{*}\otimes\bigwedge^{n+1}F$; then:

$\displaystyle d^{\prime}_{1}\circ d^{\prime}_{2}(g_{k}^{*}\otimes f_{I})$
$\displaystyle=d^{\prime}_{1}((-1)^{k+1}x_{kI_{k}}f_{I\backslash
I_{k}}+(-1)^{k+2}x_{kI_{k+1}}f_{I\backslash I_{k+1}})$
$\displaystyle=(-1)^{k+1}x_{kI_{k}}(x_{1I_{1}}\cdots\widehat{x_{kI_{k}}}x_{kI_{k+1}}\cdots
x_{nI_{n}})g_{[n]}$
$\displaystyle\quad+(-1)^{k+2}x_{kI_{k+1}}(x_{1I_{1}}\cdots
x_{kI_{k}}\widehat{x_{kI_{k+1}}}\cdots x_{nI_{n}})g_{[n]}$ $\displaystyle=0.$

Assume now that $\ell\geq 1$; the fact that $d^{\prime}_{\ell+1}\circ
d^{\prime}_{\ell+2}=0$ is a nearly identical computation to that of the
standard Eagon-Northcott differential, where one uses the fact that

$\mathcal{I}_{<}(\alpha-\epsilon_{i},I\backslash
I_{j})=\begin{cases}\mathcal{I}_{<}(\alpha,I)\backslash\\{(i,I_{j})\\}\qquad\textrm{if}\
\alpha_{i}>1\\\
\mathcal{I}_{<}(\alpha,I)\backslash\\{(i^{\prime},I_{j^{\prime}})\mid
i^{\prime}=i\\}\qquad\textrm{if}\ \alpha_{i}=1.\\\ \end{cases}$

∎

######  Definition 3.8.

Adopt notation and hypotheses as in Setup 3.1. Then the _sparse Eagon-
Northcott_ complex (with respect to $<$) is the complex of Definition 3.6

######  Remark 3.9.

The complex of Definition 3.6 is called _sparse_ because of the relationship
of the differentials to that of the classical Eagon-Northcott complex. Namely,
the differentials are obtained by simply setting some of the entries equal to
$0$ in the matrix representation.

######  Corollary 3.10.

Adopt notation and hypotheses as in Setup 3.1. Then the sparse Eagon-Northcott
complex $E^{\prime}_{\bullet}$ is a minimal free resolution of the ideal
$\operatorname{in}_{<}(I_{n}(M))$.

###### Proof.

It is clear that the image of each basis element of $E_{i}^{\prime}$ forms a
linearly independent subset of $\ker(d_{i-1})$ for each $i\geq 2$. Using
Theorem 2.8, this image must also be a generating set. ∎

The following results are quick corollaries of Corollary 3.10. For the
definition of the polarization used in Corollary 3.12, see [1].

######  Corollary 3.11.

Adopt notation and hypotheses as in Setup 3.1. Let $E_{\bullet}^{\prime}$
denote the sparse Eagon-Northcott complex with respect to $<$. Then
$E_{\bullet}^{\prime}\otimes R/\sigma$ is a minimal free resolution of the
ideal of all squarefree monomials of degree $n$ in $m$ variables, where

$\displaystyle\sigma=\\{x_{11}-x_{21},x_{11}-x_{31},\ldots,x_{11}-x_{n1}\\}\cup\\{x_{12}-x_{22},\ldots,x_{12}-x_{n2}\\}\cup\ldots$
$\displaystyle\cup\\{x_{1m}-x_{2m},\ldots,x_{1m}-x_{nm}\\}$

###### Proof.

This is immediate by Theorem 2.8. ∎

######  Corollary 3.12.

Adopt notation and hypotheses as in Setup 3.1. Let $E_{\bullet}^{\prime}$
denote the sparse Eagon-Northcott complex with respect to $<$. Then, under the
relabelling

$x_{ij}\mapsto x_{j-i+1,i},$

$E^{\prime}_{\bullet}$ is a minimal free resolution of the box polarization of
$(x_{1},\dotsc,x_{m-n+1})^{n}$.

In particular, with the above relabelling, $E_{\bullet}^{\prime}\otimes
R/\sigma$ is a minimal free resolution of $(x_{1},\dotsc,x_{m-n+1})^{n}$,
where

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

##  4\. Linear Strand of Certain DFIs

In this section, we construct an explicit linear strand for the initial ideal
of certain classes of DFIs. This linear strand is built as a _generalized_
sparse Eagon-Northcott complex, where the generators in each homological
degree are parametrized by faces of the _clique complex_ associated to the
simplicial complex $\Delta$ parametrizing the DFI. We then define the notion
of an $i$-nonface of a simplicial complex (see Definition 4.5); it turns out
that the nonexistence of $1$-nonfaces will guarantee that homology of the
associated generalized sparse Eagon-Northcott complex is trivial in
appropriate degrees. In particular, we deduce that so-called lcm-closed DFIs
have the property that

$\beta_{i,i+n}(R/J_{\Delta})=\beta_{i,i+n}(R/\operatorname{in}_{<}J_{\Delta})\
\textrm{for all}\ i\geq 0,$

for any diagonal term order $<$. To conclude this section, we verify a
conjecture of Ene, Herzog, and Hibi on the Betti numbers of binomial edge
ideals when the associated graph has at most $2$ maximal cliques.

