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\begin{document}
\mbox{}
%\vspace{-2ex}
\title[Stanley's simplicial poset conjecture]%
{Stanley's simplicial poset conjecture,\\ after M.\ Masuda}
\author{Ezra Miller and Vic Reiner}%\\
%\MakeLowercase{\texttt{ezra@math.umn.edu, reiner@math.umn.edu}}}
\thanks{\textit{Email: }\texttt{ezra@math.umn.edu},
\texttt{reiner@math.umn.edu}}
\begin{abstract}
M. Masuda recently provided the missing piece proving a conjecture
of R.P.~Stanley on the characterization of $f$-vectors for Gorenstein$\!$*
simplicial posets. We propose a slight simplification of Masuda's proof.
\end{abstract}
\maketitle
Our main result, Theorem~\ref{even}, was first proved by Masuda
\cite{masuda}, completing the missing step in a conjecture of Stanley
characterizing the $f$-vectors of Gorenstein$\!$* simplicial posets.
This note gives a simplified proof of it, using elementary methods.
We begin with some background on simplicial posets; see Stanley \cite{Sta91}
for more detail and explanations for assertions not justified here.
A {\em simplicial poset}\/~$P$ is a
finite poset with a minimal element~$\hat 0$ such that every interval
$[\hat 0,p]$ for $p \in P$ is a boolean algebra. We shall work
instead with the associated regular cell complex $\Gamma = \Gamma(P)$, whose
face poset is~$P$. The (closed) faces of~$\Gamma$ are simplices that
meet pairwise in subcomplexes of their boundaries \cite{Sta91}. For
simplicity, we identify each face $G$ of~$\Gamma$ (denoted $G \in
\Gamma$ in what follows) with the corresponding element of~$P$.
Let $S=\kk[x_G: G \in \Gamma]$ be a polynomial ring over a field $\kk$
in indeterminates indexed by the faces of $\Gamma$. The {\em face ring}\/
of~$\Gamma$ is the quotient $A_\Gamma = S/I_\Gamma$, where
$
I_\Gamma = \Big\.
$
Here the summation runs over the minimal faces $F$ among those containing
both $G$ and $G'$, and the {\em meet}\/ $G \wedge G'$
is the largest face in $\Gamma$ that is
contained in both $G$ and~$G'$; its uniqueness (when the sum is nonzero)
follows from the fact that $G$ and $G'$ lie in the boolean algebra
$[\hat 0,F]$ for any common upper bound~$F$ in the sum.
Write $f_i = f_i(\Gamma)$ for the number of
faces of dimension $i$ in~$\Gamma$, and set $f_{-1} = 1$.
Letting~\mbox{$d-1$} be the dimension of~$\Gamma$, one has an equivalent
encoding of the {\em $f$-vector}\/ $(f_{-1},f_0,f_1,\ldots,f_{d-1})$
via the {\em $h$-vector}\ $(h_0,h_1,\ldots,h_d)$, whose entries are
uniquely defined by the equation
$$
\sum_{i=0}^d f_{i-1}(t-1)^{d-i} = \sum_{i=0}^d h_i t^{d-i}.
$$
Stanley \cite{Sta91} completely characterized the possible $f$-vectors (or
$h$-vectors) when $\Gamma$ is {\em Cohen-Macaulay} over $\kk$, and almost characterized
the possible $f$-vectors when $\Gamma$ satisfies the
stronger condition of being {\em Gorenstein$\!$*}\/ over $\kk$, that is, when
$\Gamma$ triangulates a $\kk$-homology sphere. When $\Gamma$ is Cohen-Macaulay,
$(h_0,h_1,\ldots,h_d)$ can be interpreted
as the Hilbert function of the quotient ring
$A_\Gamma/\Theta$, where $\Theta$ is the ideal generated by
any linear system of parameters $\theta_1,\ldots,\theta_d$.
Consequently $h_i \geq 0$ for all $i = 0,\ldots,d$, and this nonnegativity
is sufficient to characterize these $h$-vectors \cite[Theorem 3.10]{Sta91}.
