# A new symmetric resolution for $(x_{1},\dots,x_{n})^{n}$

Đào, Hoài  Oklahoma State University<EMAIL_ADDRESS>and  Mermin,
Jeffrey  Oklahoma State University<EMAIL_ADDRESS>
###### Abstract.

Let $S=k[x_{1},\cdots,x_{n}]$ be a polynomial ring over an arbitrary field
$k$. We construct a new symmetric polytopal minimal resolution of
$(x_{1},\cdots,x_{n})^{n}$. Using this resolution, we also obtain a symmetric
polytopal minimal resolution of the ideal obtained by removing $x_{1}\cdots
x_{n}$ from the generators of $(x_{1},\cdots,x_{n})^{n}$.

We thank Reid Buchanan, Alessandra Costantini, Hailong Dao, Chris Francisco,
Craig Huneke, Susan Morey, Uwe Nagel, Irena Peeva, Jay Schweig, and Irena
Swanson for helpful conversations. Many of these conversations took place
during workshops at the Simons Laufer Institute, which also provided us with
travel support.

00footnotetext: _2020 Math Subject Classification:_ 13D02, 13F55,
05E4000footnotetext: _Keywords and Phrases:_ Minimal resolutions, polytopal
resolutions, Borel ideals

##  1\. Introduction

Let $S=k[x_{1},\cdots,x_{n}]$ and $I=\left(x_{1},\cdots,x_{n}\right)^{n}$. It
is well known that $I$ is an equigenerated Borel ideal with minimal resolution
given by either the Eliahou-Kervaire or the complex-of-boxes resolution. The
Eliahou-Kervaire construction is very nice algebraically and it is supported
by a regular cellular complex (see [mermin2010cellular]); however, this
complex is not polytopal. The complex-of-boxes resolution, introduced by Nagel
and Reiner (see [NagelReiner]), is polytopal but not symmetric.

In this paper, based on the complex-of-boxes resolution, we construct a
symmetric polytopal complex supporting the minimal resolution of $I$. Using
this complex, we also construct a symmetric polytopal complex supporting the
minimal resolution of the ideal $\widehat{I}$ obtained by removing
$m=x_{1}\cdots x_{n}$ from the generators of $I=(x_{1},\cdots,x_{n})^{n}$.

The ideas involved in these constructions are more generally applicable; one
can describe a minimal polytopal resolution of any power of the maximal ideal
with cyclic symmetry about an arbitrary (fixed) generator, or of the ideal
obtained by deleting an arbitrary monomial generator from a power of the
maximal ideal. Unfortunately, doing so is extremely notational; the details
will appear in the companion paper [Dao].

We should note that the Betti numbers of $\widehat{I}$ are already well-known
to experts, at least in folklore. (See [VdB] for an approach using trimming
complexes; alternatively, see [DE] for a proof that its resolution is linear
until the last step.) The polytopal structure of the resolution produced in
this paper allows us to write down an explicit basis and convenient formulas
for the differentials. It also allows us to consider generalizations such as
the removal of multiple generators; some analysis in this direction will
appear in [Dao].

The paper is organized as follows. In Section 2, we introduce some
terminology, notation, and tools used throughout the paper. In Section 3, we
recall the Eliahou-Kervaire and the complex-of-boxes resolutions for Borel
ideals. In Section 4, we construct a new (minimal) symmetric polytopal
resolution for $I$. In Section 5, we construct a new (minimal) symmetric
polytopal resolution for $\widehat{I}$ in the case $n=3$. In Section 6, we
generalize this construction to an arbitrary number of variables. We compute
the Betti numbers of $\widehat{I}$ in Section 7. Finally, in Section 8, we
discuss some natural questions arising from this work.

##  2\. Background and Notation

Throughout the paper, $S=k[x_{1},\cdots,x_{n}]$ is a polynomial ring over an
arbitrary field $k$.

###  2.1. Free resolutions

######  Definition 2.1.

Let $I$ be an ideal of $S$. A free resolution of $\dfrac{S}{I}$ is an exact
sequence

$\textstyle{\mathbf{F}\colon\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{i}}$$\textstyle{F_{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{i-1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\textstyle{\dfrac{S}{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where each $F_{i}$ is a free $S-$module.

If $I$ is a homogeneous ideal of $S$, then the free resolution is said to be
minimal if $d_{i+1}\left(F_{i+1}\right)\subset(x_{1},\dots,x_{n})F_{i}$ for
all $i$.

Up to isomorphism, there exists a unique minimal free resolution of
$\dfrac{S}{I}$. If $\mathbf{F}$ is the minimal free resolution of
$\dfrac{S}{I}$, the $i^{\text{th}}$ Betti number (or total Betti number) of
$\dfrac{S}{I}$ over $S$, denoted by $\beta_{i}^{S}$, is

$\displaystyle\beta_{i}^{S}\left(\dfrac{S}{I}\right)={\rm{rank}}\left(F_{i}\right),$

and if
$F_{i}=\bigoplus\limits_{p\in{\mathbb{Z}}}S\left(-p\right)^{\beta_{i,p}}$ for
each $i$, the exponents $\beta_{i,p}$ are called graded Betti numbers of
$\dfrac{S}{I}$.

######  Notation 2.2.

Following [mermin2011simplicial], we write multigradings multiplicatively.
That is, for each monomial $m$ of $S$, set $S_{m}$ equal to the $k$-vector
space spanned by $m$. Then $S=\bigoplus S_{m}$, and $S_{m}\cdot S_{n}=S_{mn}$,
so this decomposition is a grading. We say that the monomial $m$ has
multidegree $m$. We allow multidegrees to have negative exponents, so, for
example, the twisted module $S(m^{-1})$ is a free module with generator in
multidegree $m$, and $S(m^{-1})_{n}\cong S_{m^{-1}n}$ as a vector space; this
is one-dimensional if no exponent of $m^{-1}n$ is negative, and trivial
otherwise. Note that $S=S(1)$.

###  2.2. Cellular resolutions and polyhedral complexes

The idea of parametrizing a resolution by a simplicial complex was first
introduced in [BPS] by Bayer, Peeva, and Sturmfels, and then extended to more
general cellular complexes in [BS] by Bayer and Sturmfels. For details, see
the original papers or the textbooks [miller2004combinatorial, Chapter 4] or
[Peeva, Chapters 57–60]. In this paper, we are specifically interested in
resolutions supported on a polytopal complex; the definitions below loosely
follow the treatment in [miller2004combinatorial].

######  Definition 2.3.

A polyhedral cell complex $X$ is a finite collection of convex polytopes (in
${\mathbb{R}}^{n}$), called faces of $X$, satisfying two properties:

  * •

If $\mathcal{P}\in X$ and $\mathcal{F}$ is a face of $\mathcal{P}$, then
$\mathcal{F}\in X$.

  * •

If $\mathcal{P},\mathcal{Q}\in X$, then $\mathcal{P}\cap\mathcal{Q}$ (computed
as a subset of $\mathbb{R}^{n}$) is a face of both $\mathcal{P}$ and
$\mathcal{Q}$.

The embedding of $X$ in $\mathbb{R}^{n}$ equips it with an _orientation
function_ ,

$\text{sign}:X\times X\to\\{\pm 1,0\\}$

with the property that $\text{sign}(G,F)=\pm 1$ if and only if $G$ is a facet
of $F$. For all $F,G\in X$ we have $\displaystyle\sum_{H\in
X}\text{sign}(G,H)\text{sign}(H,F)=0$.

######  Definition 2.4.

The cell complex $X$ is called a _labelled_ complex if it is equipped with a
labeling function

$\ell:X\to\\{\text{monomials of $S$}\\}$

that assigns a monomial $\ell(v)$ to each vertex of $X$ and satisfies
$\ell(F)={\rm lcm}\\{\ell(v):v\in F\\}$ for every face $F$ of $X$. We refer to
$\ell(F)$ as the _label_ of $F$.

######  Definition 2.5.

Let $X$ be a labelled polytopal complex. We construct a complex of free
modules $\mathcal{F}_{X}$ with basis given by the faces of $X$ and
differential arising from the labels and the topological boundary maps. More
precisely, for an $i$-dimensional face $F\in X$, we create a basis element
(also called $F$) with homological degree $i$ and internal degree $\ell(F)$.
Then the differential $\partial$ of $\mathcal{F}_{X}$ is

$\partial(F)=\sum\limits_{\text{facets }G\text{ of
}F}{\rm{sign}}(G,F)\dfrac{\ell(F)}{\ell(G)}G$

If $\mathcal{F}_{X}$ is acyclic, we say that it is a _polytopal resolution_.

For any multidegree $m$, denote

$\displaystyle X_{\leq m}=\\{F\in X:\ell(F)\text{ divides }m\\},$
$\displaystyle X_{<m}=\\{F\in X:\ell(F)\text{ strictly divides }m\\}.$

######  Theorem 2.6.

Let $X$ be a labelled polytopal complex. The free complex $\mathcal{F}_{X}$
supported on $X$ is a (polytopal) resolution if and only if $X_{\leq m}$ is
acyclic for all $m$. In this case, it is a free resolution of the quotient
module $\dfrac{S}{I}$, where $I=(\ell(v):v\in X\text{ is a vertex})$ is
generated by the monomial labels of vertices.

###  2.3. Borel ideals

######  Definition 2.7.

We say that a monomial ideal $B$ is _Borel_ if it satisfies the condition

> If $mx_{j}\in B$ and $i<j$, then $mx_{i}\in B$.

######  Definition 2.8.

We refer to the operation sending $mx_{j}$ to $mx_{i}$ as a _Borel move_ , and
the operation sending $mx_{i+1}$ to $mx_{i}$ as an _elementary Borel move_.
Observe that a monomial ideal is Borel if and only if it is closed under
(elementary) Borel moves.

######  Definition 2.9.

Let $T=\\{m_{1},\cdots,m_{s}\\}$ be a set of monomials. Define
${\rm{Borel}}(T)={\rm{Borel}}(m_{1},\cdots,m_{s})$ to be the smallest Borel
ideal containing $T$. We say that $m_{1},\cdots,m_{s}$ are Borel generators of
${\rm{Borel}}(T)$. If $T=\\{m\\}$ has cardinality one, we say that
${\rm{Borel}}(T)$ is the principal Borel ideal generated by $m$.

######  Remark 2.10.

Let $\mathbf{m}=(x_{1},\dots,x_{n})$ be the homogeneous ideal. Then
$\mathbf{m}^{d}=\text{Borel}(x_{n}^{d})$ is a (principal) Borel ideal.