To start off, we recall some results on linear strands of general $R$-modules
due to Herzog, Kiana, and Madani.

######  Theorem 4.1 ([8], 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 $j=0,\ 1$.

######  Proposition 4.2 ([8], Corollary 1.2).

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

Let $N$ be a finitely generated $R$-module with minimal graded free resolution
$F_{\bullet}$. Assume that there exist isomorphisms making the following
diagram commute:

$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{G_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{F_{1}^{\textrm{lin}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{0}^{\textrm{lin}}.}$

Then $G_{\bullet}\cong F_{\bullet}^{\textrm{lin}}$.

Notice that the following definition is simply a subcomplex of the sparse
Eagon-Northcott complex as in Definition 3.6.

######  Definition 4.3.

Adopt notation and hypotheses as in Setup 3.1. Let $\Delta$ denote a
simplicial complex on the vertex set $[m]$. Define
$\mathcal{C}^{<}_{0}(\Delta,M):=\bigwedge^{n}G$. For $i\geq 1$, let
$\mathcal{C}^{<}_{i}(\Delta,M)\subseteq
D_{i-1}(G^{*})\otimes\bigwedge^{n+i-1}F$ denote the free submodule generated
by all elements of the form

$g^{*(\alpha)}\otimes f_{\sigma},$

where $\sigma\in\Delta$ with $|\sigma|=n+i-1$ and
$\alpha=(\alpha_{1},\dotsc,\alpha_{n})$ with $|\alpha|=i-1$.

Let $\mathcal{C}^{<}_{\bullet}(\Delta,M)$ denote the complex induced by the
differential

$d^{\prime}_{1}:\bigwedge^{n}F\to\bigwedge^{n}G$

sending $f_{I}\mapsto\operatorname{in}_{<}(M(f_{I}))$ and for $\ell\geq 2$,

$d^{\prime}_{\ell}:D_{\ell-1}(G^{*})\otimes\bigwedge^{n+\ell-1}F\to
D_{\ell-2}(G^{*})\otimes\bigwedge^{n+\ell-2}F$

is the sparse Eagon-Northcott differential

$d_{\ell}(g^{*(\alpha)}\otimes
f_{I})=\sum_{\\{i\mid\alpha_{i}>0\\}}\sum_{(i,I_{j})\in\mathcal{I}_{<}(\alpha,I)}(-1)^{j+1}x_{iI_{j}}g^{*(\alpha-\epsilon_{i})}\otimes
f_{I\backslash I_{j}}.$

######  Definition 4.4.

Adopt notation and hypotheses as in Setup 3.1. Let $\Delta$ denote a
simplicial complex on the vertex set $[m]$. The complex of Definition 4.3 will
be called the _generalized sparse Eagon-Northcott_ complex.

Notice that by the definition of a simplicial complex, the above complex is
indeed well-defined. The following definition introduces a slight
generalization of the notion of a _minimal nonface_.

######  Definition 4.5.

Let $\Delta$ be a simplicial complex. Then an $i$-nonface
$\sigma=\\{\sigma_{1}<\dots<\sigma_{\ell}\\}$ is an element
$\sigma\notin\Delta$ such that for some $j\geq 1$,
$\sigma\backslash\sigma_{j+k}\in\Delta$ for all $k=0,\dotsc,i$.

######  Example 4.6.

Consider the following graph $G$ on vertices $\\{1,2,3,4\\}$:

$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4}$$\textstyle{1.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Observe that the associated clique complex has facets $\\{1,2,4\\}$ and
$\\{1,3,4\\}$, and no minimal nonfaces. However, $\\{1,2,3,4\\}$ is a
$1$-nonface of the clique complex, since $\\{1,2,4\\}$ and $\\{1,3,4\\}$ are
both facets.

If we instead consider the graph

$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4}$$\textstyle{1,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

then the clique complex has facets $\\{1,2,3\\}$ and $\\{2,3,4\\}$. The set
$\\{1,2,3,4\\}$ is not a $1$-nonface. Likewise, there are no $1$-nonfaces of
cardinality $3$. In the graph

$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4}$$\textstyle{1,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

the associated clique complex has facets $\\{1,2,4\\}$ and $\\{2,3,4\\}$, and
has no $1$-nonfaces of cardinality $4$. However, $\\{1,3,4\\}$ is a
$1$-nonface of cardinality $3$ since $\\{1,4\\}$ and $\\{3,4\\}$ are vertices
of $G$.

######  Remark 4.7.

Notice that a minimal nonface $\sigma$ is a $\dim\sigma$-nonface in the above
definition. Moreover, any $i$-nonface is a $k$-nonface for all $k\leq i$.

In the proofs of the results in the remainder of this section, notice that we
have chosen to augment our complexes with the ring $R$. This means that we are
resolving the quotient ring $R/I$ as opposed to the module $I$; this has the
effect of shifting the indexing in the statements of Theorem 4.1 and
Proposition 4.2.

######  Lemma 4.8.

Adopt notation and hypotheses as in Setup 3.1. If the simplicial complex
$\Delta$ has no $1$-nonfaces of cardinality $\geq n+1$, then the complex
$\mathcal{C}^{<}_{\bullet}(\Delta,M)$ is the linear strand of a finitely
generated graded $R$-module with initial degree $n$.