The stronger Gorenstein$\!$* property further implies that
$A_\Gamma/\Theta$ will be a Poincar\'e duality algebra,
and hence $h_i = h_{d-i}$ for all $i$. This almost characterizes such $h$-vectors,
as shown by the following theorem of Masuda, conjectured by Stanley.
\begin{thm}\cite[Corollary 1.2]{masuda}, \cite[Remark 5]{Sta91}
Let $(h_0,\ldots,h_d) \in \NN^{d+1}$ satisfy $h_i = 1$ and $h_i =
h_{d-i}$ for all $i$. Then there is a Gorenstein$\!$* simplicial poset $P$
of rank $d$ with $h_i = h_i(\Gamma(P))$ if and only if either
\begin{enumerate}
\item[$\bullet$] $h_i > 0$ for all $i = 0,\ldots,d$, or else
\item[$\bullet$] $\sum_{i=0}^d h_i$ is even.
\end{enumerate}
\end{thm}
Stanley proved half of this theorem, by showing that the
above conditions on $(h_0,\ldots,h_d)$ are sufficient to explicitly
construct such a Gorenstein$\!$* simplicial poset. For the other
half, since $f_{d-1} = \sum_{i=0}^d h_i$,
it only remains to show that the condition $h_i=0$ for some
$i=1,\ldots,d-1$ forces $\Gamma$ to have an even number
of $(d-1)$-dimensional faces (called {\em facets}). In fact, Masuda shows
(see discussion following \cite[Eq. (5.1)]{masuda})
that the assumption $h_i=0$ implies the following stronger property.
\begin{thm} \cite{masuda}
\label{even}
If\/ $\Gamma$ is Gorenstein$\!$* and $h_i(\Gamma) = 0$ for some $i$
strictly between $0$ and $d$, then for every subset
$V=\{v_1,\ldots,v_d\}$ of vertices,
% there are an number of facets in $\Gamma$ having $V$ as their vertex
% set.
the number of facets in $\Gamma$ having vertex set~$V$ is even.
\end{thm}
\begin{proof}
Since the quotient $A_\Gamma/\Theta$ is a Poincar\'e duality algebra,
one has $h_d=1$, that is, its degree~$d$ piece (its {\em socle}\/) is
a $1$-dimensional vector space over~$\kk$. Using the relations in
$I_\Gamma$, one can see that the product $x_{v_1} \cdots x_{v_d}$ in
$A_\Gamma$ is congruent to the sum $\sum_{F} x_F$ as $F$ ranges over
all facets having $V$ as their vertex set. If $h_i = 0$, then as
observed in \cite{masuda}, the image of $x_{v_1} \cdots x_{v_d}$
modulo~$\Theta$ will be zero, because it has factors (such as $x_{v_1}
\cdots x_{v_i}$) lying in the vanishing $i^{\rm th}$-graded component
$(A_\Gamma/\Theta)_i$. Hence the image of $\sum_{F} x_F$
modulo~$\Theta$ must be zero in $(A_\Gamma/\Theta)_i$. Since this sum
takes place in the socle of $A_\Gamma/\Theta$, it is therefore enough
to prove the following general claims:
\begin{enumerate}
\item[\textsf{(i)}]~the images modulo $\Theta$ of the variables~$x_F$
for facets~$F$ containing the vertices $v_1, \ldots, v_d$ are all the
same up to $\pm$ sign in the $1$-dimensional socle, and
\item[\textsf{(ii)}]~these images $\bar x_F$ are all nonzero.
\end{enumerate}
Indeed, since the sum $\sum_{F} x_F$ modulo~$\Theta$ is zero in the
socle, \textsf{(i)} and \textsf{(ii)} would imply that there must be
an even number of terms in the sum.
Claims~\textsf{(i)} and \textsf{(ii)} follow from
Proposition~\ref{nonzero} and
%% ~\ref{ridge} along with
Corollary~\ref{pm}, below, since $\Gamma$ being a $\kk$-homology
sphere implies that it is also a pseudomanifold.
\end{proof}
The rest of this note proves results implying Claims~\textsf{(i)}
and~\textsf{(ii)}. We do not assume that $\Gamma$ is Gorenstein*
anywhere in what follows, unless explicitly stated otherwise.
% , but we use the term {\em facet}\/ to mean top-dimensional face
% (rather than the more usual `maximal face').