######  Definition 2.11.

Let $m$ and $m^{\prime}$ be two monomials of degree $d$. Factor
$m=x_{i_{1}}\cdots x_{i_{d}}$ and $m^{\prime}=x_{i^{\prime}_{1}}\cdots
x_{i^{\prime}_{d}}$ with $i_{1}\leq\cdots\leq i_{d}$ and
$i^{\prime}_{1}\leq\cdots\leq i^{\prime}_{d}$. The _Borel order_ on the set of
all monomials of degree $d$ is defined as

$\displaystyle m^{\prime}\leq_{\mathrm{Borel}}m\iff i^{\prime}_{1}\leq
i_{1},\cdots,i^{\prime}_{d}\leq i_{d}.$

######  Remark 2.12.

Let $m$ and $m^{\prime}$ be two monomials of degree $d$, with factorizations
$m=x_{i_{1}}\cdots x_{i_{d}}$ and $m^{\prime}=x_{i^{\prime}_{1}}\cdots
x_{i^{\prime}_{d}}$. Then $m^{\prime}\in{\rm{Borel}}(m)$ if and only if
$m^{\prime}\leq_{\mathrm{Borel}}m.$

######  Example 2.13.

The principal ideal generated by $abc$ is

$\operatorname{Borel}(abc)=(abc,abb,aac,aab,aaa)=(a^{3},a^{2}b,a^{2}c,ab^{2},abc)$

The notion of $Q$-Borel ideals was introduced in [FMS2] to study ideals
satisfying a weaker exchange property than Borel ideals. We need it here to
simultaneously study ideals that are Borel with different orders on the
variables.

######  Definition 2.14.

Let $Q$ be a poset on $\\{x_{1},\cdots,x_{n}\\}$ and $m$ be a monomial in $S$.
A $Q$-Borel move on $m$ is an operation that sends $m$ to a monomial
$m\cdot\dfrac{x_{i_{1}}}{x_{j_{1}}}\cdots\dfrac{x_{i_{s}}}{x_{j_{s}}}$, where
$i_{t}<_{Q}j_{t}$ for all $t$, and all $x_{j_{t}}$ divide $m$.

######  Definition 2.15.

A monomial ideal $B$ is a $Q$-Borel ideal if $B$ is closed under $Q$-Borel
moves. That is, if $m\in B$, then any monomial obtained from a $Q$-Borel move
on $m$ is also in $B$.

Let $T=\\{m_{1},\cdots,m_{s}\\}$ be a set of monomials. Define
${\rm{Borel}}_{Q}(T)={\rm{Borel}}_{Q}(m_{1},\cdots,m_{s})$ to be the smallest
$Q$-Borel ideal containing $T$. We say that $m_{1},\cdots,m_{s}$ are $Q$-Borel
generators of ${\rm{Borel}}_{Q}(T)$. If $T=\\{m\\}$ has cardinality one, we
say that ${\rm{Borel}}_{Q}(T)$ is the principal $Q$-Borel ideal generated by
$m$.

######  Example 2.16.

Let $S=k[a,b,c]$ and $I=\left(a^{2},ab,ac,c^{2}\right)$. Observe that $I$ is
not Borel since $c^{2}\in I$ but $bc\not\in I$. However, $I$ is $Q-$Borel with
$Q=(x<z<y)$, and $I={\rm{Borel}}_{Q}(ab,c^{2}).$

There are several explicit ways to resolve a Borel ideal. Two of the most
important are the Eliahou-Kervaire resolution and the complex-of-boxes
resolution, which we discuss in the next section.

##  3\. Resolving powers of the maximal ideal

In this section, we recall two famous resolutions of Borel ideals which are
the Eliahou-Kervaire and the complex-of-boxes resolutions.

For a monomial $m=x_{i_{1}}\dots x_{i_{d}}$ with $i_{1}\leq\dots\leq i_{d}$,
define $\max(m)=i_{d}$, the maximal index of a variable dividing $m$. If $I$
is a Borel ideal, the Eliahou-Kervaire resolution is a minimal resolution of
$S/I$ with basis consisting of $1$ (in homological degree 0) and the EK-
symbols $[m,\alpha]$ where $m$ is a monomial generator of $I$ and $\alpha$ is
a squarefree monomial with $\max(\alpha)<\max(m)$. The EK-symbol $[m,\alpha]$
has multidegree $m\alpha$ and homological degree $(1+\deg(\alpha))$. See
[PeevaStillman] for proofs, motivation, and the differential maps.

The Eliahou-Kervaire resolution allows us to compute the (graded) Betti
numbers of a Borel ideal by counting the basis elements associated to each
monomial generator.

######  Proposition 3.1.

([EliahouKervaire]) Let $I$ be a Borel ideal generated minimally by monomials
$m_{1},\cdots,m_{r}$. Then

$\displaystyle\beta_{p,p+q}^{S}(I)=\sum\limits_{\deg(m_{i})=q,\hskip
5.69054pt1\leq i\leq r}\binom{\max(m_{i})-1}{p},$
$\displaystyle\beta_{p}^{S}(I)=\sum\limits_{i=1}^{r}\binom{\max(m_{i})-1}{p}.$

The Eliahou-Kervaire resolution is known to be cellular but is believed to be
polytopal only in three or fewer variables [mermin2010cellular]. The
associated cellular complexes are shown in Figure 1.

$a^{3}$$a^{2}b$$ab^{2}$$b^{3}$$a^{2}c$$abc$$b^{2}c$$ac^{2}$$bc^{2}$$c^{3}$

$a^{2}$$b^{2}$$c^{2}$$d^{2}$$ab$$ac$$bc$$ad$$bd$$cd$

Figure 1. The Eliahou-Kervaire resolution of $(a,b,c)^{3}$ and
$(a,b,c,d)^{2}$.

The complex of boxes, introduced by Nagel and Reiner [NagelReiner], has the
virtue of always being polytopal.

######  Definition 3.2.

An _elementary box_ or _interval_ is a simplex whose vertices are a subset of
variables. Anticipating Notation 4.1, we set $\Delta_{[i,j]}$ equal to the
simplex on vertices $\\{x_{i},\dots,x_{j}\\}$.

######  Definition 3.3.

A _box_ is a finite product of intervals. The simplicial structure of the
intervals induces a polytopal structure on the box.

######  Example 3.4.

In Figure 2, $\Delta_{[a,c]}\times\Delta_{[c,d]}$ is a box, as is
$\Delta_{[a,b]}\times\Delta_{[b,c]}\times\Delta_{[c,d]}$.

$ac$$bc$$c^{2}$$ad$$bd$$cd$

$abc$$ac^{2}$$b^{2}c$$bc^{2}$$abd$$acd$$b^{2}d$$bcd$

Figure 2. The polytopal structures of
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Delta_{[a,c]}}\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Delta_{[c,d]}}$
and $\Delta_{[a,b]}\times\Delta_{[b,c]}\times\Delta_{[c,d]}$. The vertices are
labelled with monomials rather than ordered tuples, using the labelling of
Definition 3.5.

######  Definition 3.5.

Let $\Gamma=\displaystyle\prod_{i=1}^{n}\Delta_{i}$ be a box. Every vertex of
$\Gamma$ is an ordered $n$-tuple of variables,
$v=(\dots,x_{j_{i}}\in\mathop{{\rm supp}}\Delta_{i},\dots)$. We label the
vertices with monomials by forgetting the ordering,
$\ell(v)=\displaystyle\prod_{i=1}^{n}x_{j_{i}}$.

We say that $\Gamma$ is an _admissible box_ if all vertices have distinct
monomial labels.

######  Example 3.6.

As shown in Figure 3, $\Delta_{[a,c]}\times\Delta_{[c]}\times\Delta_{[c,d]}$
is an admissible box, whereas $\Delta_{[a,b]}\times\Delta_{[a,b]}$ is not
admissible.

$ac^{2}$$bc^{2}$$c^{3}$$acd$$bcd$$c^{2}d$

$ab$$b^{2}$$a^{2}$$ab$

Figure 3. The polytopal structures of
$\Delta_{[a,c]}\times\Delta_{[c]}\times\Delta_{[c,d]}$ and
$\Delta_{[a,b]}\times\Delta_{[a,b]}$.

######  Construction 3.7.

For a monomial $m=x_{i_{1}}\dots x_{i_{d}}$, set $\Gamma(m)$ equal to the
$(i_{d}-1)$-dimensional box

$\Gamma(m)=\Delta_{[1,i_{1}]}\times\Delta_{[i_{1},i_{2}]}\times\Delta_{[i_{2},i_{3}]}\times\dots\times\Delta_{[i_{d-1},i_{d}]}.$

Observe that $\Gamma(m)$ is admissible.

If $I$ is an equigenerated Borel ideal with minimal generating set
$I=(m_{1},\dots,m_{s})$, then the _complex of boxes_ associated to $I$ is the
polytopal complex generated by the faces
$\\{\Gamma(m_{1}),\dots,\Gamma(m_{s})\\}$. (This generating set is not
minimal, since some faces, e.g. $\Gamma(x_{1}^{d})$, are not facets.)

######  Example 3.8.

Like the Eliahou-Kervaire resolution, the complex of boxes associated to
$(x_{1},\dots,x_{n})^{d}$ is a subdivision of the topological $n$-simplex with
vertices labeled by powers of the variables. In three variables, the two
complexes coincide (modulo a permutation) as shown in Figure 4. But with more
variables, differences begin to emerge. Most notably, the complex of boxes is
polytopal, whereas the Eliahou-Kervaire resolution is not, as seen in Figure
5. For example, the complex-of-boxes resolution of $(a,b,c,d)^{2}$ can be
decomposed into four polytopes as in Figure 6.

$a^{3}$$a^{2}b$$ab^{2}$$b^{3}$$a^{2}c$$abc$$b^{2}c$$ac^{2}$$bc^{2}$$c^{3}$

$a^{3}$$a^{2}b$$ab^{2}$$b^{3}$$a^{2}c$$abc$$b^{2}c$$ac^{2}$$bc^{2}$$c^{3}$

Figure 4. The Eliahou-Kervaire resolution (left) and the complex-of-boxes
resolution (right) of $(a,b,c)^{3}$.

