%------------------------------------------------------------------
%compile using AMS-LaTeX
\documentclass{amsart}
\usepackage{amssymb,amscd,amsthm,verbatim}
% "plain" theorem style; all of these share the same counter
\newtheorem{proposition}{Proposition}
\newtheorem{theorem}[proposition]{Theorem}
\newtheorem{corollary}[proposition]{Corollary}
\newtheorem{lemma}[proposition]{Lemma}
\newtheorem{conjecture}[proposition]{Conjecture}
\newtheorem{question}[proposition]{Question}
% "definition" theorem style, a bit less dramatic than the "plain" style.
% These share the same counter as the other theorem environments.
\theoremstyle{definition}
\newtheorem{defn}[proposition]{Definition}
\newtheorem{ex}{Example}
\newtheorem{rem}[proposition]{Remark}
\numberwithin{equation}{section}
\begin{document}
\newcommand{\field}{\mathbb F}
\newcommand{\reals}{\mathbb R}
\newcommand{\A}{\mathcal A}
\newcommand{\N}{\mathbb N}
\newcommand{\dimens}{\mathrm{dim}}
\newcommand{\degr}{\mathrm{deg}}
\newcommand{\reg}{\mathrm{reg}}
\newcommand{\del}{\mathrm{del}}
\newcommand{\indeg}{\mathrm{indeg}}
\newcommand{\supp}{\mathrm{supp}}
\newcommand{\depth}{\mathrm{depth}}
\newcommand{\Poin}{\mathrm{Poin}}
\newcommand{\redhom}{\tilde{H}}
\newcommand{\naturals}{\mathbb N}
\newcommand{\link}{\mathrm{link}}
\newcommand{\str}{\mathrm{star}}
\newcommand{\Tor}{\mathrm{Tor}}
\newcommand{\mm}{\mathfrak{m}}
\newcommand{\K}{\mathbb K}
\newcommand{\dlin}{d^{\sf linear}}
\newcommand{\binomial}[2]{\left( \begin{matrix} #1 \\ #2 \end{matrix} \right)}
\title[On the linear syzygies of a Stanley-Reisner ideal]
{Linear syzygies of Stanley-Reisner ideals}
\author[V. Reiner]{V. Reiner}
\email{reiner@math.umn.edu}
\address{School of Mathematics\\
University of Minnesota\\
Minneapolis, MN 55455, USA}
\author[V. Welker]{V. Welker}
\email{welker@math.tu-berlin.de}
\address{Fachbereich Mathematik, MA 7--1 \\
Technische Universit\"at Berlin \\
10623 Berlin. Germany}
\keywords{minimal free resolution, linear strand, Stanley-Reisner
ring, square-free monomial ideal}
\thanks{First author supported by
Sloan Foundation and Univ. of Minnesota McKnight-Land Grant Fellowships.
Second author supported by Deutsche Forschungsgemeinschaft (DFG)}
\begin{abstract}
We give an elementary description of the maps in the linear strand
of the minimal free resolution of a square-free monomial ideal, that is,
the Stanley-Reisner ideal associated to a
simplicial complex $\Delta$. The description is in terms of
the homology of the canonical Alexander dual complex $\Delta^*$.
As applications we are able to
\item[$\bullet$] prove for monomial ideals and $j=1$ a conjecture of J. Herzog
giving lower bounds on the number of $i$-syzygies
in the linear strand of $j^{th}$-syzygy modules.
\item[$\bullet$] show that the maps in the linear strand can be
written using only $\pm 1$ coefficients if $\Delta^*$ is a pseudomanifold,
\item[$\bullet$] exhibit an example where multigraded maps in the linear
strand cannot be written using only $\pm 1$ coefficients.
\item[$\bullet$] compute the entire resolution explicitly
when $\Delta^*$ is the complex of independent sets of a matroid.
\end{abstract}
\maketitle
\section{Introduction}
The goal of this paper is to describe in a simple topological fashion
the maps in the linear strand of the minimal free resolution of a
Stanley-Reisner ideal. It is an outgrowth of two recent trends in the theory
of minimal free resolutions. The first is a series of results
\cite{AramovaHerzog1, AramovaHerzog2, BayerPeevaSturmfels,
BayerSturmfels, Eagon, PeevaSturmfels} giving explicit
descriptions of the maps in the minimal free resolutions for
various classes of ideals. The second is the realization
that when dealing with resolutions of Stanley-Reisner ideals $I_\Delta$
associated to a simplicial complex $\Delta$, it can be easier to
work with the {\it canonical Alexander dual} $\Delta^*$
(see \cite{BayerCharalambousPopescu, EagonReiner, HerzogReinerWelker, Terai}).
Our description of the linear strand for the resolution of $I_\Delta$
will be in terms of natural maps on the homology of links of faces
in $\Delta^*$.
We note that our description of the linear strand is in some
sense not new, as it may be derived with a little work from
known results \cite{AramovaHerzog1, AramovaHerzog2, Eagon, EisenbudGoto,
HerzogSimisVasconcelos}. However, the exact form
in which we describe the linear strand seems not to appear elsewhere,
and we give an elementary proof of its correctness here.
This form is extremely useful for certain applications. In
particular, in Section \ref{EGH-conjecture}
we use it to prove for monomial ideals
a special case of a conjecture of J. Herzog \cite{Herzog},
asserting that when the linear strand of a homogeneous
$j^{th}$-syzygy module contains
$p$-syzygies for some $p>0$ then it must contain at least
$\binomial{p+j}{i+j}$ $i$-syzygies for each $i < p$.
In Section \ref{pseudomanifolds} we show that the maps
in the linear strand can be written using only $\pm 1$ coefficients
whenever $\Delta^*$ is a pseudomanifold, and exhibit a small example
of a non-pseudomanifold where this fails. Section \ref{matroids}
gives the entire resolution explicitly when $\Delta^*$
is the complex of independent sets of a matroid.
\section{Minimal free resolutions and notation}
We first review minimal free resolutions and their linear strand.
Let $A=\field[x_1,\ldots,x_n]$ be a polynomial ring over a field $\field$,
and $I \subset A$ an ideal which is homogeneous with
respect to the standard grading setting $\degr(x_i)=1$. Regarding $I$ as an
$A$-module, it has a finite minimal free resolution
\begin{equation}
\label{resolution}
0 \longrightarrow \bigoplus_{m} A(-m)^{\beta_{p,m}} \quad
{\overset{d_p}{\longrightarrow}} \quad \cdots \quad
%{\overset{d_2}{\rightarrow}} \quad
%\bigoplus_{m} A(-m)^{\beta_{1,m}} \quad
{\overset{d_1}{\longrightarrow}}
\bigoplus_{m} A(-m)^{\beta_{0,m}}
{\overset{d_0}{\longrightarrow}} \,\, I \,\,
\rightarrow 0
\end{equation}
in which the notation $A(-m)$ denotes a free $A$-module with
basis element in degree $m$, with the $m$'s chosen
to make the maps homogeneous.