###### Proof.

Employ Theorem 4.1. To avoid trivialities, assume $n\leq\dim(\Delta)+1$.
Observe first that $H_{i}(\mathcal{C}_{\bullet}^{<}(\Delta,M))_{i+n-1}=0$ for
all $i\geq 1$ trivially.

To finish the proof, it will be shown that

$H_{i}(\mathcal{C}_{\bullet}^{<}(\Delta,M))_{i+n}\neq 0\ \textrm{for all}\
i\geq 1$ $\implies$ $\textrm{there exists a}\ 1\textrm{-nonface of
cardinality}\ n+i,\ \textrm{for all}\ i\geq 1.$

For convenience, use the notation
$\mathcal{C}_{\bullet}^{<}(M)=:E_{\bullet}^{\prime}$, where
$\mathcal{C}_{\bullet}^{<}(M)$ is as in Definition 4.3.

Assume $H_{i}(\mathcal{C}_{\bullet}^{<}(\Delta,M))_{i+n}\neq 0$. Let
$z\in\mathcal{C}_{i}^{<}(\Delta,M)$ be a cycle that is not a boundary; without
loss of generality, assume $z$ is multihomogeneous. The complex
$\mathcal{C}_{\bullet}^{<}(M)$ is exact, whence $z=d(y)$ for some
$y\in\mathcal{C}_{i+1}^{<}(M)$. By multihomogeneity, $y=\lambda
g^{*(\alpha)}\otimes f_{\sigma}$ for some $\lambda\in k^{\times}$ with
$|\sigma|=n+i$, $|\alpha|=i$. The assumption that $z$ is not a boundary
implies that $\sigma\notin\Delta$, since otherwise
$y\in\mathcal{C}_{i+1}(\Delta,M)$. By definition of the differential of
$\mathcal{C}_{\bullet}^{<}(M)$,

$z=\lambda\cdot\sum_{\\{\ell\mid\alpha_{\ell}>0\\}}\sum_{(\ell,\sigma_{j})\in\mathcal{I}_{<}(\alpha,\sigma)}(-1)^{j+1}x_{\ell\sigma_{j}}g^{*(\alpha-\epsilon_{\ell})}\otimes
f_{\sigma\backslash\sigma_{j}}.$

Since $z\neq 0$, $(\ell,\sigma_{j})\in\mathcal{I}_{<}(\alpha,\sigma)$ for some
$\ell$, $j$. By definition of $\mathcal{I}_{<}(\alpha,\sigma)$, this means
$(\ell,\sigma_{k})\in\mathcal{I}_{<}(\alpha,\sigma)$ for all
$|\alpha_{\leq\ell-1}|\leq j\leq|\alpha_{\leq\ell}|$. This translates to the
fact that $\sigma$ is an $\alpha_{\ell}$-nonface of cardinality $n+i$. Since
$\alpha_{\ell}\geq 1$, the result follows. ∎

######  Remark 4.9.

The proof of Lemma 4.8 allows one to construct explicit examples of nonzero
homology on the complex $\mathcal{C}_{\bullet}^{<}(\Delta,M)$. Let
$\Delta^{\textrm{clique}}$ be the simplicial complex associated to the first
graph of Example 4.6. Then the element

$z=x_{22}f_{1,3,4}-x_{23}f_{1,2,4}$

is a cycle which is not a boundary in
$\mathcal{C}_{\bullet}^{<}(\Delta^{\textrm{clique}},M)$.

######  Lemma 4.10.

Adopt notation and hypotheses as in Setup 3.1. Then the following are
equivalent:

  1. (1)

$H_{1}(\mathcal{C}_{\bullet}^{<}(\Delta,M))_{n+1}=0$,

  2. (2)

$\Delta$ has no $1$-nonfaces of cardinality $n+1$.

###### Proof.

The implication $(2)\implies(1)$ is Lemma 4.8. Conversely, assume that
$\sigma\notin\Delta$ is a $1$-nonface of cardinality $n+1$. By definition,
there exists some $j$ such that $\sigma\backslash\sigma{j}$ and
$\sigma\backslash\sigma_{j+1}\in\Delta$. This means that
$z=(-1)^{j+1}(x_{j\sigma_{j}}f_{\sigma\backslash\sigma_{j}}-x_{j\sigma_{j+1}}f_{\sigma\backslash\sigma_{j+1}})$
is a cycle in $\mathcal{C}_{1}^{<}(\Delta,M)$ that is not a boundary, since
$z=d_{2}(g_{j}^{*}\otimes f_{\sigma})$, and $g_{j}^{*}\otimes
f_{\sigma}\notin\mathcal{C}_{2}(\Delta,M)$ by construction. ∎

Recall that the standard Eagon-Northcott complex inherits a
$\mathbb{Z}^{n}\times\mathbb{Z}^{m}$-grading, as described in Notation 2.10.
Since the sparse Eagon-Northcott complexes of Definition 4.3 are obtained by
simply setting certain entries in the differentials equal to $0$, these maps
will remain multigraded in an identical manner. We tacitly use this
multigrading for the remainder of this section.