For a monomial $m = x_{G_1} \cdots x_{G_r}$ in~$A_\Gamma$, define $m$
to be {\em standard}\/ if $G_1 \subseteq \cdots \subseteq G_r$ is a
(weak) chain in~$\Gamma$. The following is a slight strengthening of
Stanley's observation \cite[Lemma~3.9]{Sta91} that $A_\Gamma$ is
integral over its subalgebra generated by $(A_\Gamma)_1$.
\begin{lemma} \label{module}
The variables~$x_G$ for $G \in \Gamma$ generate $A_\Gamma$ as a\/ {\em
module}\/ over the subalgebra of~$A_\Gamma$ generated by
$(A_\Gamma)_1$. In fact, $A_P$ is spanned $\kk$-linearly by monomials
$m x_G$ in which $m$ is a monomial in the variables $x_v$ for the
vertices~$v$ of $G$.
\end{lemma}
\begin{proof}
Every element of~$A_\Gamma$ is a sum of standard monomials by
\cite[Lemma~3.4]{Sta91}. Let \mbox{$x_{G_1} \cdots x_{G_r}$} be a
standard monomial, so \mbox{$G_1 \subseteq \cdots \subseteq G_r$}.
For each face $G \in \Gamma$, denote by $x^G$ the product of all
variables $x_v$ for vertices $v$ of~$G$. Now, for every index $i <
r$, use the defining relations of~$A_\Gamma$ to replace $x_{G_i}$ with
$x^{G_i} - \sum x_{G'}$, the sum being over all minimal faces $G' \neq
G_i$ containing the vertices of~$G_i$. Observe that $x_{G'} x_{G_r} =
0$ in~$A_\Gamma$ for the faces~$G'$ in the sum, because no face
contains both $G_r$ and a face other than~$G_i$ with the same vertices
as~$G_i$. Hence $x_{G_1} \cdots x_{G_r}$ is equal to $x_{G_r}$ times
a monomial in the variables $x_v$ for vertices $v$ of $G$.%
\end{proof}
Abusing notation slightly, let $\Theta$ denote a linear system of
parameters $\theta_1,\ldots,\theta_d$ for $A_\Gamma$, and
$\kk[\Theta]$ the polynomial subalgebra of $A_\Gamma$ that they
generate. After choosing an ordering on the vertices of~$\Gamma$, one
can express $\Theta$ as a $d \times n$ matrix whose rows are
$\theta_1,\ldots,\theta_d$. As observed by Masuda \cite[Lemma
3.1]{masuda}, given any facet $F$ of $\Gamma$, one can compose the
finite extension $\kk[\Theta] \into A_\Gamma$ with the surjection
$A_\Gamma \onto A_{[\hat 0, F]}$ that sends all variables $x_G$ for $G
\not\subset F$ to zero. Because the composite \mbox{$\kk[\Theta]
\rightarrow A_{[\hat 0, F]}$} must also be finite, and since $A_{[\hat
0, F]}$ is a polynomial ring on the variables $\{ x_v\}_{v_\in F}$,
the $d \times d$ submatrix $\Theta_F$ of~$\Theta$ with columns indexed
by vertices in~$F$ has nonzero determinant $\det(\Theta_F)$. For any
$y \in A_\Gamma$, denote by $\bar y$ the image of~$y$ \mbox{in the
quotient ring $A_\Gamma/\Theta$}.
% In particular, if $\Gamma$ is Gorenstein* and $F$ is a facet, then
% $\bar x_F$ lies in the socle of~$A_\Gamma/\Theta$.
\begin{lemma} \label{algebraic-moving-lemma}(cf. \cite[\S 5.2 Lemma, p. 107]{fulton})
Suppose that\/ $\Gamma$ is {\em pure}, meaning that its facets all
have dimension~$d-1$. Let $m$ be a monomial in the variables $x_v$
for vertices $v$ in a facet~$F$. Then, for any face~$G$ of~$F$, the
image $\bar m \bar x_G$ of~$mx_G$ in $A_\Gamma/\Theta$ equals a sum of
terms $\bar m' \bar x_G$ in which each monomial $m'$ is a product of
variables $x_v$ for distinct vertices $v$ outside of~$G$.
\end{lemma}
\begin{proof}
Invertibility of $\Theta_F$ implies that for any vertex $v$ of $F$,
the $\kk$-span of $\theta_1,\ldots,\theta_d$ contains a linear form
$\theta'= x_v + \sum_{w \not\in F} c_w x_w$ for some constants
$c_w \in \kk$.