%, and $\sum_m \beta_{p,m} > 0$.
It is well-known that the number
$\beta_{i,m}$ is the dimension of the $m^{th}$ graded piece $\Tor_i^A(I,\field)_m$
of the graded $\field$-vector space $\Tor_i^A(I,\field)$.
For an ideal $I$ let $t(I)$ be the
minimal degree of a homogeneous generator of $I$. Note that
$\Tor_i^A(I,\field)_m$ vanishes unless $\degr(m) \geq i+t(I)$.
{From} grading considerations, the map
$$
d_i:\bigoplus_{m}A(-m)^{\beta_{i,m}}
\rightarrow
\bigoplus_{m} A(-m)^{\beta_{i-1,m}}
$$
has a direct summand
$$
\dlin_i: A(-(i+t(I)))^{\beta_{i,i+t(I)}}
\rightarrow
A(-(i-1+t(I)))^{\beta_{i-1,i-1+t(I)}}.
$$
We call the collection of maps $\{\dlin_i\}_{i \geq 0}$
the {\it linear strand} of the resolution. The maps
in the linear strand inherit a uniqueness property from the
uniqueness of maps in the minimal free resolution.
To be more precise, since the maps in
the resolution are unique up to a simultaneous $A$-linear
change of bases in the free modules that they map between,
the maps in the linear strand are unique up to a simultaneous
$\field$-linear change of bases in the lowest
graded components of these free modules. We say that $I$ {\it has
linear (or $t(I)$-linear) resolution} if $d_i =\dlin_i$
for all $i$, or equivalently, if $\Tor_i^A(I,\field)_m = 0$
for $m \neq i+t(I)$.
When $I$ is generated by square-free
monomials, it is traditional to associate with it a certain
simplicial complex $\Delta$, for which $I=I_\Delta$ is the
{\it Stanley-Reisner ideal} of $\Delta$ and $A/I_\Delta$ is the
{\it Stanley-Reisner ring}. The definition of
$\Delta$ as a simplicial complex on vertex set $[n]:=\{1,2,\ldots,n\}$
is straightforward: the minimal non-faces of $\Delta$ are defined
to be the supports of the minimal square-free monomial generators of $I$.
Since $I$ respects the fine $\naturals^n$-grading by monomials on $A$, one
can find an $\naturals^n$-graded minimal free resolution for $I$
as an $A$-module, and define for any monomial ${\bf x}^\alpha$
$$
\beta_{i,{\bf x}^\alpha} :=
\dimens_\field \Tor_i^A(I,\field)_{{\bf x}^\alpha}.
$$
Instead of working with $\Delta$ (as in \cite{Hochster}),
we will instead work with a certain canonical Alexander dual
$\Delta^*$,
$$
\Delta^* : = \{F \subseteq [n]: [n]-F \not\in \Delta\}.
$$
which in \cite{HerzogReinerWelker} we called the {\it Eagon complex}
associated to $I$.
Alternatively one can describe $\Delta^*$ in terms of the square-free
monomial ideal as follows: The maximal faces (or {\it facets}) of
$\Delta^*$ are the complements of the supports of the minimal
square-free monomial generators of $I=I_\Delta$.
\section{Description of the linear strand}
In this section we describe the linear strand in the minimal
free resolution of a Stanley-Reisner ideal $I_\Delta$ in the
polynomial ring $A = \field[x_1,\ldots,x_n]$. The description
will be in terms of the homology of links of faces in the Eagon
complex $\Delta^*$.
In \cite{EagonReiner}, the following reformulation of a
famous result of Hochster \cite{Hochster} was proved:
\begin{theorem}
\label{Eagon-R}
$$
\Tor_i^A(I_\Delta,\field)_{{\bf x}^\alpha} \cong
\redhom_{i-1}(\link_{\Delta^*}F; \field)
$$
if ${\bf x}^{\alpha} = {\bf x}^V := \prod_{i \in V} x_i$
for some subset $V$ of $[n]$ whose complement
$F=[n]-V$ is a face of $\Delta^*$, and otherwise the above
Tor-group vanishes.
\end{theorem}
Here $\redhom_{\bullet}(-;\field)$ denotes reduced simplicial
homology with coefficients in $\field$. Also the {\it link},
{\it star}, and {\it deletion} of a face $F$ in a simplicial complex $K$
are subcomplexes of $K$ defined by
$$
\begin{aligned}
\link_K F : &= \{G \in K: G \cup F \in K, G \cap F = \varnothing \}\\
\str_K F : &= \{G \in K: G \cup F \in K \}\\
\del_K F: &= \{G \in K: G \cap F = \varnothing \}.
\end{aligned}
$$
We begin by noting a consequence of Theorem \ref{Eagon-R}:
Only {\it top}-dimensional homology
of links of faces in $\Delta^*$ can contribute to the linear strand.
To see this, note that the linear $i$-syzygies are measured by
$\Tor^A_i(I_\Delta, \field)_{i+t}$, where $t = t(I_{\Delta})$ is
the minimal
degree of a generator of $I_\Delta$.
In the notation of Theorem \ref{Eagon-R}, this implies
$$
n-|F|=|V| = i+t = i+n - \dimens(\Delta^*)-1
$$
or equivalently,
$$
\begin{aligned}
i-1 &=\dimens(\Delta^*) - |F| \\
&\geq \dimens(\link_{\Delta^*}F).
\end{aligned}
$$
Therefore, the group $\redhom_{i-1}(\link_{\Delta^*}F;\field)$
appearing in Theorem \ref{Eagon-R} will
vanish unless $i-1 = \dimens(\link_{\Delta^*}F)$, and in this
case it is the top-dimensional homology of this link.
Thus to specify the $\naturals^n$-graded maps
$\dlin_\bullet$ in the linear strand, it suffices to
specify for every face $F$ in $\Delta^*$ and every
vertex $v$ in $[n]$ a map as follows:
$$
\partial_{\link_{\Delta^*} F, v}:
\redhom_{i-1}(\link_{\Delta^*} F ; \field) \rightarrow
\redhom_{i-2}(\link_{\Delta^*} (F \cup \{v\}) ; \field)
$$
where $i-1 = \dimens(\link_{\Delta^*} F)$ as before. Given
such a family of maps, one can then postulate the following
candidate for the $({\bf x}^V, {\bf x}^{V-v})$-graded
component of $\dlin_i$, that is, the component of the
linear syzygies of multidegree ${\bf x}^V$ which are
$A$-linear combinations of the syzygies of multidegree
${\bf x}^{V-v}$:
\begin{equation}
\label{map-description}
\begin{matrix}
(\dlin_i)_{({\bf x}^V, {\bf x}^{V-v})}: &
A \otimes_\field \Tor^A_i(I_\Delta;\field)_{{\bf x}^V} &\longrightarrow &
A \otimes_\field \Tor^A_i(I_\Delta;\field)_{{\bf x}^{V-v}} \\
& f \otimes z & \longmapsto &
x_v f \otimes \partial_{\link_{\Delta^*} F, v} (z)
\end{matrix}
\end{equation}
where $z$ represents a cycle in
$\redhom_{i-1}(\link_{\Delta^*} F ; \field)$. Since
$$
\link_{\Delta^*} (F \cup \{v\}) =
\link_{\link_{\Delta^*} F} ( v )
$$
we can rename the complex $\link_{\Delta^*} F$ by the name $K$, and
it suffices to define in general for any simplicial complex $K$ and any
of its vertices $v$ a map:
$$
\partial_{K,v}: \redhom_{\dimens K} (K ; \field) \rightarrow
\redhom_{\dimens K -1} (\link_K v; \field).