######  Theorem 4.11.

Adopt notation and hypotheses as in Setup 3.1. Assume that $\Delta$ is an
$(n-1)$-pure simplicial complex such that $\Delta^{\textrm{clique}}$ has no
$1$-nonfaces of cardinality $n+1$. Let $F_{\bullet}$ denote the minimal graded
free resolution of $\operatorname{in}_{<}(J_{\Delta})$; then

$F_{\bullet}^{\textrm{lin}}\cong\mathcal{C}_{\bullet}^{<}(\Delta^{\textrm{clique}},M).$

###### Proof.

Let $Z^{\textrm{lin}}:=(\ker d_{1})_{n+1}$, where $d_{1}$ is the first
differential of the complex
$\mathcal{C}_{\bullet}^{<}(\Delta^{\textrm{clique}},M)$. By construction,
$\mathcal{C}_{1}^{<}(\Delta^{\textrm{clique}},M)$ is generated in degree $n+1$
and hence induces a homogeneous map

$\partial:\mathcal{C}_{1}^{<}(\Delta^{\textrm{clique}},M)\to
Z^{\textrm{lin}}.$

Let $0\neq z\in Z^{\textrm{lin}}$ be an element of multidegree
$(\epsilon_{s}+1,\epsilon_{i_{1}}+\cdots+\epsilon_{i_{n+1}})$ (where $1$
denotes the appropriately sized vector of all $1$’s). Set
$\tau:=\\{i_{1}<\cdots<i_{n+1}\\}$; by multihomogeneity, there are constants
$\lambda_{k}\in k$ such that

$z=\sum_{k=1}^{n+1}\lambda_{k}x_{si_{k}}f_{\tau\backslash i_{k}}.$

Since $z$ is a cycle of $\mathcal{C}_{1}^{<}(M)$ (where
$\mathcal{C}_{\bullet}^{<}(M):=E_{\bullet}^{\prime}$ is as in Definition 4.3),
there exists $y\in\mathcal{C}_{2}^{<}(M)$ such that $d_{2}(y)=z$. By
multihomogeneity, $y=\lambda g_{s}\otimes f_{\tau}$ for some constant
$\lambda$, whence
$z=\lambda(-1)^{s+1}(x_{s\sigma_{s}}f_{\sigma\backslash\sigma_{s}}-x_{s\sigma_{s+1}}f_{\sigma\backslash\sigma_{s+1}})$.
This implies that $\sigma\in\Delta^{\textrm{clique}}$, since otherwise
$\Delta^{\textrm{clique}}$ would have a $1$-nonface of cardinality $n+1$,
contradicting our assumptions. Thus $Z^{\textrm{lin}}$ is generated by the set

$\\{r_{s}(\sigma):=(-1)^{s+1}(x_{s\sigma_{s}}f_{\sigma\backslash\sigma_{s}}-x_{s\sigma_{s+1}}f_{\sigma\backslash\sigma_{s+1}})\mid
1\leq s\leq n,\ \sigma\in\Delta^{\textrm{clique}},\ |\sigma|=n+1\\}.$

Moreover, since
$\textrm{mdeg}(r_{s}(\sigma))\neq\textrm{mdeg}(r_{s^{\prime}}(\sigma^{\prime})$
for $s\neq s^{\prime}$ or $\sigma\neq\sigma^{\prime}$, the above is a basis.
Finally, $d_{2}(g_{s}^{*}\otimes f_{\sigma})=r_{s}(\sigma)$, whence the
induced map $\partial$ is an isomorphism of vector spaces. ∎

######  Remark 4.12.

Let $\Delta$ be an $(n-1)$-pure closed simplicial complex. Then
$\Delta^{\textrm{clique}}$ has no minimal nonfaces in cardinality $\geq n+1$,
since any minimal nonface is in particular a $1$-nonface. This means that
$\Delta^{\textrm{clique}}$ satisfies the hypotheses of Theorem $3.1$ of [8].

The following theorem says that the Betti numbers of a DFI and its initial
ideal coincide in the case that the associated clique complex has no
$1$-nonfaces. Recall that in general, the Betti numbers of the initial ideal
are merely an upper bound and it is quite rare to have equality throughout.

######  Theorem 4.13.

Adopt notation and hypotheses as in Setup 3.1. Assume that $\Delta$ is an
$(n-1)$-pure simplicial complex with no $1$-nonfaces of cardinality $n+1$.
Then for all $i\geq 1$,

$\beta_{i,n+i}(J_{\Delta})=\beta_{i,n+i}(\operatorname{in}_{<}(J_{\Delta})).$

###### Proof.

Notice that the linear strand of $J_{\Delta}$ is
$\mathcal{C}_{\bullet}(\Delta^{\textrm{clique}},M)$ where
$\mathcal{C}_{\bullet}$ is the generalized Eagon-Northcott complex of [8].
Then, $\mathcal{C}_{\bullet}$ and $\mathcal{C}_{\bullet}^{<}$ have the same
underlying free modules, so the result follows. ∎

######  Proposition 4.14.

Let $\Delta$ be pure $(n-1)$-dimensional simplicial complex on $m$ vertices,
and let $J_{\Delta}$ be its associated $n$-determinantal facet ideal. If
$J_{\Delta}$ is lcm-closed, then there are no $1$-nonfaces in
$\Delta^{\textrm{clique}}$.