$a^{2}$$b^{2}$$c^{2}$$d^{2}$$ab$$ac$$bc$$ad$$bd$$cd$

$a^{2}$$b^{2}$$c^{2}$$d^{2}$$ab$$ac$$bc$$ad$$bd$$cd$

Figure 5. The complexes supporting the Eliahou-Kervaire resolution and the
complex-of-boxes resolution of $(a,b,c,d)^{2}$. The Eliahou-Kervaire complex
is not polytopal, since the cell containing $d^{2}$ contains $ac$ and $bc$ but
none of the segment connecting them. The box complex consists of two
tetrahedra and two triangular prisms.

$d^{2}$$ad$$bd$$cd$

$a^{2}$$ab$$ac$$ad$

$b^{2}$$ab$$ac$$bc$$ad$$bd$

$c^{2}$$ac$$bc$$ad$$bd$$cd$

Figure 6. The four boxes comprising the complex-of-boxes resolution of
$(a,b,c,d)^{2}$.

######  Theorem 3.9.

[NagelReiner, Theorem 3.12] Let $I$ be an equigenerated Borel ideal. Then the
complex of boxes associated to $I$ is a polytopal complex, and supports a
minimal resolution of $I$.

##  4\. A new symmetric polytopal resolution of $(x_{1},\cdots,x_{n})^{n}$

In this section, we build a resolution of
$I=\mathfrak{m}^{n}=(x_{1},\dots,x_{n})^{n}$ which is a polytopal subdivision
of the $(n-1)$-dimensional simplex, and is also symmetric about its center
$m=x_{1}\cdots x_{n}$. (In particular, no variable is preferenced over the
others, so all corners of the simplex will look the same.) Our strategy is to
let the cyclic group act on the principal Borel ideal
$I_{1}=\operatorname{Borel}(m)$ and its complex of boxes. This produces $n$
polytopal complexes which intersect in positive codimension; gluing them
together yields the desired resolution.

Unfortunately the gluing process is quite technical. We begin with a string of
lemmas.

######  Lemma 4.1.

[Sinefakopoulos2007, Lemma 3.1] Let $I$ and $J$ be two monomial ideals in $S$
such that $G(I+J)=G(I)\cup G(J)$ set-theoretically. Suppose that

  1. (i)

$X$ and $Y$ are regular cell complexes in ${\mathbb{R}}^{N}$ that support a
(minimal) free resolution of $I$ and $J$, respectively, and

  2. (ii)

$X\cap Y$ is a regular cell complex that supports a (minimal) free resolution
of $I\cap J$.

Then $X\cup Y$ supports a (minimal) free resolution of $I+J$.

######  Remark 4.2.

[Sinefakopoulos2007, Remark 3.2] For any two monomial ideals $I$ and $J$, we
have $G(I+J)\subseteq G(I)\cup G(J)$. Our assumption that $G(I+J)=G(I)\cup
G(J)$ guarantees the right labelling of the cell complex $X\cup Y$. A case
where equality holds is when all elements of $G(I)\cup G(J)$ are of the same
degree.

######  Remark 4.3.

From assumption (ii) of Lemma 4.1, $I\cap J$ is generated by the labels of the
vertices of $X\cap Y$. We conclude that $G(I\cap J)=G(I)\cap G(J)$. This is an
extremely strong assumption, as intersecting ideals almost always produces
generators in higher degree. But it does hold in the cases of interest, where,
among other things, the ideals in question all have linear quotients, and the
boundaries of the complexes coincide.

######  Lemma 4.4.

Let $M_{1},M_{2},\cdots,M_{n}$ be equigenerated monomial ideals, all generated
in the same degree. Assume that

  1. (i)

$W_{1},\cdots,W_{n}$ are regular cell complexes in ${\mathbb{R}}^{N}$ that
support a (minimal) free resolution of $M_{1},\cdots,M_{n}$, respectively, and

  2. (ii)

For all subsets $\\{i_{1},\dots,i_{\ell}\\}$, $W_{i_{1}}\cap
W_{i_{2}}\cap\cdots\cap W_{i_{\ell}}$ is a regular cell complex that supports
a (minimal) free resolution of $M_{i_{1}}\cap M_{i_{2}}\cap\cdots\cap
M_{i_{\ell}}$.

Then $W_{1}\cup W_{2}\cup\cdots\cup W_{n}$ supports a (minimal) free
resolution of $M_{1}+M_{2}+\cdots+M_{n}$.

###### Proof.

Since $M_{1},M_{2},\cdots,M_{n}$ are all equigenerated monomial ideals,

$\displaystyle
G\left(M_{i_{1}}+M_{i_{2}}+\cdots+M_{i_{\ell}}\right)=G\left(M_{i_{1}}\right)\cup
G\left(M_{i_{2}}\right)\cup\cdots\cup G\left(M_{i_{\ell}}\right).$

We will prove the statement by induction on the number of ideals, $n$.

The case $n=1$ is trivial and the case $n=2$ is given by Lemma 4.1. Assume
that the statement holds for $n$. We shall show that it is also true for
$n+1$. Let

$\displaystyle M_{1}^{\prime}$ $\displaystyle=M_{1}+M_{n+1},$ $\displaystyle
W_{1}^{\prime}$ $\displaystyle=W_{1}\cup W_{n+1},$ $\displaystyle
M_{2}^{\prime}$ $\displaystyle=M_{2}+M_{n+1},$ $\displaystyle W_{2}^{\prime}$
$\displaystyle=W_{2}\cup W_{n+1},$ $\displaystyle\vdots$ $\displaystyle\vdots$
$\displaystyle M_{n}^{\prime}$ $\displaystyle=M_{n}+M_{n+1},$ $\displaystyle
W_{n}^{\prime}$ $\displaystyle=W_{n}\cup W_{n+1}.$

It follows from Lemma 4.1 that $W_{i}^{\prime}$ is a (minimal) free resolution
of $M_{i}^{\prime}$, for $1\leq i\leq n$, and that

$\displaystyle W^{\prime}_{i_{1}}\cap W^{\prime}_{i_{2}}\cap\cdots\cap
W^{\prime}_{i_{\ell}}=\left(W_{i_{1}}\cap W_{i_{2}}\cap\cdots\cap
W_{i_{\ell}}\right)\cup W_{n+1}$

is a (minimal) free resolution of

$\displaystyle\left(M_{i_{1}}\cap M_{i_{2}}\cap\cdots\cap
M_{i_{\ell}}\right)+M_{n+1}=M_{1}^{\prime}\cap M_{2}^{\prime}\cap\cdots\cap
M_{i_{\ell}}^{\prime}.$

By the inductive hypothesis (applied to $M_{i}^{\prime}$ and
$W_{i}^{\prime}$), we obtain that

$\displaystyle W_{1}\cup W_{2}\cdots\cup W_{n+1}=W_{1}^{\prime}\cup
W_{2}^{\prime}\cup\cdots\cup W_{n}^{\prime}$

is a (minimal) free resolution of

$\displaystyle M_{1}+M_{2}+\cdots
M_{n+1}=M_{1}^{\prime}+M_{2}^{\prime}+\cdots+M_{n}^{\prime}.$

∎

######  Corollary 4.5.

Let $M_{1},M_{2},\cdots,M_{n}$ be equigenerated monomial ideals, all generated
in the same degree. Assume that

  1. (i)

$W_{1},\cdots,W_{n}$ are regular cell complexes in ${\mathbb{R}}^{N}$ that
support a (minimal) free resolution of $M_{1},\cdots,M_{n}$, respectively, and

  2. (ii)

$W_{i_{1}}\cap W_{i_{2}}\cap\cdots\cap W_{i_{\ell}}$ is a regular cell complex
that supports a (minimal) free resolution of $M_{i_{1}}\cap
M_{i_{2}}\cap\cdots\cap M_{i_{\ell}}$.

Then $\left(W_{1}\cup W_{2}\cup\cdots\cup W_{n-1}\right)\cap W_{n}$ supports a
(minimal) free resolution of

$\displaystyle\left(M_{1}+M_{2}+\cdots+M_{n-1}\right)\cap M_{n}.$

###### Proof.

Apply Lemma 4.4 to $M_{1}\cap M_{n},M_{2}\cap M_{n},\cdots,M_{n-1}\cap M_{n}$.
∎

######  Lemma 4.6.

[Sinefakopoulos2007, Lemma 3.3] Let $I\subset k[x_{1},\cdots,x_{s}]$ and
$J\subset k[x_{s+1},\cdots,x_{n}]$ be two monomial ideals. Suppose that $X$
and $Y$ are regular cell complexes in ${\mathbb{R}}^{N}$ of dimension $s-1$
and $n-s-1$, respectively, that support a (minimal) free resolution of $I$ and
$J$, respectively. Then the regular cell complex $X\times Y$ supports a
(minimal) free resolution for $IJ.$

######  Lemma 4.7.

[Sinefakopoulos2007, Lemma 3.5] Let $I\subset k[x_{1},\cdots,x_{s}]$ and
$J\subset k[x_{s},\cdots,x_{n}]$ be two monomial ideals such that $\lvert
G(IJ)\rvert=\lvert G(I)\rvert\cdot\lvert G(J)\rvert$. Suppose that $X$ and $Y$
are regular cell complexes in some ${\mathbb{R}}^{N}$ of dimension $s-1$ and
$n-s$, respectively, that support a (minimal) free resolution of $I$ and $J$,
respectively. Then the regular cell complex $X\times Y$ supports a (minimal)
free resolution for $IJ.$

Next, we need notation to describe the various ideals and complexes obtained
from the action of the cyclic group. We begin with the notion of a cyclic
interval. Let $\sigma$ be the standard generator of the cyclic group on $n$
letters; that is, $\sigma=(1\quad 2\quad\cdots\quad n)$

######  Notation 4.8.

For any $t\in\\{1,\dots,n\\}$, the _initial interval_ $[1,t]$ refers to the
set $\\{1,2,\dots,t\\}$. For any $i,j\in\\{1,\dots,n\\}$, the _cyclic
interval_ $[i,j]$ refers to the interval obtained by applying $\sigma^{i-1}$
to the appropriate initial interval. That is, $[i,j]=\\{i,i+1,\dots,j-1,j\\}$
with all elements understood to be taken modulo $n$.

######  Notation 4.9.

For any cyclic interval $[i,j]$, we define the _simplicial complex supported
on $[i,j]$_, $\Delta_{[i,j]}=\langle\\{x_{s}:s\in[i,j]\\}\rangle$, the
_monomial supported on $[i,j]$_ $m_{[i,j]}=\prod_{s\in[i,j]}x_{s}$, and the
_prime ideal supported on $[i,j]$_, $\mathbf{m}_{[i,j]}=(x_{s}:s\in[i,j])$.