$$
Setting $D:=\dimens K$, a natural candidate for such a map $\partial_{K,v}$
is the connecting homomorphism in the Mayer-Vietoris exact sequence
\begin{equation}
\label{Mayer-Vietoris}
0 \longrightarrow \redhom_{D}(\str_K v) \oplus
\redhom_{D}(\del_K v)
\longrightarrow \redhom_{D}(K)
\overset{\partial_{K,v}}{\longrightarrow}
\redhom_{D-1}(\link_K v) \rightarrow \cdots
\end{equation}
arising from the decomposition
$$
\begin{aligned}
K &= \str_K v \cup \del_K v \\
\link_K v &= \str_K v \cap \del_K v.
\end{aligned}
$$
\noindent
Note that in the exact sequence (\ref{Mayer-Vietoris}) we have suppressed the
field coefficients $\field$ for notational convenience, and we will
continue to do so when it causes no confusion.
We point out for future use that if one is
given an explicit representing cycle $z \in \redhom_{\dimens K}(K)$,
then one can obtain $\partial_{K,v} z$ very explicitly in two
different ways. On the one hand, from the definition of
the maps in any Mayer-Vietoris sequence,
$$
\partial_{K,v} (z) = \partial \left( z|_{\str_K v} \right)
$$
i.e. one obtains $\partial_{K,v} z$ by applying the simplicial
boundary operator $\partial$ to the chain one gets by
restricting $z$ to the faces supported in $\str_K v$.
On the other hand, if one defines for each vertex $v$ of $K$
an $\field$-linear map $\delta_v$ on oriented simplicial chains by
$$
\delta_v [i_1, i_2, \ldots, i_r] = \left\{
\begin{matrix}
(-1)^j [i_1, i_2, \ldots, \hat{i_j},\ldots, i_r] & \text{ if } v=i_j \\
0 & \text{ if } v \not\in \{i_1, i_2, \ldots, i_r\}
\end{matrix}
\right.
$$
then we claim that $\partial_{K,v}(z) = \delta_v (z)$.
To deduce this description of $\partial_{K,v}(z)$ from
the previous one, note that any
terms in $\partial \left( z|_{\str_K v} \right)$
supported on faces of $K$ containing $v$ must cancel each
other, since the result has to be a cycle supported in $\link_K v$.
The remaining terms in the simplicial boundary map are exactly those
in $\delta_v$.
We can now prove:
\begin{theorem}
\label{main-result}
The maps $\dlin_i$ in the linear strand of the $\naturals^n$-graded
minimal free resolution of a Stanley-Reisner ideal $I_\Delta$ are given
by equation (\ref{map-description}), where for a complex $K$ and vertex $v$,
the map $\partial_{K,v}$ is described above.
\end{theorem}
\begin{proof}
Let $d'_i$ be the map asserted by the theorem to coincide with
$\dlin_i$. We first show that $d'_{i-1} d'_i=0$, so that these
maps do form a complex, and then show that this complex must be the
linear strand.
The fact that $d'_{i-1} d'_i=0$ comes from a separate analysis of
each $\naturals$-graded component.
The $({\bf x}^V, {\bf x}^{V-\{v,w\}})$-component of $d'_{i-1} d'_i$
is determined by a map $\gamma$ having the form
$$
\gamma: \redhom_{\dimens K}(K) \longrightarrow
\redhom_{\dimens K-2}(\link_K \{v,w\})
$$
where here $K=\link_{\Delta^*} F$ for $F=[n]-V$.
The map $\gamma$ is the sum of two composite maps
$$
\begin{aligned}
\redhom_{\dimens K}(K) \overset{\partial_{K,v}}{\longrightarrow}
&\redhom_{\dimens K-1}(\link_K \{v\})
\overset{\partial_{\link_K v,w}}{\longrightarrow}
\redhom_{\dimens K-2}(\link_K \{v,w\}) \\
\redhom_{\dimens K}(K) \overset{\partial_{K,w}}{\longrightarrow}
&\redhom_{\dimens K-1}(\link_K \{w\})
\overset{\partial_{\link_K w,v}}{\longrightarrow}
\redhom_{\dimens K-2}(\link_K \{v,w\}).
\end{aligned}
$$
{From} our second description of $\partial_{K,v}$ we have
$$
\gamma = \partial_{\link_K v,w} \circ \partial_{K,v} +
\partial_{\link_K w,v} \circ \partial_{K,w}
=\delta_w \delta_v + \delta_v \delta_w
$$
and the righthand side is easily checked to be $0$. Hence
$d'_{i-1} d'_i = 0$ in each $\naturals^n$-graded component.
To show that $d'_i$ coincides with $\dlin_i$, we use induction
on $i$. For the case $i=0$, note that
$\redhom_{-1}(\link_{\Delta^*}F)$ vanishes unless $F$ is maximal
face of $\Delta^*$, and then it is one-dimensional. When $F$ is
a maximal face $F$ of $\Delta^*$, setting $V = [n]-F$ we have
that ${\bf x}^V$ is a minimal generator
of $I_\Delta$, and it is easy to
check that the ${\bf x}^V$-component of $d'_0$ is simply
the monomial ${\bf x}^V$, just as in $\dlin_0$.
For the inductive step, assume that $d'_{i-1}$ coincides with
$\dlin_{i-1}$. Since $d'_{i-1} d'_i = 0$, it follows that
the $A$-linear combinations of the $(i-1)$-syzygies defined
by $d'_i$ are all genuine $i$-linear syzygies. Hence they must
form an $\field$-subspace in the space of all linear $i$-syzygies.
By construction $d'_i$ produces exactly the same number
of ${\bf x}^V$-graded linear $i$-syzygies as predicted by
Theorem \ref{Eagon-R} for the space of {\it all}
${\bf x}^V$-graded linear $i$-syzygies, although some of these
syzygies might, a priori, be linearly dependent over $\field$.