###### Proof.

It suffices to show that there are no $1$-nonfaces of cardinality $n+1$ in
$\Delta^{\textrm{clique}}$. Suppose, seeking contradiction, that
$\mathbf{f}=\\{f_{1}<\dots<f_{n+1}\\}$ is a $1$-nonface of cardinality $n+1$
in $\Delta^{\textrm{clique}}$. By definition, there exists some $f_{i}$ such
that $\mathbf{a}=\\{f_{1},\dots,\widehat{f_{i}},f_{i+1},\dots,f_{n+1}\\}$ and
$\mathbf{b}=\\{f_{1},\dots,f_{i},\widehat{f_{i+1}},\dots,f_{n+1}\\}$ are
facets of $\Delta$. Let $\Delta_{\mathbf{a}}$ and $\Delta_{\mathbf{b}}$ be
maximal cliques of $\Delta$ containing $\mathbf{a}$ and $\mathbf{b}$,
respectively, with nontrivial intersection. By assumption,
$\Delta_{\mathbf{a}}\neq\Delta_{\mathbf{b}}$. Because $J_{\Delta}$ is lcm-
closed, there exists some facet
$\mathbf{c}\in\Delta_{\mathbf{a}}\cap\Delta_{\mathbf{b}}$ such that
$\operatorname{in}[\mathbf{c}]$ divides

$\operatorname{lcm}(\operatorname{in}[\mathbf{a}],\operatorname{in}[\mathbf{b}])=x_{1f_{1}}x_{2f_{2}}\dots
x_{if_{i}}x_{if_{i+1}}x_{i+1f_{i+2}}\dots x_{n,f_{n+1}}.$

The only possible facets $\mathbf{c}$ of $\Delta$ satisfying this property are
$\mathbf{a}$ and $\mathbf{b}$ themselves, so they must be in the same maximal
clique of $\Delta$ and $\mathbf{f}$ is indeed a face in
$\Delta^{\textrm{clique}}$, giving the desired contradiction. ∎

######  Corollary 4.15.

Adopt notation and hypotheses as in Setup 3.1. Assume that $J_{\Delta}$ is an
lcm-closed DFI. Then for all $i$,

$\beta_{i,n+i}(J_{\Delta})=\beta_{i,n+i}(\operatorname{in}_{<}(J_{\Delta})).$

Using Corollary 4.15 as initial evidence, we pose the following:

######  Conjecture 4.16.

Adopt notation and hypotheses as in Setup 3.1. Assume that $J_{\Delta}$ is an
lcm-closed DFI. Then

$\beta_{ij}(R/J_{\Delta})=\beta_{ij}(R/\operatorname{in}_{<}J_{\Delta})\quad\textrm{for
all}\ i,j.$

######  Remark 4.17.

It is important to notice that the condition on $1$-nonfaces is not sufficient
for the minimal generators to form a Gröbner basis, and is hence more general
than the property of being lcm-closed. For example, let $\Delta$ be the
simplicial complex with facets ${1,2,3}$, ${1,4,5}$, and ${1,6,7}$. One can
verify directly that there are no $1$-nonfaces of cardinality $4$, but
calculations in Macaulay2 show that the minimal generating set of the
associated determinantal facet ideal does _not_ form a Gröbner basis.

To conclude this section, we gather some necessary facts to prove (in a
special case) a conjecture of Ene, Herzog, and Hibi. The following result, due
to Conca and Varbaro, shows that ideals with squarefree initial ideals (with
respect to some term order) exhibit homological behavior similar to that of
the associated generic initial with respect to revlex.

######  Theorem 4.18 ([4, Follows from Theorem 1.3]).

Let $I$ be a homogeneous ideal in a standard graded polynomial ring $R$ with
term order $<$. If $\operatorname{in}_{<}(I)$ is squarefree, then the extremal
Betti numbers of $R/I$ and $R/\operatorname{in}_{<}(I)$ coincide. In
particular,

$\operatorname{reg}(R/I)=\operatorname{reg}(R/\operatorname{in}_{<}(I))\quad\textrm{and}\quad\operatorname{pd}_{R}(R/I)=\operatorname{pd}_{R}(R/\operatorname{in}_{<}(I)).$

######  Theorem 4.19 ([15, Theorem 1.1]).

Let $I$ be a graded ideal and let $<$ be any term order. Then the graded Betti
numbers $\beta_{i,j}(R/I)$ can be obtained from the graded Betti numbers
$\beta_{i,j}(R/\operatorname{in}_{<}(I))$ by a sequence of consecutive
cancellations.

######  Theorem 4.20 ([10, Corollary of Theorem 2.3]).

Let $G$ be a closed graph with at most $2$ maximal cliques. Then

$\operatorname{reg}(R/J_{G})\leq 2.$

Finally, we arrive at the last result of this section.