######  Notation 4.10.

The poset $Q_{1}$ is the usual chain on $[1,n]$,
$Q_{1}:x_{1}<x_{2}<\dots<x_{n}$. For any $i$, we define the poset $Q_{i}$ to
be $Q_{i}=\sigma^{i-1}(Q_{1})$. That is, the covering relations in $Q_{i}$ are
$x_{i}<x_{i+1}<\dots<x_{n}<x_{1}<\dots<x_{i-1}$.

Equivalently, all $Q_{i}$ are obtained by removing the relation
$x_{i-1}<x_{i}$ from the overdetermined set of cyclic covering relations
$x_{1}<x_{2}<\dots<x_{n-1}<x_{n}<x_{1}$.

Finally, the ideal $I_{1}$ is the principal Borel ideal generated by
$m=x_{1}\dots x_{n}$, and for all $i$, $I_{i}$ refers to the cyclic
permutation of $I_{1}$, that is, $I_{i}=\sigma^{i-1}I_{1}$. Equivalently,
$I_{i}=\operatorname{Borel}_{Q_{i}}(m)$.

######  Notation 4.11.

For any $t$, denote by $\Gamma_{[1,t]}$ the complex of boxes minimally
resolving $\operatorname{Borel}(m_{[1,t]})$. For any $i,j$, denote by
$\Gamma_{[i,j]}$ the complex obtained by applying $\sigma^{i-1}$ to the
appropriate $\Gamma_{[1,t]}$. (Specifically, $t=j-i+1$ or $j-i+1+n$, whichever
is in $[1,n]$.) We abuse notation by referring to $\Gamma_{[i,j]}$ as the
_complex of boxes resolving $\operatorname{Borel}_{Q_{i}}(m_{[i,j]})$_.

######  Lemma 4.12.

We have

$\displaystyle I_{1}$
$\displaystyle=(x_{1})(x_{1},x_{2})\cdots(x_{1},\cdots,x_{n})$
$\displaystyle={\mathbf{m}}_{[1,1]}{\mathbf{m}}_{[1,2]}\cdots{\mathbf{m}}_{[1,n]},$

Similarly, for any $i$,

$\displaystyle
I_{i}={\mathbf{m}}_{[i,i]}{\mathbf{m}}_{[i,i+1]}\cdots{\mathbf{m}}_{[i,i-1]}$

###### Proof.

This is immediate from [Sinefakopoulos2007, Proposition 2.7], which states
that $\operatorname{Borel}(\prod
x_{i}^{d_{i}})=\prod(\operatorname{Borel}(x_{i}))^{d_{i}}$. (Here all $d_{i}$
are equal to $1$.) ∎

######  Lemma 4.13.

We have $I=\displaystyle\sum_{i}^{n}I_{i}$.

###### Proof.

It suffices to show that, for an arbitrary monomial $f$ of degree $n$, we have
$f\in I_{i}$ for some $i$.

Write $f=\prod x_{j}^{e_{j}}$, and set
$\delta_{[r,s]}(f)=\sum_{j\in[r,s]}(e_{j}-1)$. Then $f\in I_{i}$ if and only
if $\delta_{[i,s]}(f)\geq 0$ for all $s$ (including $s=i$, which is the
statement that $x_{i}$ divides $f$).

Choose $t$ such that $\delta_{[1,t]}(f)$ is minimal. If $\delta_{[1,t]}(f)\geq
0$, then $f\in I_{1}$. If not, we claim that $f\in I_{t+1}$. Indeed, setting
$r=t+1$, we have

$\delta_{[r,s]}(f)=\left\\{\begin{matrix}\delta_{[1,s]}(f)-\delta_{[1,t]}(f)&\text{if}&r\leq
s\leq n\\\ \delta_{[r,n]}(f)+\delta_{[1,s]}(f)&\text{if}&1\leq
s<r\end{matrix}\right.$

Observe that
$\delta_{[1,t]}(f)+\delta_{[r,n]}(f)=\delta_{[1,n]}(f)=\deg(f)-n=0$. Thus
$\delta_{[r,n]}(f)=-\delta_{[1,t]}(f)$ and in either case we have
$\delta_{[r,s]}(f)=\delta_{(1,s)}(f)-\delta_{[1,t]})(f)$, which is nonnegative
by the choice of $t$. ∎

######  Theorem 4.14.

For $1\leq i_{1}<\cdots<i_{\ell}\leq n$, set $i_{\ell+1}=i_{1}$ and then
denote by $\mathbf{p}_{j}$ the product of cyclic primes
$\mathbf{p}_{j}:=\mathbf{p}_{[i_{j},i_{j+1}-1]}:=\mathbf{m}_{[i_{j},i_{j}]}\mathbf{m}_{[i_{j},i_{j}+1]}\dots\mathbf{m}_{[i_{j},i_{j+1}-1]}$.

Then:

  1. (1)

For all $j$, we have
$\mathbf{p}_{j}=\operatorname{Borel}_{Q_{i_{j}}}(m_{[i_{j},i_{j+1}-1]})$.

  2. (2)

We have

$I_{i_{1}}\cap\cdots\cap
I_{i_{\ell}}=\mathbf{p}_{{1}}\mathbf{p}_{{2}}\dots\mathbf{p}_{{\ell}}$

  3. (3)

$I_{i_{1}}\cap\cdots\cap I_{i_{\ell}}$ is minimally resolved by the product of
the box complexes resolving the $\mathbf{p}_{j}$. That is

$X_{i_{1},\dots,i_{\ell}}:=\Gamma_{[i_{1},i_{2}-1]}\times\Gamma_{[i_{2},i_{3}-1]}\times\dots\times\Gamma_{[i_{\ell},i_{1}-1]}$

gives a minimal resolution of $I_{i_{1}}\cap\cdots\cap I_{i_{\ell}}$.

  4. (4)

$I_{i_{1}}\cap\cdots\cap I_{i_{\ell}}$ is minimally resolved by the
intersection of the box complexes resolving the
$\operatorname{Borel}_{Q_{i_{j}}}(m)$,

$X_{i_{1},\dots,i_{\ell}}=\displaystyle\bigcap_{j=1}^{\ell}\Gamma_{[i_{j},i_{j}-1]}.$

###### Proof.

To show that $I_{i_{1}}\cap\dots\cap
I_{i_{\ell}}\subseteq\prod\mathbf{p}_{j}$, observe that
$I_{i_{j}}\subset\mathbf{p}_{j}$ for all $j$ by Lemma 4.12. Since the
$\mathbf{p}_{j}$ have disjoint support, we have
$\bigcap\mathbf{p}_{j}=\prod\mathbf{p}_{j}$. Thus

$\bigcap_{j=1}^{\ell}I_{i_{j}}\subset\bigcap_{j=1}^{\ell}\mathbf{p}_{j}=\prod_{j=1}^{\ell}\mathbf{p}_{j}.$

To show that $\prod\mathbf{p}_{j}\subseteq I_{i_{1}}\cap\dots\cap
I_{i_{\ell+1}}$, it suffices to show that, without loss of generality,
$\prod\mathbf{p}_{j}\subseteq I_{i_{1}}$. Observe that

$\displaystyle\mathbf{p}_{j}$
$\displaystyle=\mathbf{m}_{[i_{j},i_{j}]}\mathbf{m}_{[i_{j},i_{j}+1]}\dots\mathbf{m}_{[i_{j},i_{j+1}-1]}$
$\displaystyle\subseteq\mathbf{m}_{[i_{1},i_{j}]}\mathbf{m}_{[i_{1},i_{j}+1]}\dots\mathbf{m}_{[i_{1},i_{j+1}-1]}.$

Thus,

$\prod\mathbf{p}_{j}\subseteq\biggl{[}\mathbf{m}_{[i_{1},i_{1}]}\dots\mathbf{m}_{[i_{1},i_{2}-1]}\biggr{]}\biggl{[}\mathbf{m}_{[i_{1},i_{2}]}\dots\mathbf{m}_{[i_{1},i_{3}-1]}\biggr{]}\dots\biggl{[}\mathbf{m}_{[i_{1},i_{\ell}]}\dots\mathbf{m}_{[i_{1},i_{1}-1]}\biggr{]}=I_{i_{1}}$

as desired.

This proves (2). For (3), observe that the supports of the
$\mathbf{p}_{i_{j}}$ are pairwise disjoint and induct on Lemma 4.6; we
conclude that $X_{i_{1},\dots,i_{\ell}}$ yields a minimal resolution.

For (4), set
$\mathcal{I}=\displaystyle\bigcap_{j=1}^{\ell}\Gamma_{[i_{j},i_{j}-1]}$ and
$\mathcal{P}=\Gamma_{[i_{1},i_{2}-1]}\times\dots\times\Gamma_{[i_{\ell,i_{1}-1}]}$.
We will show $\mathcal{I}=\mathcal{P}$.

To show $\mathcal{P}\subseteq\mathcal{I}$, let $\Lambda$ be a facet of
$\mathcal{P}$. We may write $\Lambda=\prod_{j}\Gamma_{i_{j}}(f_{j})$, with
each $f_{j}\in\mathbf{p}_{j}$. Observe that the poset on
$\\{x_{i_{j}},\dots,x_{i_{j+1}-1}\\}$ defined by the covering relations
$x_{i_{j}}<x_{i_{j}+1}<\dots<x_{i_{j+1}-1}$ is a sub-poset of $Q_{i_{j}}$ for
all $j$. Setting $f=\prod_{j}f_{j}$, we immediately have, for all
$t\in\\{1,\dots,\ell\\}$, $f\in\operatorname{Borel}_{Q_{i_{t}}}(m)$ and thus
$\Gamma_{i_{j}}(f_{j})\subseteq\Gamma_{i_{t}}(f_{j})$. We conclude
$\Lambda\subseteq\Gamma_{i_{t}}(f)\in\Gamma_{[i_{t},i_{t+1}-1]}$. Since $t$ is
arbitrary, we have $\Lambda\in\mathcal{I}$ as desired.

To show $\mathcal{I}\subseteq\mathcal{P}$, assume without loss of generality
that $i_{1}=1$, and let $\Lambda$ be a face of $\mathcal{I}$. Since
$\Lambda\in\Gamma_{[1,n]}(m)$, there exists a monomial
$f\in\operatorname{Borel}(m)$ such that $\Lambda\subseteq\Gamma_{Q_{1}}(f)$.
Observe that we may choose $f$ to be a vertex of $\Lambda$, and so
$\\{f\\}\in\Gamma_{[i_{j},i_{j}-1]}$ for all $j$. In particular, $f\in I_{j}$
for all $j$.