Therefore, it suffices to show for each set $V$ that
all of the ${\bf x}^V$-graded linear $i$-syzygies produced by $d'_i$
are linearly independent over $\field$. Setting $K = \link_{\Delta^*} F$
for $F=[n]-V$ as before, this amounts to showing injectivity of the map
$$
\redhom_{\dimens K}(K)
\overset{\oplus_v \partial_{K,v}}{\longrightarrow}
\bigoplus_v \redhom_{\dimens K - 1}( \link_K v).
$$
To prove injectivity, we use the following lemma.
\begin{lemma}
\label{M-V-trick}
Let $z$ be a cycle in $\redhom_{\dimens K}(K) (= \tilde{Z}_{\dimens K}(K))$.
Then $\partial_{K,v}(z) = 0$ if and only if $z$ does not
involve the vertex $v$.
\end{lemma}
\begin{proof}
Consider the Mayer-Vietoris exact sequence (\ref{Mayer-Vietoris}).
Since $\str_K v$ is a cone, it is contractible and
$\redhom_{\dimens K}( \str_K v)$ vanishes.
Hence exactness at $\redhom_{\dimens K}(K)$
shows that if $\partial_{K,v}(z) = 0$,
then $z$ is represented by a cycle in $\del_K v$. But since
there are no ($\dimens K$)-boundaries to create ambiguity, this means
that $z$ does not involve $v$. The converse is obvious.
\end{proof}
Injectivity of the map $\oplus_v \partial_{K,v}$
is now clear: Any cycle $z$ in
$\redhom_{\dimens K}(K)$ which is sent to $0$ must involve no
vertices of $K$ by Lemma \ref{M-V-trick}.
This concludes the proof of Theorem \ref{main-result}.
\end{proof}
\section{A conjecture by Herzog in the case of monomial ideals}
\label{EGH-conjecture}
In this section we prove for monomial ideals a conjecture by
J. Herzog \cite{Herzog} about linear syzygies. Recall that
a {\it $j^{th}$-syzygy module} is a module that occurs as the kernel
in the $(j-1)^{st}$-homological degree in the resolution of
a finitely generated module $N$ over the polynomial ring $A$.
For example any ideal $I$ in the polynomial ring $A$ is a
first syzygy module, since $I$ is the kernel of the natural
surjection $A \rightarrow N = A/I$.
\begin{conjecture}[Herzog \cite{Herzog}]
\label{EGH}
Let $M$ be a $j^{th}$-syzygy module for some graded $A$-module $N$,
where $A=\field[x_1,\ldots,x_n]$ is given the standard grading.
If $M$ has non-zero $p$-syzygies in its linear strand for some $p \geq 0$,
then it will have at least $\binomial{p+j}{i+j}$ $i$-syzygies in its
linear strand for each $i$.
\end{conjecture}
\noindent
A homogeneous ideal $I$ in $A$ is a graded first syzygy module.
So when $M$ is such an ideal $I$ we can reformulate the
conjecture as follows: if $\Tor^A_p(I,\field)_{p+t(I)}$ does not
vanish for some $p \geq 0$, then
$$
\dimens_\field \Tor^A_i(I,\field)_{i+t(I)} \geq \binomial{p+1}{i+1}
$$
for each $i$.
Herzog's conjecture is motivated by a result of M. Green \cite{Green}
(see also \cite{EisenbudKoh}) that
contains the case $i=0$, $j=1$, and his own results that show the
conjecture in full generality for $j=0$ \cite{Herzog}.
By the technique of polarization \cite{Froberg}, the $j=1$ case
of Conjecture \ref{EGH} follows for all monomial ideals $I (=M)$
once it is proven for square-free monomial ideals.
For square-free monomial ideals, we will in fact show
something slightly stronger:
\begin{theorem}
\label{square-free-Eisenbud-Green-Herzog}
Let $I$ be a square-free monomial ideal in $A$.
% , with $t$ the minimal degree of its generators.
If $\Tor^A_p(I,\field)_{{\bf x}^V} \neq 0$ for some set
$V$ with $|V|=p+t(I)$, then for each $i$ there exist at least $\binom{p+1}{i+1}$
subsets $V' \subseteq V$ having $|V'|=i+t(I)$ and
$\Tor^A_i(I,\field)_{{\bf x}^{V'}} \neq 0$.
\end{theorem}
\begin{proof}
The result will be deduced from the following lemma, whose
statement involves only simplicial topology, but whose proof
relies heavily on the maps $\partial_{K,v}$ which appear in
the linear strand:
\begin{lemma}
\label{EGH-crux}
Let $K$ be a $(q-1)$-dimensional simplicial complex and $\field$
any field. If $\redhom_{q-1}(K;\field) \neq 0$, then there exist
at least $q+1$ vertices $v$ of $K$ with the property that
$\redhom_{q-2}(\link_K v;\field) \neq 0$.
\end{lemma}
\begin{proof}
Let $z$ be a non-trivial cycle in $\redhom_{q-1}(K;\field)$.
Because it is a non-trivial $(q-1)$-cycle, $z$ must involve
at least $q+1$ vertices. Hence by Lemma \ref{M-V-trick},
at least $q+1$ of the maps $\partial_{K,v}$ must have
$\partial_{K,v}(z) \neq 0$.
Therefore at least $q+1$ vertices must have
$\redhom_{q-2}(\link_K v;\field) \neq 0$.
\end{proof}
\noindent
To deduce Theorem \ref{square-free-Eisenbud-Green-Herzog}, we use induction
on $p-i$. It is trivially true for $p-i=0$.
In the inductive step, assume that there are at least $\binomial{p+1}{i+1}$
subsets $V'' \subseteq V$ having $|V''|=i+t(I)$ and
$\Tor^A_{i}(I;\field)_{{\bf x}^{V''}} \neq 0$.
For each such $V''$, we can apply Lemma~\ref{EGH-crux} and
Theorem \ref{Eagon-R} to $K=\link_{\Delta^*} F''$ where
$F'' = [n]-V''$. In this way, we obtain
a collection ${\mathcal C}_{V''}$ of at least $i+1$ subsets
$V' \subset V''$ with $|V'| = i-1+t(I)$ and
$\Tor^A_{i-1}(I;\field)_{{\bf x}^{V''}} \neq 0$.
It remains then to show that the cardinality of the union
$\bigcup_{V''} {\mathcal C}_{V''}$ is at least $\binomial{p+1}{i}$.
If we fix attention on a particular subset $V' \subset V$
having $|V'|=i-1+t(I)$ elements, then it can occur in at most
$p+1-i$ different
collections ${\mathcal C}_{V''}$ since $|V - V'|=p+1-1$.
Therefore
$$
\begin{aligned}
\left| \bigcup_{V''} {\mathcal C}_{V''} \right|
&\geq \frac{1}{p+1-i} \sum_{V''} |{\mathcal C}_{V''}| \\
&\geq \frac{1}{p+1-i} \binom{p+1}{i+1} (i+1)\\
&=\binom{p+1}{i}.
\end{aligned}
$$
This concludes the proof of Theorem \ref{square-free-Eisenbud-Green-Herzog}.