######  Corollary 4.21.

Let $G$ be a closed graph with at most $2$ maximal cliques. Let $J_{G}$ denote
the associated binomial edge ideal $J_{G}$ and let $<$ denote any diagonal
term order. Then for all $i,j$,

$\beta_{i,j}(R/J_{G})=\beta_{i,j}(R/\operatorname{in}_{<}(J_{G})).$

###### Proof.

Since $G$ is a closed graph, $J_{G}$ is lcm-closed and hence the associated
clique complex of $G$ has no $1$-nonfaces (by Proposition 4.14). It is well
known that every binomial edge ideal has squarefree Gröbner basis with respect
to $<$ (see [7]); in particular, Theorem 4.18 conbimed with Theorem 4.20 shows
that $\operatorname{reg}(R/\operatorname{in}_{<}J_{G})\leq 2$. Theorem 4.19
asserts that the Betti numbers of $R/J_{G}$ are obtained by those of
$R/\operatorname{in}_{<}J_{G}$ by a sequence of consecutive cancellations.
However, Theorem 4.13 implies that no cancellations are possible. ∎

##  5\. Linear Strands Supported on Cellular Complexes

In this section, we introduce the notion of a linear strand supported on a
polyhedral cell complex (Definition 5.6), generalizing the well-studied
phenomenon of cellular resolutions. We show that the first linear strand of
the initial ideal of any determinantal facet ideal with respect to a diagonal
term order is supported on a polyhedral cell complex. In particular, this cell
complex is an induced subcomplex of the _complex of boxes_ introduced by Nagel
and Reiner (see Construction 5.8), which supports a minimal linear free
resolution of squarefree strongly stable and strongly stable ideals [13].

We begin by recalling some basic notions from the theory of cellular
resolutions. For a more detailed exposition, see [11, Chapter 4].

######  Definition 5.1.

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 $\mathcal{P}$.

######  Construction 5.2.

Set $S=k[x_{1},\dots,x_{n}]$ to be a polynomial ring over a field $k$. Let
$\mathcal{P}$ be an oriented polyhedral complex and let
$(\alpha_{H})_{H\in\mathcal{P}}\in\mathbb{Z}^{n}$ be a labeling of the cells
of $\mathcal{P}$ such that

$\alpha_{H}=\operatorname{lcm}\\{\mathbf{x}^{\alpha_{G}}\mid G\subset H\\}.$

The labeled complex $(\mathcal{P},\alpha)$ gives rise to an algebraic complex
of free $\mathbb{Z}^{n}$-graded $S$-modules in the following way. Let
$(C_{\bullet},\partial_{\bullet})$ be the cellular chain complex for
$\mathcal{P}$. For two cells $G,H\in\mathcal{P}$ with $\dim H=\dim G+1$ denote
by $\epsilon(H,G)\in\\{0,\pm 1\\}$ the coefficient of $G$ in the cellular
boundary of $H$. Define the free modules

$F_{i}\coloneqq\bigoplus_{\begin{subarray}{c}H\in\mathcal{P}\\\ \dim
H=i+1\end{subarray}}S(-\alpha_{H}).$

The differentials $d_{i}:F_{i}\rightarrow F_{i-1}$ are given by

$d(e_{H})\coloneqq\sum_{\dim G=\dim
H-1}\epsilon(H,G)\mathbf{x}^{\alpha_{H}-\alpha_{G}}e_{G}.$

One can verify this defines an algebraic complex $\mathcal{F}_{\mathcal{P}}$.
For $\beta\in\mathbb{Z}^{n}$, denote by $\mathcal{P}_{\leq\beta}$ the
subcomplex of $\mathcal{P}$ given by all cells $H\in\mathcal{P}$ with
$\alpha_{H}\leq\beta$ coordinatewise. Let
$I=\langle\mathbf{x}^{\alpha_{v}}\mid v\in\mathcal{P}\text{ a vertex}\rangle$.

######  Lemma 5.3 ([11, Proposition 4.5]).

Adopt notation and hypotheses of Construction 5.2. Let
$\mathcal{F}_{\mathcal{P}}$ be the algebraic complex obtained from the labeled
polyhedral complex $(\mathcal{P},\alpha)$. If for every
$\beta\in\mathbb{Z}^{n}$ the subcomplex $\mathcal{P}_{\leq\beta}$ is acyclic
over $k$, then $\mathcal{F}_{\mathcal{P}}$ resolves the quotient $S/I$.
Furthermore, the resolution is minimal if $\alpha_{H}\neq\alpha_{G}$ for any
two faces $G\subset H$ with $\dim H=\dim G+1$.

######  Definition 5.4.

Adopt notation and hypotheses of Construction 5.2. The complex
$\mathcal{F}_{\mathcal{P}}$ is a _cellular resolution_ if it meets the
criteria of Lemma 5.3, and the polyhedral complex $\mathcal{P}$ _supports_ the
resolution.

We extend the notion of cellular resolution to study multigraded linear
strands supported on a polyhedral cell complex. The following lemma follows
naturally from Theorem 4.1.

######  Lemma 5.5.

Adopt notation and hypotheses of Construction 5.2 and assume the complex
$\mathcal{F}_{\mathcal{P}}$ is $d$-linear, i.e., all the generators of $I$
have degree $d$ and all higher syzygy maps are given by linear forms. Then
$\mathcal{F}_{\mathcal{P}}$ is the first linear strand of $S/I$ if, for any
$\alpha\in\mathbb{Z}^{n}$ with $\lvert\alpha\rvert=d+k$ and $k>0$,
$\tilde{H}_{i}(\mathcal{P}_{\leq\alpha})=0$ for $i=k$ and $k-1$.