We may factor $f=x_{\alpha_{1}}x_{\alpha_{2}}\dots x_{\alpha_{n}}$ with
$\alpha_{t}\leq t$ for all $t$; since $f\in I_{2}$ we also have
$\alpha_{i_{2}},\dots,\alpha_{n}\in[i_{2},\dots,n]$. By construction, we may
write $\Gamma_{[1,n]}(f)=\mathrm{Z}_{1}\times\dots\times\mathrm{Z}_{n}$, with
$\mathrm{Z}_{1}=\\{x_{1}\\}$ and all other
$\mathrm{Z}_{t}\subseteq[x_{\alpha_{t-1}},x_{\alpha_{t}}]$. Set
$f_{1}=x_{\alpha_{1}}x_{\alpha_{2}}\dots x_{\alpha_{i_{2}-1}}$ and
$\Lambda_{1}=\mathrm{Z}_{1}\times\dots\times\mathrm{Z}_{i_{2}-1}$; observe
that $\Lambda_{1}\in\Gamma_{Q_{1}}(f_{1})$. For each $j$, define $f_{j}$ and
$\Lambda_{j}$ similarly. We have $f=\prod f_{j}$ and
$\Lambda=\prod\Lambda_{j}\in\mathcal{P}$ as desired. ∎

We have now shown that the box complexes resolving the $I_{i}$ satisfy the
hypotheses of Lemma 4.4. We conclude that their union resolves $I$. More
precisely, we have:

######  Theorem 4.15.

Set $X_{i}=\Gamma_{[i,i-1]}$ equal to the complex of boxes resolving $I_{i}$.
Then for any $i_{1},\cdots,i_{\ell}$, the polytopal complex
$X_{i_{1}}\cup\cdots\cup X_{i_{\ell}}$ supports a minimal (linear) resolution
of $I_{i_{1}}+\cdots+I_{i_{\ell}}$.

######  Corollary 4.16.

Set $X_{i}$ equal to the complex of boxes resolving $I_{i}$, and $X=\bigcup
X_{i}$. Then $X$ supports a minimal resolution of $\mathbf{m}^{n}=I$.

######  Example 4.17.

The complex in Figure 7 supports the minimal linear resolution of
$I_{a}+I_{b}=(a^{3},a^{2}b,ab^{2},b^{3},a^{2}c,abc,b^{2}c,bc^{2})$.

$a^{3}$$a^{2}b$$ab^{2}$$b^{3}$$a^{2}c$$abc$$b^{2}c$$bc^{2}$ Figure 7. A
minimal resolution of $I_{a}+I_{b}\subset k[a,b,c]$.

######  Example 4.18.

The complex in Figure 8 supports a minimal linear resolution of $(a,b,c)^{3}$.

$a^{3}$$a^{2}b$$ab^{2}$$b^{3}$$a^{2}c$$abc$$b^{2}c$$ac^{2}$$bc^{2}$$c^{3}$
Figure 8. A minimal linear resolution of $(a,b,c)^{3}$ .

##  5\. The link and star of $abc$ in $k[a,b,c]$

We have established in Corollary 4.16 a polytopal resolution for
$(x_{1},\dots,x_{n})^{n}$ which is symmetric around the monomial $m=x_{1}\dots
x_{n}$. Our next goal is to remove $m$ from the generating set and describe
the resolution of the resulting ideal, which we call the _pinched power ideal_
by analogy to its toric ideal, which in geometric contexts is called the
pinched Veronese. Our strategy is to replace the $n$ boxes containing $m$ with
their union, a single polytope which does not have $m$ as a vertex. This is
easier said than done; the notation necessary to make everything rigorous is a
mess of subscripts and hats. Thus we begin with a carefully worked example in
three variables, modifying the two-dimensional picture in Example 4.18 to the
one in Figure 9. The general case begins in the next section.

$a^{3}$$a^{2}b$$ab^{2}$$b^{3}$$a^{2}c$$b^{2}c$$ac^{2}$$bc^{2}$$c^{3}$ Figure
9. The polytopal complex supporting the minimal resolution of the pinched
power $\widehat{I}\subset k[a,b,c]$.

In this section, we use the same notation and proofs that we’ll use in
general, even though the two-dimensional picture allows everything to be
verified by inspection.

######  Notation 5.1.

We denote the star of $m=abc$ by $Y$. $Y$ is the cubical complex consisting of
the three boxes $Y_{1}=\Gamma_{Q_{1}}(m)=[a]\times[a,b]\times[b,c]$,
$Y_{2}=\Gamma_{Q_{2}}(m)=[b]\times[b,c]\times[c,a]$, and
$Y_{3}=\Gamma_{Q_{3}}(m)=[c]\times[c,a]\times[a,b]$. In Figure 10, $Y_{1}$ is
the red box, $Y_{2}$ is the yellow box, and $Y_{3}$ is the blue box.

The ideal corresponding to $Y$ is denoted by $J$, and the ideal corresponding
to each $Y_{i}$ is denoted by $J_{i}$.

$a^{2}b$$ab^{2}$$a^{2}c$$abc$$b^{2}c$$ac^{2}$$bc^{2}$ Figure 10. The star of
$abc$ is the polytopal complex $Y$ above, with three facets $Y_{1}$, $Y_{2}$,
and $Y_{3}$.

######  Remark 5.2.

Observe that $J$ is equal to
$\mathbf{m}_{[1,2]}\mathbf{m}_{[2,3]}\mathbf{m}_{[3,1]}$. (It has seven
monomial generators rather than eight because $m$ occurs in this product as
both $(a)(b)(c)$ and $(b)(c)(a)$.) Equivalently, it is the ideal generated by
all monomials which can be attained from $m$ in at most two elementary
($Q$-)Borel moves.

######  Proposition 5.3.

$Y$ supports a minimal polytopal resolution of $J$.

Our next step in studying $\widehat{I}$ is to remove $m$ from our generating
sets. In general, this is notationally accomplished by placing a hat on the
objects we have already studied.

######  Notation 5.4.

Let $\widehat{X_{1}}$, $\widehat{X_{2}}$, and $\widehat{X_{3}}$ be the
polytopal complexes obtained by removing all faces containing $m$ from
$X_{1}$, $X_{2}$, and $X_{3}$, respectively, and let $\widehat{I_{1}}$,
$\widehat{I_{2}}$, and $\widehat{I_{3}}$ be the ideals obtained by removing
$m$ from the lists of minimal generators for $I_{1}$, $I_{2}$, and $I_{3}$,
respectively.

######  Remark 5.5.

The $\widehat{X_{i}}$ are acyclic complexes, but are not pure, because the
squares containing $m$ are reduced in dimension rather than eliminated. Thus,
in Figure 9, $\widehat{X_{1}}$ is the union of the red triangle with the edge
connecting $ab^{2}$ to $a^{2}b$, $\widehat{X_{2}}$ is the union of the yellow
triangle with the edge connecting $bc^{2}$ to $b^{2}c$, and $\widehat{X_{3}}$
is the union of the blue triangle with the edge connecting $ca^{2}$ to
$c^{2}a$. On the other hand, each $\widehat{I_{i}}$ is a $Q_{i}$-Borel ideal,
and $\widehat{X_{i}}$ is the complex of boxes minimally resolving
$\widehat{I_{i}}$.

######  Notation 5.6.

We denote the grey hexagon in Figure 9 by $\widehat{Y}$. More carefully,
$\widehat{Y}$ is the convex hull of the vertices of $Y$. Equivalently,
$\widehat{Y}$ is obtained by treating $Y$ as a subset of $\mathbb{R}^{3}$
(forgetting its structure as a polytopal complex), then observing that it is a
hexagon, and treating this as a polytope.

Let $\widehat{J}$ be the ideal generated by the vertices of $\widehat{Y}$, or
equivalently by all the generators of $J$ except $m$.

######  Remark 5.7.

We realize $\widehat{Y}$ as a polytope inside the plane $x+y+z=3$. Its six
defining half-planes are $x\leq 2$, $x+y\leq 3$, $y\leq 2$, $y+z\leq 3$,
$z\leq 2$, $z+x\leq 3$. These arise from the sub-boxes of the $Y_{i}$ not
containing $m$. For example, $Y_{1}=[a]\times[a,b]\times[b,c]$ has two such
sub-boxes, namely $[a]\times[a]\times[b,c]$ and $[a]\times[a,b]\times[b]$.
These give us the hyperplanes $a=2$ and $a+b=3$, respectively.

Since $\widehat{Y}$ lives inside the plane $x+y+z=3$, there are many
equivalent ways to write these half-spaces. It may be easier to think of
$x\leq 2$ as $y+z\geq 1$, and of $x+y\leq 3$ as $z\geq 0$, and similarly for
their cyclic permutations.

######  Proposition 5.8.

$\widehat{Y}$ supports a minimal resolution of $\widehat{J}$.

The proof of Proposition 5.8 involves demonstrating that, for every
multidegree $\alpha$, the subcomplex $\widehat{Y}_{\leq\alpha}$ is acyclic.
Our strategy is to show that each $\widehat{Y}_{\leq\alpha}$ is homotopic to
$Y_{\alpha}$, which we know to be acyclic since $Y$ supports a minimal
resolution. For some multidegrees this is straightforward, but for others we
need to define the homotopy. We begin with a lemma.

######  Lemma 5.9.

Let $\alpha\neq m$ be a multidegree which is a multiple of $m$ but not of
$m^{2}$. Then $Y_{\leq\alpha}$ has a unique maximal face containing $m$, and
this face is a subcomplex of some $Y_{i}$. Furthermore, $i$ may be chosen so
that, for some $j$ and $k$, the deformation retract $\phi_{i,j,k}$ of $Y_{i}$
, defined by pushing along the vector from $x_{j}$ to $x_{k}$, induces a
homotopy from $Y_{\leq\alpha}$ to $\widehat{Y}_{\leq\alpha}$.

###### Proof.

We may assume that $\alpha$ divides the ${\rm lcm}$ of all generators. Thus,
up to cyclic permutation, there are only two such $\alpha$, namely $ab^{2}c$
and $ab^{2}c^{2}$.

If $\alpha=ab^{2}c$, then $Y_{\leq\alpha}$ consists of two edges, from $abc$
to $ab^{2}$ and from $ab^{2}$ to $b^{2}c$. Meanwhile,
$\widehat{Y}_{\leq\alpha}$ consists of a single edge from $ab^{2}$ to
$b^{2}c$. Everything is contained in the yellow box $Y_{2}$; we choose $i=2$,
$j=3$, and $k=2$ so that the homotopy is simply contracting the first edge to
its vertex at $ab^{2}$.