\end{proof}
\section{Pseudomanifolds and a counterexample}
\label{pseudomanifolds}
In this section we give some consequences of Theorem \ref{main-result}.
It is easy to see for any monomial ideal $I$, the matrix entries in
the maps in the multigraded minimal free resolution will always be
single terms, that is a monomial times some coefficient.
We show that the coefficients occurring in the linear strand
can always be chosen to be $\pm 1$ whenever $\Delta^*$ is a pseudomanifold,
and give a non-pseudomanifold example where this property fails.
Say that a $d$-dimensional simplicial complex $K$ is a
{\it pseudomanifold without boundary}, or just a {\it pseudomanifold},
if every $(d-1)$-face lies in exactly two
$d$-faces. Examples of such complexes are triangulations of manifolds
or singular spaces whose singularities have real codimension
at least two, such as complex varieties.
We emphasize that our definition differs somewhat
from the definition of pseudomanifolds sometimes given in the
literature (e.g. \cite[p. 261]{Munkres}),
where it is further assumed $K$ is pure, and that any two $d$-faces are
{\it gallery-connected}, i.e. connected by a path of
$d$-faces in which every pair of $d$-faces forming a step in the
path share a common $(d-1)$-face. Note however, that dropping the
assumption of purity gives us only a spurious extra generality
with regard to results on the linear strand:
The linear strand in the minimal free resolution of $I_\Delta$
depends only on the minimal generators of $I_\Delta$ of lowest degree,
and hence depends only on the pure subcomplex of $\Delta^*$ generated by
its faces of maximum dimension.
Recall that $\redhom_d(K;\field)$ for a $d$-dimensional pseudomanifold
$K$ takes on a particularly simple form and has a canonical basis
(up to negating basis vectors) of {\it orientation cycles}
(\cite[p. 394]{Munkres}).
To be precise, let $\Gamma_1, \ldots,
\Gamma_m$ be the {\it gallery-equivalence} classes of $d$-faces of
$K$, where gallery-equivalence is the equivalence relation
generated by the relation $F \sim F'$ if the two $d$-faces $F,F'$
share a common $(d-1)$-face $F \cap F'$. For each class $\Gamma_i$
it is easy to see that either there exists a unique way (up to an
overall sign change) to choose signs so that
$$
z_{\Gamma_i} = \sum_{F \in \Gamma_i} \pm [F]
$$
forms an cycle (called an orientation cycle), or else $\Gamma_i$
contributes no orientation cycle to the basis of
$\redhom_d(K;\field)$. Note that the latter cannot
happen if $\field$ has characteristic $2$. Also note that any
cycle $z$ in $\redhom_d(K;\field)$ is completely determined
by its coefficients $c_{[F_i]}$ on any system of representatives
of $d$-faces $F_i \in \Gamma_i$ of the classes
$\Gamma_i$: If $z_{\Gamma_i}$ is normalized so that the
oriented simplex $[F_i]$ has coefficient $+1$, then
$$
z = \sum_i c_{[F_i]} z_{\Gamma_i}.
$$
As a consequence of this discussion, we have the following:
\begin{proposition}
\label{pseudomanifold}
If $\Delta^*$ is a pseudomanifold, then the maps in the linear
strand of the minimal free resolution of $I_{\Delta^*}$ can be
written using only $\pm 1$ coefficients.
\end{proposition}
\begin{proof}
Note that if $\Delta^*$ is a $d$-dimensional pseudomanifold and
$F$ is a face
of $\Delta$ with
$$
\dimens(\link_{\Delta^*} F) = d - |F|
$$
(i.e. $F$ is a face whose link in $\Delta^*$ has top homology that
might contribute to the linear strand), then $\link_{\Delta^*} F$ is also a
pseudomanifold.
Therefore by Theorem \ref{main-result}, it suffices to show that
for any pseudomanifold $K$ and vertex $v$, the map $\partial_{K,v}$
is a $\pm 1$-matrix when written with respect to the basis of
orientation cycles for
$\redhom_d(K;\field)$ and
$\redhom_{d-1}(\link_K v;\field)$.
To see this, let $\Gamma_i$ (resp. $\Gamma_j'$) be a
gallery-equivalence class of $d$-faces (resp. $(d-1)$-faces) for
$K$ (resp. $\link_k v$), which gives rise to an orientation cycle
$z_{\Gamma_i}$ (resp. $z_{\Gamma_j'}$) over $\field$. Then it is
easy to check from the description $\partial_{K,v} = \delta_v$
that the coefficient of $z_{\Gamma_j'}$ in $\partial_{K,v}(z_{\Gamma_i})$
will be $0$ unless any chosen $(d-1)$-face $F'$ in $\Gamma_j'$
has the property that $F=F' \cup \{v\}$ is in $\Gamma_i$. In the
latter case, the coefficient will be $\pm 1$, depending upon the
sign of $[F]$ in $z_{\Gamma_i}$ and of $[F']$ in $z_{\Gamma_j'}$.
\end{proof}
The proof of Proposition \ref{pseudomanifold} shows that the
maps in the linear strand when $\Delta^*$ is a pseudomanifold
are essentially
``incidence matrices" for the orientation cycles of all of the
links of faces in $\Delta^*$. An interesting special case of
this occurs when $\Delta^*$ is a {\it homology
sphere over }$\field$, that is when every face
$F$ of $\Delta^*$ has
$$
\redhom_i(\link_{\Delta^*} F;\field) =
\left\{
\begin{matrix}
\field & \text{ if } i=\dimens \Delta^* - |F| \\
0 & \text{ else }.
\end{matrix}
\right.
$$
This condition may also be phrased in terms of local
homology groups $\redhom( |\Delta^*|, |\Delta^*|-x; \field)$,
and hence is topologically invariant.
In \cite{EagonReiner} it was pointed out that for homology spheres
$\Delta^*$, the Betti numbers in the (linear) resolution of $I_\Delta$
coincide with the $f$-{\it vector} listing the number of
faces of various dimensions in $\Delta^*$. In \cite[page 3]{Sturmfels}
Sturmfels further remarked that this linear resolution is essentially the
{\it coboundary complex} for the simplicial complex $\Delta^*$.
Although this can be deduced from Theorem \ref{main-result},
it is easy enough to prove directly so we omit the proof.
The previous results raise the question of whether there exists a
monomial ideal $I$ for which the maps {\it cannot} be written using only
$\pm 1$ coefficients. The following example illustrates that this
can happen, even when the whole resolution is linear, and even when
it is linear over an arbitrary field $\field$.
% due to shellability of $\Delta^*$.