######  Definition 5.6.

Adopt notation and hypotheses of Construction 5.2. The complex
$\mathcal{F}_{\mathcal{P}}$ is a _cellular linear strand_ if it satisfies the
criteria of Lemma 5.5, and the polyhedral complex $\mathcal{P}$ _supports the
linear strand_ of $S/I$.

######  Setup 5.7.

Let $R=k[x_{ij}\mid 1\leq i\leq n,1\leq j\leq m]$ and $M=(x_{ij})_{1\leq i\leq
n,1\leq j\leq m}$ denote a generic $n\times m$ matrix, where $n\leq m$. Denote
by $I_{n}(M)$ the ideal of maximal minors of $M$. Let $<$ denote any diagonal
term order on $R$, and for any ideal $I\subset R$, denote by
$\operatorname{in}(I)=\operatorname{in}_{<}(I)$ the initial ideal of $I$ with
respect to $<$.

Partition the variables of $R$ into $n$ subsets
$\check{X}_{i}=\\{x_{i1},\dots,x_{im}\\}$. Set
$K=\\{\mathbf{a}\mid\mathbf{a}\text{ is an $n$ subset of }[m]\\}$, so the
elements of $K$ are in one-to-one correspondence with the generators of
$\operatorname{in}_{<}(I_{n}(M))$ via
$\mathbf{a}=\\{a_{1}<\dots<a_{n}\\}\longleftrightarrow\mathbf{x}_{\mathbf{a}}=x_{1a_{1}}\cdots
x_{na_{n}}$.

The following construction by Nagel and Reiner can be defined more generally
for squarefree strongly stable and strongly stable ideals, but for our
purposes we only consider the case when the ideal in question is
$\operatorname{in}(I_{n}(M))$.

######  Construction 5.8.

(see [13]) Adopt notation and hypotheses of Setup 5.7. Call a subset of $K$
which is a Cartesian product $X_{1}\times\dots\times X_{n}$ for subsets
$X_{j}\subseteq\check{X}_{j}$ a _box_ inside $K$, and define the _complex of
boxes_ of $K$ to be the polyhedral subcomplex of the product of simplices
$2^{\check{X}_{1}}\times\dots\times 2^{\check{X}_{n}}$ having faces indexed by
the boxes inside $K$.

######  Theorem 5.9 ([13, Theorem 3.12]).

Adopt notation and hypotheses of Construction 5.8. Labelling a vertex
$\mathbf{a}$ in the complex of boxes by the monomial $\mathbf{x}_{\mathbf{a}}$
gives a minimal linear cellular resolution of $R/\operatorname{in}(I_{n}(M))$.

Hence $\beta_{i,j}(R/\operatorname{in}(I_{n}(M)))=1$ where $i=\sum_{k}\lvert
X_{k}\rvert-n$, $j=X_{1}\sqcup\dots\sqcup X_{n}$ for every box
$X_{1}\times\dots\times X_{n}$ inside $K$, and all other Betti numbers vanish.

######  Notation 5.10.

Let $\Delta$ be a pure $(n-1)$-dimensional simplicial complex on $m$ vertices,
and let $\mathcal{P}$ denote the complex of boxes supporting a minimal linear
cellular resolution of $R/\operatorname{in}(I_{n}(M))$. Denote by
$\mathcal{P}(\Delta)$ the induced polyhedral subcomplex of $\mathcal{P}$ on
the vertex set labeled by $\\{\mathbf{x}_{\mathbf{a}}\mid\mathbf{a}\text{ a
facet of }\Delta\\}.$

######  Theorem 5.11.

Adopt notation and hypotheses of Setup 5.7. Let $\Delta$ be a pure
$(n-1)$-dimensional simplicial complex on $m$ vertices. Then the linear strand
of $R/\operatorname{in}(J_{\Delta})$ is supported on $\mathcal{P}(\Delta)$.

###### Proof.

First, observe that although $\operatorname{in}(J_{\Delta})$ may have
generators in higher degree, its linear strand is completely determined by
syzygies among the generators of degree $n$.

Let $\ell=f_{n-1}(\Delta)$, the number of facets of $\Delta$, and proceed by
downward induction on $\ell$. The base case $\ell=\binom{m}{n}$ corresponds to
the case where $J_{\Delta}=I_{n}(M)$ and is clear. Fix $\ell\geq 0$ and assume
that for any $\Delta$ with $\ell$ generators, $\mathcal{P}(\Delta)$ supports
the first linear strand of $\operatorname{in}(J_{\Delta})$. Let
$\Delta^{\prime}\subset\Delta$ be the subcomplex with a single facet
$\mathbf{a}$ removed. Then
$\mathcal{P}(\Delta^{\prime})\subset\mathcal{P}(\Delta)$ is the induced
subcomplex $\mathcal{P}(\Delta)\setminus v_{\mathbf{a}}$, where
$v_{\mathbf{a}}$ is the vertex labeled by the generator
$\mathbf{x}_{\mathbf{a}}$ of $\operatorname{in}(J_{\Delta})$.