If $\alpha=ab^{2}c^{2}$, then $Y_{\leq\alpha}$ consists of the (yellow) two-
dimensional face $Y_{2}$ and the (blue) edge from $ac^{2}$ to $bc^{2}$.
Meanwhile, $\widehat{Y}_{\leq\alpha}$ consists of three edges: the (blue) edge
from $ac^{2}$ to $bc^{2}$ and the two (yellow) edges from $bc^{2}$ to $b^{2}c$
and from $b^{2}c$ to $ab^{2}$. We want to collapse the yellow box to these two
edges while leaving the blue edge alone, so we choose $i=2$, $j=1$, and $k=2$.
∎

###### Proof of Proposition 5.8.

Let $\alpha$ be an arbitary multidegree; we will show that
$\widehat{Y}_{\leq\alpha}$ is acyclic.

Observe first that every cell of $Y$, except for those containing $m$, is
identical to a cell of $\widehat{Y}$; conversely, every cell of $\widehat{Y}$,
except for the hexagon itself, is identical to a cell of $Y$.

If $\alpha$ is not a multiple of $m$, none of these problematic cells appear,
so $\widehat{Y}_{\leq\alpha}=Y_{\leq\alpha}$ is acyclic.

If $\alpha$ is a multiple of $m^{2}$, all of these problematic cells appear,
so $\widehat{Y}_{\leq\alpha}$ and $Y_{\leq\alpha}$ are equal when viewed as
topological spaces, and in particular $\widehat{Y}_{\leq\alpha}$ is acyclic.

If $\alpha=m$, then $\widehat{Y}_{\leq\alpha}$ is the empty complex.

Otherwise, Lemma 5.9 applies, and we conclude that $\widehat{Y}_{\leq\alpha}$
is homotopic to $Y_{\leq\alpha}$ and in particular is acyclic. ∎

We are now ready to prove that the picture in Figure 9 supports a minimal
polytopal resolution of the pinched power.

######  Notation 5.10.

Denote by $\widehat{X}$ the union of the four polytopal complexes
$\widehat{X}_{1}$, $\widehat{X}_{2}$, $\widehat{X}_{3}$, and $\widehat{Y}$,
and recall that $\widehat{I}$ is the ideal generated by all degree 3 monomials
except $m=abc$.

######  Theorem 5.11.

With notation as above, $\widehat{X}$ supports a minimal resolution of
$\widehat{I}$.

###### Proof.

We will show that, for all multidegrees $\alpha$, the induced subcomplex
$\widehat{X}_{\leq\alpha}$ is acyclic. As always, it suffices to check
multidegrees $\alpha$ dividing $m^{3}$, the least common multiple of the
generators of $\widehat{I}$.

As in the proof of Proposition 5.8, if $\alpha$ does not divide $m$ or is a
multiple of $m^{2}$, then $\widehat{X}_{\leq\alpha}=X_{\leq\alpha}$ as
topological spaces, and all of these are acyclic since $X$ resolves $I$.

Thus, we may assume that $\alpha$ divides $m$ but not $m^{2}$. Up to cyclic
permutation, there are seven remaining multidegrees. We compute the
corresponding $\widehat{X}_{\leq\alpha}$ in the following table.

$\mathbf{\alpha}$ | $\mathbf{\widehat{X}_{\leq\alpha}}$ | $\mathbf{\alpha}$ | $\mathbf{\widehat{X}_{\leq\alpha}}$
---|---|---|---
$abc$ | (empty) |  |
$a^{2}bc$ | $a^{2}b$$a^{2}c$ | $a^{2}b^{2}c$ | $a^{2}b$$ab^{2}$$a^{2}c$$b^{2}c$
$a^{3}bc$ | $a^{3}$$a^{2}b$$a^{2}c$ | $a^{3}b^{2}c$ | $a^{3}$$a^{2}b$$a^{2}c$$ab^{2}$$b^{2}c$
$a^{3}bc^{2}$ | $a^{3}$$a^{2}b$$a^{2}c$$ac^{2}$$bc^{2}$$b^{2}c$ | $a^{3}b^{3}c$ | $a^{3}$$a^{2}b$$ab^{2}$$b^{3}$$a^{2}c$$b^{2}c$

Since each $\widehat{X}_{\leq\alpha}$ is acyclic, we conclude that
$\widehat{X}$ does in fact resolve $\widehat{I}$. ∎

##  6\. A minimal resolution of the ideal obtained by removing $x_{1}\cdots
x_{n}$ from the generators of $(x_{1},\cdots,x_{n})^{n}$.

######  Notation 6.1.

We generalize all notation from the previous section and denote

  * •

$\widehat{I}$ the ideal obtained by removing $m$ from the generators of
$I=(x_{1},\cdots,x_{n})^{n}$.

  * •

$J=\mathbf{m}_{[1,2]}\mathbf{m}_{[2,3]}\dots\mathbf{m}_{[n-1,n]}\mathbf{m}_{[n,1]}$,
the ideal supported by $Y$.

  * •

$\widehat{J}$ the ideal obtained by removing $m$ from the generators of $J$.

  * •

$\widehat{X_{i}}$ the complex obtained by removing all faces containing $m$
from $X_{i}$.

  * •

$Y_{i}=\Gamma_{i}(m)$ and $Y=\bigcup Y_{i}$, the collection of boxes
containing $m$ inside $X$.

  * •

$\widehat{Y}$ the polytope obtained by forgetting the interior structure
(i.e., the faces containing $m$) of $Y$. That is, $\widehat{Y}$ has a single
$(n-1)$-dimensional face, which is the (interior of the) union of the $Y_{i}$;
for $d<n-1$ its $d$-dimensional faces are the $d$-dimensional faces of the
$Y_{i}$ which don’t contain $m$.

  * •

$\widehat{I_{i}}$ the ideal obtained by removing $m$ from the generators of
$I_{i}$. Then $\widehat{I_{i}}$ is a $Q_{i}$-Borel ideal and $\widehat{X_{i}}$
is the complex-of-boxes resolving $\widehat{I_{i}}.$

  * •

$\widehat{J_{i}}$ the ideal obtained by removing $m$ from the generators of
$J_{i}$, and $\widehat{Y_{i}}$ the complex obtained by deleting all faces of
$Y_{i}$ containing $m$.

  * •

$\widehat{X}=\widehat{Y}\cup\bigcup_{i}\widehat{X_{i}}$, the complex obtained
from $X$ by replacing $Y$ with $\widehat{Y}$.

######  Remark 6.2.

  1. (1)

For all $i$, $Y\cap\widehat{X}_{i}=\widehat{Y}_{i}$ and
$J\cap\widehat{I}_{i}=\widehat{J}_{i}$. Since $Y_{i}$ is a box, it follows
from Lemma 6.8 that $\widehat{Y_{i}}$ supports the minimal resolution of
$\widehat{J_{i}}$.

  2. (2)

The defining half-spaces of $\widehat{Y}$ are the cyclic permutations of
$\\{x_{1}+x_{2}+\dots+x_{j}\geq j-1\\}$ for $j\in\\{1,\dots,n-1\\}$. Because
$\widehat{Y}$ is contained in the hyperplane $x_{1}+\dots+x_{n}=n$, this is
equivalent to the cyclic permutations of the half-spaces
$\\{x_{1}+x_{2}+\dots+x_{j}\leq j+1\\}$

Observe that all $Y_{i}=\Gamma_{i}(m)$ and $J_{i}$ satisfy the hypotheses of
Lemma 4.4, so their union resolves $J$. More generally, we have:

######  Theorem 6.3.

The complex $Y_{i_{1}}\cup Y_{i_{2}}\cup\cdots\cup Y_{i_{\ell}}$ resolves
$J_{i_{1}}+J_{i_{2}}+\cdots+J_{i_{\ell}}$ minimally.

######  Corollary 6.4.

The complex $Y$ supports the minimal resolution of $J$.

######  Example 6.5.

The complex in Figure 11 supports a minimal linear resolution of
$J=\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}abcd},a^{2}bc,abd^{2},a^{2}bd,ac^{2}d,a^{2}c^{2},acd^{2},a^{2}cd,b^{2}cd,ab^{2}c,b^{2}d^{2},ab^{2}d,bc^{2}d,abc^{2},bcd^{2}\right)$.

$abc^{2}$$b^{2}d^{2}$$acd^{2}$$ac^{2}d$$abd^{2}$$b^{2}cd$$bc^{2}d$$bcd^{2}$$ab^{2}d$$a^{2}bd$$a^{2}cd$
$a^{2}c^{2}$$a^{2}bc$$ab^{2}c$ Figure 11. A minimal resolution of
$J=(a,b)(b,c)(c,d)(d,a)$, inside $I=(a,b,c,d)^{4}$. The red vertex in the
center represents the monomial $abcd$. The polytopal complex $Y$ consists of
four cubes sharing this vertex; the diametrically opposite vertices are,
respectively, $a^{2}bc$, $b^{2}cd$, $c^{2}da$, and $d^{2}ab$.

######  Remark 6.6.

The complex $Y$ is combinatorially equivalent to the unit “corner” of the
first orthant of $\mathbb{R}^{n}$, realized as the union of the $n$ unit
$(n-1)$-hypercubes in the mutually orthogonal hyperplanes $x_{i}=0$. (Because
all monomials of interest have degree $n$, we can obtain $Y$ by projection
onto the probability hyperplane $\sum x_{i}=n$.) In the language of
[Bayer1996duality], this is a corner whose probe cube consists of a single
vertex.

We find it a useful heuristic to think of $Y$ as a subcomplex of the
_inadmissible_ box
$[x_{n},x_{1}]\times[x_{1},x_{2}]\times[x_{2},x_{3}]\times\dots\times[x_{n-1},x_{n}]$.
From this perspective, faces of $Y$ are products $F=F_{1}\times\dots\times
F_{n}$ with each $F_{j}$ a nonempty subset of $\\{x_{j-1},x_{j}\\}$; in this
notation $F\subset Y_{i}$ if and only if $F_{i}=\\{x_{i}\\}$ so we must
further require that all faces satisfy this equality for some $i$.