\smallskip
\noindent
{\bf Example:}\\
Let $I_\Delta$ be the ideal in $A = \field[x_1,x_2,x_3,x_4,x_5,x_6,x_v]$
$$
\begin{matrix}
(x_3 x_4 x_5 x_6, & x_2 x_4 x_5 x_6, & x_1 x_4 x_5 x_6, & & \\
x_3 x_4 x_5 x_v, & x_2 x_4 x_5 x_v, & x_1 x_3 x_5 x_v, & x_1 x_2 x_4 x_v, & x_1 x_4 x_6 x_v, \\
x_1 x_5 x_6 x_v, & x_3 x_4 x_6 x_v, & x_2 x_5 x_6 x_v, & x_2 x_3 x_6 x_v, &
x_1 x_2 x_3 x_v)
\end{matrix}
$$
where $\field$ does not have characteristic $2$ or $3$.
Then $\Delta^*$ is the simplicial complex on vertex set
$\{1,2,3,4,5,6,v\}$ with maximal faces
$$
\{12v,13v,23v, 126,136,246,356,235,234,125,134,145,456\},
$$
and can be described in the following fashion: The induced
subcomplex of $\Delta^*$ on the vertices $\{1,2,3,4,5,6\}$ is a minimal
triangulation of the real projective plane having the property
that $123$ is {\it not} a $2$-face, and $\Delta^*$ is obtained
from this subcomplex by adding the three triangles $v12, v13, v23$.
One can think of this addition as first adding in the missing $2$-face
$123$ and then subdividing this $2$-face with a vertex $v$ into
three smaller triangles.
It is easy to check that the order in which the maximal faces of $\Delta^*$
are listed above is a {\it shelling order}, and hence $\Delta^*$ is
Cohen-Macaulay over any field $\field$, so that $I_\Delta$
always has a $4$-linear resolution whose Betti numbers are independent of
$\field$ (see \cite{EagonReiner}). The complex $\Delta^*$ has the
interesting property that even though it is shellable and
homotopy equivalent to a $2$-sphere, the orientation cycle in
$\redhom_2(\Delta^*,\field)$ cannot be written as a linear combination
of $2$-faces with $\pm 1$ coefficients. This property was
shown to us by G. M. Ziegler in a similar example, where the triangle
$123$ is not subdivided into three smaller triangles. Unfortunately,
this original example had a resolution with only $\pm 1$ coefficients
in the maps, requiring the subdivision by vertex $v$.
A computation with the computer algebra package Macaulay shows
that the $4$-linear resolution of $I_\Delta$ has the form
$$
0 \longrightarrow A(-7)^1 \longrightarrow A(-6)^{10} \longrightarrow A(-5)^{21}
\longrightarrow A(-4)^{13} \longrightarrow I_\Delta \longrightarrow 0.
$$
This resolution has all $\pm 1$ coefficients in the maps except for the
last map
$$
A(-7)^1 \longrightarrow A(-6)^{10}
$$
which contains some coefficients of $\pm 2$. Call this Macaulay resolution
${\mathcal M}_\bullet$. We now assume that
we have such a multigraded minimal free resolution ${\mathcal F}_\bullet$
which uses only coefficients $\pm 1$, in its maps, and
will reach a contradiction.
Obviously, the map $A(-4)^{13} \longrightarrow I_\Delta$
in ${\mathcal F}_\bullet$ must simply list the minimal generators of
$I_\Delta$, possibly with $\pm 1$ coefficients in front, so without
loss of generality, we may alter the basis for $A(-4)^{13}$ in
${\mathcal F}_\bullet$ by $\pm$ signs so that the coefficients are all
$+1$, as in ${\mathcal M}_\bullet$. Since the edges
$e = v1, v2, v3, 16, 26, 36, 46, 56$ have links in $\Delta^*$ which are
$0$-spheres, they each give rise to a unique $1$-syzygy having
multidegree complementary to $e$ by Theorem \ref{Eagon-R}.
In ${\mathcal M}_\bullet$
these $1$-syzygies have only $\pm 1$ coefficients, and
so ${\mathcal F}_\bullet$ must choose a $\pm 1$ multiple of these same
syzygies, hence we may assume that the columns of the map
$A(-5)^{21} \longrightarrow A(-4)^{13}$ in these multidegrees are the same
as in ${\mathcal M}_\bullet$. A similar remark applies to the
vertices $v , 6$ whose links in $\Delta^*$ are $1$-spheres,
once we note for reasons of multidegree
that the unique $2$-syzygies to which they correspond
will only involve the $1$-syzygies which correspond to the edges
$e$ listed previously, and hence are the same in ${\mathcal F}_\bullet$
as in ${\mathcal M}_\bullet$.
Let us name the basis elements of $A(-6)^{10}$ corresponding to these
two $2$-syzygies by $e_v, e_6$, respectively, in both ${\mathcal M}_\bullet$
and ${\mathcal F}_\bullet$.
Lastly, we compare the unique $3$-syzygy in ${\mathcal M}_\bullet$
and ${\mathcal F}_\bullet$. In ${\mathcal M}_\bullet$, this $3$-syzygy
has coefficient $x_6$ on $e_6$ and $2x_v$ on $e_v$. Since $\field$
does not have characteristic $2$ or $3$, there is no way for
${\mathcal F}_\bullet$ to rescale this unique $3$-syzygy to have
both a $\pm x_6$ on $e_6$ and $\pm x_v$ on $e_v$. Contradiction. \qed
\section{The resolution for matroidal ideals}
\label{matroids}
In this section we use Theorem \ref{main-result} to explicitly
describe the minimal free resolution when $I_\Delta$ is {\it matroidal},
i.e. if $m,m'$ are two minimal monomial generators of $I$
and $x_i$ divides $m$, then there exists a $j$ such that $x_j$
divides $m'$ and $\frac{x_i}{x_j}m'$ is in $I_\Delta$. In this case, the
supports of the square-free monomial generators of $I$ form the bases of a
{\it matroid} $M$ on ground set $[n]$ (see \cite{Oxley} for definitions
and facts about matroids). It was observed in \cite{EagonReiner}
that in this case $\Delta^*$ is the complex of independent sets $IN(M^*)$
or {\it matroid complex} associated to the {\it dual matroid} $M^*$.
Since matroid complexes are pure and shellable (see \cite{Bjorner})
this implies that $I_\Delta$
has a linear resolution for any field $\field$. To make this
resolution explicit, we recall some terminology from matroid theory
from \cite{Bjorner}.
Given a base
$B$ of a matroid $M$ and an element $e$ not in $B$, there is a unique
circuit $ci_M(B,e)$ contained in $B\cup\{e\}$ called the {\it basic circuit}
for $e$ and $B$. Dually for any $b$ in $B$ there is a
unique {\it bond} $bo_M(B,b)$ contained in $([n]-B) \cup \{b\}$ called
the {\it basic bond} for $b$ and $B$. An element $e$ in $[n]-B$
(resp. $b$ in $B$) is called {\it externally (resp. internally) active}
with respect to $B$ if it is the smallest element of its basic
circuit (resp. bond) in the usual order on $[n]$. The {\it external
(resp. internal) activity} of a base $B$ is the number of elements of
$[n]$ which are externally (resp. internally) active with respect to
$B$.