By Lemma 5.5, it suffices to check that for any multidegree $\alpha$ with
$\lvert\alpha\rvert=n+k$,
$\tilde{H}_{k}(\mathcal{P}(\Delta^{\prime})_{\leq\alpha})=\tilde{H}_{k-1}(\mathcal{P}(\Delta^{\prime})_{\leq\alpha})=0$.
Observe that any face of dimension $k$ in
$\mathcal{P}(\Delta^{\prime})_{\leq\alpha}$ has multidegree $\beta$ such that
$\lvert\beta\rvert=\lvert\alpha\rvert=n+k$. Therefore, if
$\dim\mathcal{P}(\Delta^{\prime})_{\leq\alpha}=k$, it is the unique box in
$\mathcal{P}$ with multidegree $\alpha$ by Theorem 5.9, which is contractible.
If $\dim\mathcal{P}(\Delta^{\prime})_{\leq\alpha}<k$, then
$H_{k}(\mathcal{P}(\Delta^{\prime})_{\leq\alpha})=0$ trivially.

By the inductive hypothesis,
$\tilde{H}_{k-1}(\mathcal{P}(\Delta^{\prime})_{\leq\alpha})=0$ as long as
$\mathbf{x}_{\mathbf{a}}$ does not divide $\mathbf{x}^{\alpha}$, so suppose it
does. Let $C$ be a cycle of dimension $k-1$ in
$\mathcal{P}(\Delta^{\prime})_{\leq\alpha}$. Since
$\tilde{H}_{k-1}(\mathcal{P}(\Delta)_{\leq\alpha})=0$ by the inductive
hypothesis, there is some boundary $B$ of dimension $k$ in
$\mathcal{P}(\Delta)_{\leq\alpha}$ of degree $\alpha$ containing the vertex
$v_{\mathbf{a}}$. By Theorem 5.9, there is a unique box in $\mathcal{P}$ with
multidegree $\alpha$, so this must be $B$. But then $\partial(B)$ will be a
linear combination of its codimension $1$ faces, including those containing
$v_{\mathbf{a}}$, so it cannot be $C$. ∎

As a consequence of Theorem 5.11, we obtain the following means of computing
the Betti numbers in the first linear strand of
$\operatorname{in}(J_{\Delta})$.

######  Corollary 5.12.

Adopt notation and hypotheses of Setup 5.7. Let $\Delta$ be a pure
$(n-1)$-dimensional simplicial complex on $m$ vertices and let
$\mathcal{P}(\Delta)$ be as in Notation 5.10. Then
$\beta_{i,i+1}(R/\operatorname{in}(J_{\Delta}))=f_{i}(\mathcal{P}(\Delta))$.

$x_{12}x_{23}$$x_{12}x_{24}$$x_{13}x_{24}$$x_{11}x_{23}$$x_{11}x_{24}$$x_{11}x_{22}$
Figure 1. Complex of boxes $\mathcal{P}$ for $\operatorname{in}(I_{2}(M))$
where $M$ is a $2\times 4$ matrix.
$x_{12}x_{24}$$x_{13}x_{24}$$x_{11}x_{23}$$x_{11}x_{24}$$x_{11}x_{22}$ Figure
2. $\mathcal{P}(G)$ where $G$ has clique decomposition
$\\{1,2,4\\}\cup\\{1,3,4\\}$ as in Example 4.6.

######  Example 5.13.

Let $G$ be the graph in Example 4.6 with clique decomposition
$\\{1,2,4\\}\cup\\{1,3,4\\}$. The complex of boxes $\mathcal{P}$ for
$\operatorname{in}(I_{2}(M))$ where $M$ is a $2\times 4$ matrix is shown in
Figure 1. The induced subcomplex of $\mathcal{P}(G)$ of $\mathcal{P}$ on the
vertices corresponding to edges in $G$ is depicted in Figure 2. The $f$-vector
for this subcomplex is $(5,6,2)$, which indeed corresponds to the Betti
numbers in the linear strand of $\operatorname{in}(J_{G})$. In particular,
$\mathcal{P}(G)$ has $1$-cells corresponding to the $1$-nonfaces
$\\{1,2,3\\}$, $\\{1,2,4\\}$ and $2$-cells corresponding to the $1$-nonface
$\\{1,2,3,4\\}$ in Example 4.6.

Note, however, that this cell complex cannot support the full free resolution
of the ideal generated by the labels on its vertices, since
$\mathcal{P}(\Delta)_{\leq\alpha}$ is not in general acyclyic for any
multidegree $\alpha$. Consider, for example, the multidegree
$\alpha=x_{11}x_{12}x_{23}x_{24}$. Then $\mathcal{P}(G)_{\leq\alpha}$ consists
of the disjoint vertices labeled by $x_{11}x_{23}$ and $x_{12}x_{24}$, so
$\tilde{H}_{0}(\mathcal{P}(G)_{\leq\alpha})=1$ and this complex does not
satisfy the hypotheses of Lemma 5.3.

######  Remark 5.14.

Nagel and Reiner show that other strongly stable and squarefree strongly
stable ideals have a minimal, linear, cellular resolution given by the complex
of boxes $\mathcal{P}$. However, ideals generated by a subset of generators in
these other cases do not, in general, have a linear strand supported on the
induced subcomplex of $\mathcal{P}$ in the same manner. The key issue is that
in other cases, the multigraded Betti numbers may not be $0$ or $1$, so the
proof of Theorem 5.11 does not apply.

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