With this notation, the dimension of a face $F$ is the number of $F_{i}$ with
cardinality $2$, and $F$ contains $m$ if and only if $F_{i}$ contains $x_{i}$
for all $i$. This allows us to compute the $f$-vectors of $Y$ and of
$\widehat{Y}$: For $d<n$, $Y$ contains $\binom{n}{d}(2^{n-d}-1)$ faces of
dimension $d$ (since the $n-d$ singleton factors each have two choices, but
cannot all choose $\\{x_{i-1}\\}$). Of these, only $\binom{n}{d}$ contain $m$
(since all singletons must choose $\\{x_{i}\\}$). Thus $\widehat{Y}$ contains
$\binom{n}{d}(2^{n-d}-2)$ faces of dimension $d$.

######  Lemma 6.7.

The complex $\widehat{Y}$ supports a minimal resolution of $\widehat{J}$.

We will prove Lemma 6.7 by showing that $\widehat{Y}_{\leq\alpha}$ is acyclic
for all multidegrees $\alpha$. The argument in general is to show that
$\widehat{Y}_{\leq\alpha}$ is homotopic to $Y_{\alpha}$. For the difficult
multidegrees, we rely on the following lemma of Nagel and Reiner
[NagelReiner], rather than defining the homotopy ourselves.

######  Lemma 6.8.

[NagelReiner, Lemma 6.4] Let $\mathcal{C}$ be a polytopal complex and $v$ a
vertex in $\mathcal{C}$ that lies in a uinique facet $P$, and assume that $P$
has strictly positive dimension.

Then the vertex-induced subcomplex $\mathcal{C}\smallsetminus\\{v\\}$,
obtained by deleting $v$ and all faces that contain it, is homotopy equivalent
to $\mathcal{C}$.

###### Proof of Lemma 6.7.

Let $\alpha=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}$ be a monomial. We
will show that $\widehat{Y}_{\leq\alpha}$ is acyclic.

We first observe that if $m=x_{1}\cdots x_{n}$ does not divide $\alpha$, then
$\widehat{Y}_{\leq\alpha}=Y_{\leq\alpha}$ which is acyclic because $Y$
supports a minimal resolution by Corollary 6.4.

Next observe that, if $\alpha$ is a multiple of $m^{2}$, the least common
multiple of the generators of $J$, then
$\widehat{Y}_{\leq\alpha}=\widehat{Y}$, which is homeomorphic to $Y$. Thus,
since $Y=Y_{\leq\alpha}$ and $Y$ supports a minimal resolution by Corollary
6.4, we have that $\widehat{Y}_{\leq\alpha}$ is acyclic.

Finally, assume that $\alpha$ is divisible by $m$ but not by $m^{2}$. We claim
that $m$ is contained in a unique facet of $Y_{\leq\alpha}$. Indeed, let
$\mathcal{P}_{m,\alpha}$ be the box

$\mathcal{P}_{m,\alpha}=\left(\prod_{x_{i}^{2}\text{ divides
}\alpha}\\{x_{i},x_{i+1}\\}\right)\times\left(\prod_{x_{i}^{2}\text{ does not
divide }\alpha}\\{x_{i+1}\\}\right).$

We claim that $\mathcal{P}_{m,\alpha}$ is the unique facet of ${Y}_{\alpha}$
containing $m$. First, observe that $m\in\mathcal{P}_{m,\alpha}$ since one can
choose $x_{i+1}$ from every interval in the product. Next, observe that the
multidegree of $\mathcal{P}_{m,\alpha}$ is $\gcd(\alpha,m^{2})$, which is
divisible by $\alpha$. Thus $\mathcal{P}_{m,\alpha}$ is a face of
${Y}_{\leq\alpha}$.

It remains to show that every face of $Y_{\leq\alpha}$ containing $m$ is
contained in $\mathcal{P}_{m,\alpha}$. But every such face must have the form

$\left(\prod_{i\in A}\\{x_{i},x_{i+1}\\}\right)\times\left(\prod_{i\in
B}\\{x_{i+1}\\}\right).$

for some sets $A$ and $B$ satisfying $A\sqcup B=\\{1,\dots,n\\}$. If
$A\subset\\{i:x_{i}^{2}\text{ divides }\alpha\\}$, then
$F\subset\mathcal{P}_{m,\alpha}$ as desired. If not, then $F\not\in
Y_{\leq\alpha}$.

If $\alpha\neq m$, then $\mathcal{P}_{m,\alpha}$ has positive dimension and
satisfies the hypotheses of Lemma 6.8, so $\widehat{Y}_{\leq\alpha}$ is
acyclic. If $\alpha=m$, then $\widehat{Y}_{\leq\alpha}$ is the empty complex,
which is acyclic.

∎

######  Example 6.9.

The complex in Figure 12 supports a minimal resolution of
$\widehat{J}$ inside $k[a,b,c,d]$.

$abc^{2}$$b^{2}d^{2}$$acd^{2}$$ac^{2}d$$abd^{2}$$b^{2}cd$$bc^{2}d$$bcd^{2}$$ab^{2}d$$a^{2}bd$$a^{2}cd$
$a^{2}c^{2}$$a^{2}bc$$ab^{2}c$ Figure 12. A minimal resolution of
$\widehat{J}=J\smallsetminus\\{abcd\\}$, inside $I=(a,b,c,d)^{4}$.

We are now ready to prove that $\widehat{I}$ is minimally resolved by
$\widehat{X}$. The proof is the same as the proof of Lemma 6.7.

######  Theorem 6.10.

The pinched power ideal $\widehat{I}$ has polytopal minimal resolution
supported on the complex $\widehat{X}$.

###### Proof.

We verify that $\widehat{X}_{\leq\alpha}$ is acyclic. As in the proof of Lemma
6.7, we show that in every case $\widehat{X}_{\leq\alpha}$ is homeomorphic to
$X_{\leq\alpha}$.

If $\alpha$ is not divisible by $m$, then
$\widehat{X}_{\leq\alpha}=X_{\leq\alpha}$. If $\alpha$ is divisible by
$m^{2}$, then $\widehat{X}_{\leq\alpha}$ contains $\widehat{Y}$ and
$X_{\leq\alpha}$ contains $Y$, so the two complexes are homeomorphic.
Otherwise, as in the proof of Lemma 6.7, $m$ is contained in a unique facet of
$Y_{\leq\alpha}$. This facet is also the unique facet of $X_{\leq\alpha}$
containing $m$, so by Lemma 6.8, $X_{\leq\alpha}$ is homotopic to
$\widehat{X}_{\leq\alpha}$. ∎

######  Example 6.11.

The complex $\widehat{X}$, obtained by gluing the five complexes in Figure 13,
supports a minimal resolution of $\widehat{I}$ inside $I=(a,b,c,d)^{4}$.

$a^{3}d$$a^{2}bd$$ab^{2}d$$a^{2}cd$$acd^{2}$$ad^{3}$$a^{2}d^{2}$$abd^{2}$$b^{2}d^{2}$$d^{4}$$bd^{3}$$cd^{3}$$bcd^{2}$

$a^{4}$$a^{3}b$$a^{2}b^{2}$$ab^{3}$$ab^{2}c$$a^{2}bc$$a^{3}c$$a^{3}d$$a^{2}bd$$ab^{2}d$$a^{2}cd$$abc^{2}$$a^{2}c^{2}$

$abc^{2}$$b^{2}d^{2}$$acd^{2}$$ac^{2}d$$abd^{2}$$b^{2}cd$$bc^{2}d$$bcd^{2}$$ab^{2}d$$a^{2}bd$$a^{2}cd$
$a^{2}c^{2}$$a^{2}bc$$ab^{2}c$

$a^{2}cd$$abc^{2}$$a^{2}c^{2}$$ac^{3}$$ac^{2}d$$acd^{2}$$c^{2}d^{2}$$c^{4}$$bc^{3}$$cd^{3}$$c^{3}d$$bcd^{2}$$bc^{2}d$

$ab^{3}$$ab^{2}c$$ab^{2}d$$abc^{2}$$b^{4}$$b^{2}d^{2}$$b^{2}c^{2}$$b^{3}d$$b^{3}c$$bc^{3}$$bcd^{2}$$b^{2}cd$$bc^{2}d$

Figure 13. A minimal resolution of $\widehat{I}$ in $k[a,b,c,d]$.

##  7\. The Betti numbers

In this section, we represent the Betti numbers of $\widehat{I}$ in terms of
the Betti numbers of $I$. From the construction in Theorem 6.10, the
difference between the minimal resolutions of $I$ and $\widehat{I}$ is the
same as the difference between the resolutions of $J$ and $\widehat{J}$,
namely the faces containing $m$ in $Y$ (which are collapsed to a single face
in $\widehat{Y}$). We first compute the number of boxes of dimension $i$
containing $m=x_{1}\cdots x_{n}$ in $Y$.

######  Lemma 7.1.

The number of boxes of dimension $p$ containing $x_{1}\cdots x_{n}$ in $Y$ is
$\gamma_{p}=\displaystyle\binom{n}{p}$ for $0\leq p\leq n-1$. (Otherwise,
$\gamma_{p}=0$.)

###### Proof.

Any box of dimension $0\leq p\leq n-1$ containing $x_{1}\cdots x_{n}$ in $Y$
is of the form

$F_{1}\times\dots\times F_{n}$

where there are exactly $p$ factors of the form $F_{j}=\\{x_{j},x_{j+1}\\}$
and $(n-p)$ factors of the form $F_{j}=\\{x_{j+1}\\}$. Therefore,
$\gamma_{p}=\displaystyle\binom{n}{p}$, for $0\leq p\leq n-1$.

Since $Y$ has dimension $n-1$, it does not contain any boxes of higher
dimension. ∎

Now we are able to compute the Betti numbers of $\widehat{I}$, $J$, and
$\widehat{J}$.

######  Proposition 7.2.

The graded Betti numbers of $J$ are

$\displaystyle\beta_{p,p+n}(J)=\sum\limits_{\ell=1}^{n}(-1)^{\ell+1}2^{n-\ell-p}\binom{n}{\ell}\binom{n-\ell}{p},$
$\displaystyle\text{if }0\leq p\leq n-1$ $\displaystyle\beta_{p,q}(J)=0,$
$\displaystyle\text{otherwise }.$

###### Proof.

Following Theorem 4.14 (3) and Remark 6.6, set $M_{j}=\\{x_{j-1},x_{j}\\}$ for
all $j$ (so $M_{1}=\\{x_{n},x_{1}\\}$). Then every box in $Y$ has the form

$\displaystyle B=L_{1}\times L_{2}\times\cdots\times L_{n},$

where each $L_{j}$ is a nonempty subset of $M_{j}$.