Given a subset $V \subseteq [n]$ such that its complement $F=[n]-V$
is a face of $\Delta^*=IN(M^*)$, it is easy to check that
$$
\link_{\Delta^*}F = IN(M^*/F)
$$
where $M^*/F$ denotes the {\it quotient
matroid} of $M^*$ by $F$. Alternatively, we have
$$
M^*/F \cong (M|_V)^*
$$
where $M|_V$ is the restriction of $M$ to the ground set $V$.
In order to apply Theorem \ref{main-result}, we need a
basis for the top homology of $\link_{\Delta^*}F = IN(M^*/F)$.
There are two (conjecturally equivalent) choices for us to use.
Given any matroid $M$,
\begin{enumerate}
\item[$\bullet$]
Bj\"orner \cite[Prop. 7.8.4]{Bjorner} constructs a basis $\{\sigma_B\}$ for the top homology of $IN(M)$ which is indexed by bases
$B$ of $M$ having internal activity $0$, or
\item[$\bullet$]
from Theorem 14 and Remark 15 of \cite{KookReinerStanton},
one obtains a basis $\{\tau_C\}$ for the top homology of
of $IN(M)$ which is indexed by bases $C$ of $M^*$ having
external activity $0$, sometimes called {\it nbc-bases} for $M^*$.
\end{enumerate}
We choose to use the latter basis for convenience.
Fortunately, bases of internal activity $0$ for $M$ are
the same as bases of external activity $0$ for $M^*$ by complementation
within the ground set $[n]$, since the internal activity of $b$ with respect
to $B$ in $M$ equals the external activity of $b$ with respect
to $[n]-B$ in $M^*$. Conjecturally, $\tau_{C} = \sigma_{[n]-C}$,
but we will not need this.
For the purpose of stating the theorem, let us fix the following
notation. Let $I_\Delta$ be matroidal for a matroid $M$,
so that $\Delta^* = IN(M^*)$. Given $V \subseteq [n]$ and
$v \in V$ and $F=[n]-V$ as usual, choose as bases for
the top homology of
$$
\begin{aligned}
\link_{\Delta^*}F &= IN(M^*/F) \\
\link_{\Delta^*}F+v &= IN(M^*/(F+v))
\end{aligned}
$$
the sets $\{\tau_C\}, \{\tau_{C'}\}$
where $C, C'$ run through the bases of external activity
$0$ for $M|_V, M|_{V-v}$. With respect to the above bases,
let $d_{C,C'}$ denote the $(C,C')$-entry of the
$({\bf x}^V, {\bf x}^{V-v})$-graded
component of the map in the minimal free resolution of $I_\Delta$.
\begin{theorem}
The matrix entry $d_{C,C'}$ is $0$ unless either
\begin{enumerate}
\item[$\bullet$] $C = C'$, or
\item[$\bullet$] $C=C'-\{w\} \cup \{v\}$
for some $w$ in $ci_{M|_{V-v}}(C',v)$.
\end{enumerate}
In either of these cases, $d_{C,C'}=(-1)^j x_v$
where $v$ is the $j^{th}$ smallest element of $C$,
\end{theorem}
\begin{proof}
The basis elements $\tau_C$ constructed in the proof of
\cite[Theorem 14]{KookReinerStanton} have the following properties:
\begin{enumerate}
\item[$\bullet$] The coefficient of $\tau_C$ on any particular
oriented simplex is $0$ or $\pm 1$,
\item[$\bullet$] The coefficient of $\tau_C$ on the oriented simplex
$[[n]-C]$ is $+1$,
\item[$\bullet$] The coefficient of $\tau_D$ on $[[n]-C]$ is $0$
for any other basis $D$ of external activity $0$.
\end{enumerate}
As a consequence, from Theorem \ref{main-result} and the
description $\partial_{K,v}=\delta_v$, computing
$d_{C,C'}$ comes down to checking whether $[([n]-v)-C'+v]=[[n]-C']$
appears with non-zero coefficient in $\tau_C$. It
remains to show that this happens exactly under the circumstances described
in the theorem (and under those circumstances the appropriate
$\pm 1$ coefficient then follows easily).
To simplify the analysis, we let $y_C=f(\tau_C)$ where
$f$ maps chains in $IN((M|_V)^*)$ to chains in $IN(M|_V)$
by sending the oriented simplex $[B]$ to the oriented simplex $[V-B]$.
We then need to show that $C'$ occurs with non-zero coefficient in $y_C$ only in
the circumstances of the theorem. {From} the proof of
\cite[Theorem 14]{KookReinerStanton}, the chain $y_C$ has
a fairly simple description:
If one writes the elements $c_1 < c_2 < \cdots < c_r$ of $C$
in increasing order, then
$$
y_C = \sum_{(e_1,\ldots,e_r)} [e_1,\ldots,e_r]
$$
where $(e_1,\ldots,e_r)$ runs over all sequences of $r$ elements
in $[n]$ such that $e_i$ is in the {\it flat} $\overline{C_i}$
spanned by $C_i:=\{c_r,c_{r-1},\ldots,c_{i+1},c_{i}\}$
but not in the flat $\overline{C_{i+1}}$.
Renaming the restricted matroid $M|_V$ by $N$,
we are left with proving the following
claim:
\begin{enumerate}
\item[] {\bf Claim}: If $N$ is a matroid, $v$ an element of its ground set,
and $C,C'$ bases of external activity $0$ for $N,N-v$ respectively,
then $[C']$ occurs with non-zero coefficient in $y_C$ if and only
if $C=C'$ or $C=C'-w+v$
for some $w$ in $ci_N(C',v)$.
\end{enumerate}
Before proving this claim, we recall two important
properties of bases of external activity $0$ from the proof of
\cite[Theorem 14]{KookReinerStanton}:
\begin{enumerate}
\item[(a)] If $C$ is a base of external activity $0$, then any
non-empty subset $C_0 \subseteq C$ is a base of external activity $0$
for the flat $\overline{C_0}$.
\item[(b)] If $C=\{c_1 < \ldots < c_r\}$ and $C_i = \{c_r,c_{r-1},\ldots,c_i\}$
as above, then $c_i$ is the smallest element of
$\overline{C_i} - \overline{C_{i+1}}$.
\end{enumerate}
\noindent
To prove the claim, there are two cases to check depending on
$v$'s external activity with respect to $C$.
\noindent
{\bf Case 1:} The element $v$ is not externally active for $C'$.
In this case, $C'$ still has external activity $0$ when considered
as a base for $N$ rather than $N-v$. It then follows from the
bulleted facts at the beginning of the proof that
$C'$ occurs with non-zero coefficient in $y_C$ if and only
if $C=C'$.
\noindent
{\bf Case 2:} The element $v$ is externally active for $C'$.