With this notation, $B\in Y_{i}$ if and only if $L_{i}=\\{x_{i}\\}$. For
example, boxes in $Y_{1}$ have the form $\\{x_{1}\\}\times L_{2}\times
L_{3}\times\cdots\times L_{n}$.

A box has dimension $p$ if $p$ of the $L_{j}$ are equal to $M_{j}$. Thus
$Y_{1}$ contains $\displaystyle\binom{n-1}{p}2^{n-1-p}$ boxes of dimension
$p$, since there are $\displaystyle\binom{n-1}{p}$ choices for which $L_{j}$
to set equal to $M_{j}$ and two choices for each of the singletons except
$L_{1}$ (which must equal $\\{x_{1}\\}$).

For an $\ell-$tuple $\left(i_{1},\cdots,i_{\ell}\right)$, there are
$\displaystyle\binom{n-\ell}{p}2^{n-p-\ell}$ boxes of dimension $p$ in the
intersection $Y_{i_{1}}\cap\cdots\cap Y_{i_{\ell}}$, since every such box must
have $L_{i_{t}}=M_{i_{t}}$ for all $t$, $p$ of the $(n-\ell)$ remaining
$L_{j}$ must equal $M_{j}$, and there are two choices for each of the
surviving $(n-p-\ell)$ singletons. The formulas are now obtained by the
inclusion–exclusion principle. ∎

It follows from the construction of $\widehat{Y}$ that

$\displaystyle\beta_{p,p+n}(\widehat{J})=\beta_{p,p+n}(J)-\gamma_{p},$
$\displaystyle\beta_{n-1,2n}(\widehat{J})=1.$

Then we obtain the following corollary.

######  Corollary 7.3.

The Betti numbers of $\widehat{J}$ are

$\displaystyle\beta_{n-1,2n}(\widehat{J})=1,$
$\displaystyle\beta_{p,p+n}(J)=\left(\sum\limits_{\ell=1}^{n}(-1)^{\ell+1}2^{n-\ell-p}\binom{n}{\ell}\binom{n-\ell}{p}\right)-\binom{n}{p},$
$\displaystyle\text{if }0\leq p\leq n-1$
$\displaystyle\beta_{p,q}(\widehat{J})=\beta_{p,q}(J)=0,$ otherwise.

######  Theorem 7.4 (Folklore).

The Betti numbers of $I$ are

$\displaystyle\beta_{p,p+n}=\sum\limits_{i=p+1}^{n}\binom{i-1}{p}\binom{n+i-2}{n-1},$
$\displaystyle\beta_{p,q}=0,$ $\displaystyle\text{otherwise}.$

######  Corollary 7.5.

The graded Betti numbers of $\hat{I}$ are

$\displaystyle\beta_{n-1,2n}\left(\hat{I}\right)=1,$
$\displaystyle\beta_{p,p+n}\left(\hat{I}\right)=\beta_{p,p+n}\left(I\right)-\binom{n}{p},$
$\displaystyle\text{if }0\leq p\leq n-1,$
$\displaystyle\beta_{p,q}\left(\hat{I}\right)=\beta_{p,q}\left(I\right)=0,$
$\displaystyle\text{otherwise}.$

##  8\. Further questions

We have chosen in this paper to study the deletion of the central monomial
$m=x_{1}\dots x_{n}$ from the $n^{\text{th}}$ power ideal
$(x_{1},\dots,x_{n})^{n}$ even though the techniques appear to apply to the
deletion of any monomial from any power of the maximal ideal, primarily
because the notation is extremely technical even in this special case.
Nevertheless, questions about other deletions from the power arise very
naturally.

######  Question 8.1.

Let $d=d_{1}+\dots+d_{n}$. With the same techniques as in section 6 of this
paper, can we construct a polytopal complex supporting a minimal resolution of
$\widehat{I}$ obtained by removing $m=x_{1}^{d_{1}}\cdots x_{n}^{d_{n}}$ from
the generators of $I=(x_{1},\cdots,x_{n})^{d}$?

There does not appear to be a natural obstruction to deleting a generator from
the "interior" of the power. For example, the complex in Figure 14 supports
the resolution of $(a,b,c)^{4}\smallsetminus\\{abc^{2}\\}$.

$a^{4}$$a^{3}b$$a^{2}b^{2}$$ab^{3}$$b^{4}$$a^{3}c$$a^{2}bc$$ab^{2}c$$b^{3}c$$a^{2}c^{2}$$b^{2}c^{2}$$ac^{3}$$bc^{3}$$c^{4}$
Figure 14. A minimal resolution of $(a,b,c)^{4}\smallsetminus\\{abc^{2}\\}$.

On the other hand, if the removed generator has a missing variable in its
support, the (analog of) the grey hexagon changes and loses some of its
symmetry. For example, the complexes in Figure 15 support the resolutions of
$(a,b,c)^{4}\smallsetminus\\{a^{2}c^{2}\\}$ and
$(a,b,c,d)^{3}\smallsetminus\\{bcd\\}$.

The forthcoming paper [Dao] deals with the necessary notational issues and
answers Question 8.1 in the positive.

$a^{4}$$a^{3}b$$a^{2}b^{2}$$ab^{3}$$b^{4}$$a^{3}c$$a^{2}bc$$ab^{2}c$$b^{3}c$$abc^{2}$$b^{2}c^{2}$$ac^{3}$$bc^{3}$$c^{4}$

$a^{3}$$a^{2}b$$ab^{2}$$b^{3}$$b^{2}c$$abc$$a^{2}c$$a^{2}d$$abd$$b^{2}d$$acd$$bc^{2}$$ac^{2}$$c^{3}$$c^{2}d$$cd^{2}$$d^{3}$$ad^{2}$$bd^{2}$

Figure 15. The complexes support the resolutions of
$(a,b,c)^{4}\smallsetminus\\{a^{2}c^{2}\\}$ and
$(a,b,c,d)^{3}\smallsetminus\\{bcd\\}$ .

######  Question 8.2.

To what extent can the techniques of this paper be applied to an ideal
obtained by removing a generator from the minimal generators of a Borel ideal?

At first blush this seems like an obvious extension, but there are two issues
to consider. First, the boundary of the box complex resolving a Borel ideal is
complicated (whereas for a power of the maximal ideal it’s simply a simplex),
so if the removed generator is almost a Borel generator of the ideal, the
(analog of) the grey hexagon might extend outside the original box complex.
Second, the Borel ideal is not usually $Q_{i}$-Borel for all $Q_{i}$, so the
shapes on the other sides of $\widehat{Y}$ may not be tiled by boxes of the
appropriate orientation.

######  Question 8.3.

To what extent can the techniques of this paper handle the deletion of
multiple generators?

It is known by a result of Velasco [Velasco] that monomial ideals exist whose
minimal resolutions cannot be supported on any CW complex. Consequently, our
techniques, which produce polytopal complexes, cannot be expected to hold up
in arbitrary generality.

It is clear that we cannot delete generators arbitrarily, since the complexes
$\widehat{J}$ corresponding to the removed generators may overlap or the
orientations of the surviving boxes may not be consistent, causing the boxes
to disagree outside of the gray regions as in Figures 16 and 17.

$a^{5}$$a^{4}b$$a^{3}b^{2}$$a^{2}b^{3}$$ab^{4}$$a^{4}c$$b^{4}c$$a^{2}b^{2}c$$ab^{3}c$$a^{3}c^{2}$$b^{3}c^{2}$$a^{2}bc^{2}$$ab^{2}c^{2}$$a^{2}c^{3}$$b^{2}c^{3}$$abc^{3}$$ac^{4}$$bc^{4}$$c^{5}$$c^{5}$

$a^{5}$$a^{4}b$$a^{3}b^{2}$$a^{2}b^{3}$$ab^{4}$$a^{4}c$$b^{4}c$$a^{3}bc$$a^{2}b^{2}c$$ab^{3}c$$a^{3}c^{2}$$b^{3}c^{2}$$a^{2}bc^{2}$$ab^{2}c^{2}$$a^{2}c^{3}$$b^{2}c^{3}$$ac^{4}$$bc^{4}$$c^{5}$$c^{5}$

Figure 16. Minimal resolutions of $(a,b,c)^{5}\setminus\\{a^{3}bc\\}$ and
$(a,b,c)^{5}\setminus\\{abc^{3}\\}$.

However, if the deleted monomials are sufficiently sparse, we can usually fix
the orientation issue by ad-hoc methods, at least in three variables (as in
Figure 17). This becomes equivalent to a tiling problem: After deleting copies
of $\hat{J}$ around each of the removed vertices, can the rest of the simplex
be tiled with admissible boxes? (In three variables, this means the legal
tiles are diamonds of all orientations and upward-pointing triangles. With
more variables, the menagerie of available shapes is considerably larger.) The
pictures in [DochtermannEtAl], using tropical geometry, make us hopeful that
the problem can be solved in some generality.

In [DE], Dao and Eisenbud classify the equigenerated monomial ideals with
almost linear resolution. This classification suggests that a similar
construction to ours, but using larger versions of the grey hexagons, is
likely to resolve many such ideals.

The forthcoming paper [Dao] addresses this question and provides some partial
results.

$a^{5}$$a^{4}b$$a^{3}b^{2}$$a^{2}b^{3}$$ab^{4}$$a^{4}c$$b^{4}c$$a^{2}b^{2}c$$ab^{3}c$$a^{3}c^{2}$$b^{3}c^{2}$$a^{2}bc^{2}$$ab^{2}c^{2}$$a^{2}c^{3}$$b^{2}c^{3}$$ac^{4}$$bc^{4}$$c^{5}$$c^{5}$

$a^{5}$$a^{4}b$$a^{3}b^{2}$$a^{2}b^{3}$$ab^{4}$$a^{4}c$$b^{4}c$$a^{2}b^{2}c$$ab^{3}c$$a^{3}c^{2}$$b^{3}c^{2}$$a^{2}bc^{2}$$ab^{2}c^{2}$$a^{2}c^{3}$$b^{2}c^{3}$$ac^{4}$$bc^{4}$$c^{5}$$c^{5}$

Figure 17. The triangle on the left suggests that we cannot resolve
$(a,b,c)^{5}\smallsetminus\\{a^{3}bc,abc^{3}\\}$ by simply merging the two
polytopal complexes in Figure 16. The orange and purple regions show where
those complexes disagree about the orientation of the boxes. It is not clear
_a priori_ how to choose between the orientations, or even that it is possible
to tile these regions with boxes of any orientation. On the other hand, the
triangle on the right shows one of several tilings supporting the minimal
resolution of this ideal.

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