We wish to show in this case that $C=C'-\{w\} \cup \{v\}$
for some $w$ in $ci_{(M|_{V-v})^*}(C',v)$. Recalling the
notation
$C_i = \{c_r,c_{r-1},\ldots,c_i\}$, let $s$ be the unique index
such that $v$ lies in $\overline{C_s}-\overline{C_{s+1}}$.
Since $C'$ occurs with non-zero coefficient in $y_C$,
by the definition of $y_C$ we can also number
the elements $\{c'_1,\ldots,c'_r\}=C'$ in such
a way that $C'_i:=\{c'_r,c'_{r-1},\ldots,c'_i\}$ has
$C'_i, C_i$ span the same flat of $N$ for all $i$.
We will now show that $w=c'_s$ has property asserted in
the claim.
Since $v$ is not in $\overline{C_{s+1}} (= \overline{C'_{s+1}}$ ),
it follows from assertion (a) that $C_{s+1}, C'_{s+1}$ are
both bases of external activity $0$ for this flat. But by construction,
$[C'_{s+1}]$ appears as a term in $y_{C_{s+1}}$, so we must have
$C'_{s+1} = C_{s+1}$. Now $v,c_s,c'_s$ all lie in
$\overline{C_s}- \overline{C_{s+1}}
( = \overline{C'_s}- \overline{C'_{s+1}} ),$
so by assertion (b), $c_s$ must be the smallest element of
$\overline{C_s}- \overline{C_{s+1}}$ . On the other hand,
the fact that $C'$ has external activity $0$ for $N-v$ but not for $N$
implies that $C'_s$ has external activity $0$ for $\overline{C_s}-v$
but not for $\overline{C_s}$.
Therefore, $v$ must be externally active for $C'_s$ in $\overline{C'_s}$,
which means that it is the smallest element of
$\overline{C_s}- \overline{C_{s+1}}$ and so we must have $v=c_s$.
Consequently, $C_s = C'_s-c'_s+v = C'_s-w+v$. The fact that
$c'_i = c_i$ for $i > s$ follows again from assertion (b),
since $\overline{C_i}- \overline{C_{i+1}} = \overline{C'_i}- \overline{C'_{i+1}}$ for $i > s$. Therefore $C = C'-w+v$ as desired.
\end{proof}
\section{Acknowledgments}
The authors thank Jack Eagon, J\"urgen Herzog, Irena Peeva, and
G\"unter M. Ziegler for helpful comments and suggestions.
\newcommand{\journalname}[1]{\textrm{#1}}
\newcommand{\booktitle}[1]{\textrm{#1}}
\bibliographystyle{amsplain}
\begin{thebibliography}{}
\bibitem{AramovaHerzog1}
A. Aramova and J. Herzog.
\textit{Free resolutions and Koszul homology.}
\journalname{J. Pure Appl. Algebra} {\textbf{105}} (1995), 1--16.
\bibitem{AramovaHerzog2}
A. Aramova and J. Herzog.
\textit{Koszul cycles and Eliahou-Kervaire type resolutions.}
\journalname{J. Algebra} {\textbf{181}} (1996), 347--370.
\bibitem{BayerCharalambousPopescu} D. Bayer, H. Charalambous, and S. Popescu.
\textit{Betti numbers of monomial ideals.} preprint (1998).
\bibitem{BayerPeevaSturmfels}
D. Bayer, I. Peeva and B. Sturmfels.
\textit{Monomial resolutions.}
\journalname{Math. Res. Lett.}
{\textbf{5}} (1998), 31--46.
\bibitem{BayerSturmfels}
D. Bayer and B. Sturmfels.
\textit{Cellular resolutions.} preprint (1997).
\bibitem{Bjorner}
A. Bj\"orner.
\textit{Homology and shellability of matroids and geometric lattices.}
in \textit{Matroid Applications (ed. by N. White)},
Cambridge Univ. Press, Cambridge (1992).
\bibitem{Eagon} J. A. Eagon.
\textit{Minimal resolutions of ideals associated
to triangulated homology manifolds.} preprint (1995).
\bibitem{EagonReiner} J. A. Eagon and V. Reiner.
\textit{Resolutions of Stanley-Reisner rings and Alexander duality.}
to appear in \journalname{J. Pure Appl. Algebra}.
\bibitem{EisenbudGoto} D. Eisenbud and S. Goto.
\textit{Linear free resolutions and minimal multiplicity.}
\journalname{J. Algebra} {\textbf{88}} (1984), 89--133.
\bibitem{EisenbudKoh} D. Eisenbud and J. Koh.
\textit{Some linear syzygy conjectures.}
\journalname{Adv. Math.} {\textbf{90}} (1991), 47--76.
\bibitem{Froberg} R.~Fr\"oberg.
\textit{A study of graded extremal rings and of monomial rings.}
\journalname{Math. Scand.} {\textbf{51}} (1982), 22--34.
\bibitem{Green} M. Green.
\textit{Koszul homology and the geometry of projective varieties.}
\journalname{J. Algebra} {\textbf{19}} (1984), 125--171.
\bibitem{Herzog} J.~Herzog.
\textit{The linear strand of a graded free resolution.}
preprint (1998).
\bibitem{HerzogReinerWelker} J. Herzog, V. Reiner, V. Welker.
\textit{Componentwise linear ideals and Golod rings.}
preprint (1997).
\bibitem{HerzogSimisVasconcelos}
J. Herzog, A. Simis, and W. Vasconcelos.
\textit{Approximation complexes and blowing-up rings II.}
\journalname{J. Algebra} {\textbf{82}} (1983), 53--83.
\bibitem{Hochster}
M. Hochster.
\textit{Cohen-Macaulay rings, combinatorics, and simplicial complexes}
in \textit{Ring Theory II: Proc. of the Second Oklahoma Conference (B.
McDonald and A. Morris, ed.).} Dekker, New York (1977), 171--223
\bibitem{KookReinerStanton}
W. Kook, V. Reiner, and D. Stanton.
\textit{Combinatorial Laplacians of matroid complexes.}
preprint (1997).
\bibitem{Munkres}
J. Munkres.
\booktitle{Elements of Algebraic Topology.}
Benjamin/Cummings, Menlo Park CA (1984).
\bibitem{Oxley}
J.G. Oxley.
\booktitle{Matroid Theory.}
Oxford University Press, Oxford (1992).
\bibitem{PeevaSturmfels}
I. Peeva and B. Sturmfels.
\textit{Generic lattice ideals.}
\journalname{J. Amer. Math. Soc.}
{\textbf{11}} (1998), 363--373.
%\bibitem{Stanley} R. P. Stanley.
%\booktitle{Combinatorics and commutative algebra, 2nd. ed.}.
%Birkh{\"a}user, Boston (1996).
\bibitem{Sturmfels}
B. Sturmfels.
\textit{Four counterexamples in combinatorial algebraic
geometry.} preprint (1998).
\bibitem{Terai} N. Terai.
\textit{Generalization of Eagon-Reiner theorem and h-vectors
of graded rings.} preprint (1997).
\end{thebibliography}
\end{document}