We first find a sum as in the lemma in which each $m'$ is only
squarefree (but may involve vertices of $G$), by induction on the sum
of all exponents $\geq 2$ on variables~$x_v$ in~$m$. Suppose $x_v^2$
divides~$m$, and write $m = x_v \ell$. Use the linear form~$\theta'$
to write
$$
\bar m \bar x_G = -\sum_{w \not\in F} c_w \bar x_w \bar\ell
\bar x_G.
$$
If $\bar x_w \bar\ell \bar x_G \neq 0$, then some facet~$F'$
% (perhaps different from~$F$)
containing~$G$ also contains all vertices appearing in $x_w \ell$. By
induction, $\bar x_w \bar\ell \bar x_G$ can be rewritten as~desired.
Now assume that $m$ is squarefree, and use a similar argument, this
time by induction on the number of variables $x_v$ dividing $m$ for
vertices $v \in G$. The fact that each $w$ in~$\theta'$ is not in~$F$
ensures that we re-create the squarefree hypothesis at each stage.
\end{proof}
\begin{prop} \label{nonzero}
If\/ $\Gamma$ is pure, the images in $A_\Gamma/\Theta$ of the
variables $x_G$ for faces $G \in \Gamma$ span $\kk$-linearly. In
particular, if\/~$\Gamma$ is Gorenstein$\!$*, then $\bar x_F \neq 0$ in
$A_\Gamma/\Theta$ for some facet~$F$.
\end{prop}
\begin{proof}
By Lemmas~\ref{module} and~\ref{algebraic-moving-lemma}, every element
of~$A_\Gamma$ can be expressed mod~$\Theta$ as a sum of monomials of
the form~$m' x_{G'}$, where $m'$ is a product of variables~$x_v$ for
distinct vertices $v \not\in G'$. But in~$A_\Gamma$, such a monomial
$m' x_{G'}$ equals the sum of the variables~$x_G$ as $G$ runs over
all faces minimal with respect to the property that they
contain both $G'$ and all vertices~$v$ for which $x_v$
divides~$m'$.%
\end{proof}
Say that two facets $F, F'$ in a pure $(d-1)$-dimensional complex {\em
share a thin ridge}\/ if their intersection is a $(d-2)$-face
contained in no other $(d-1)$-faces.
\begin{prop} \label{ridge}
When $\Gamma$ is pure, any two facets $F, F'$ sharing a thin ridge will
have $\bar x_F = \pm
\frac{\det(\Theta_{F'})}{\det(\Theta_F)} \bar x_{F'}$ in $A_\Gamma/\Theta$.
\end{prop}
\begin{proof}
For convenience of notation, denote $\frac{\det(\Theta_{F'})}{\det(\Theta_F)}$
by $\delta$. Let $G=F \wedge F'$, and
let $v_F, v_{F'}$ be their corresponding vertices not in~$G$.
There are two possibilities for these vertices: either $v_F =
v_{F'}$ or else $v_F \neq v_{F'}$.
When $v_F = v_{F'} = v$, we get $x_v x_G = x_F + x_{F'}$. The
argument in Lemma~\ref{algebraic-moving-lemma} shows that,
modulo~$\Theta$, we can write~$\bar x_v$ as a linear combination of
degree~$1$ elements~$\bar x_v$ for vertices~$v$ not in~$F$. Since $G$
lies only in the facets $F$ and~$F'$, this implies that $\bar{x}_F
+\bar{x}_{F'} =\bar x_v \bar x_G = 0$. Hence in this case, $\bar x_F
= -\bar x_{F'}$ (which equals $-\delta \bar x_{F'}$,
since $\Theta_{F'} = \Theta_F$).
When $v_F \neq v_{F'}$, we get $x_{v_F} x_G = x_F$, and $x_{v_{F'}}
x_G = x_{F'}$. After multiplying the matrix~$\Theta$ on the left by
$\Theta_F^{-1}$, one of the rows gives (by Cramer's rule)
a linear form $x_{v_F} \mp \delta x_{v_{F'}} + \sum c_v x_v$
in the ideal $\Theta$,
where the sum is over all vertices
$v \neq v_{F'}$ that do not lie in~$F$, and the $c_v$ are
constants in $\kk$. Hence
$$
%% \label{equation-string}
\begin{array}{rcccccccl}
\displaystyle \bar x_F
&=&
\displaystyle \bar x_{v_F} \bar x_G
&=&
\displaystyle\Big(\hbox{$\pm$} \delta x_{v_{F'}} -
\sum c_v x_v\Big) \bar x_G
&=&
\displaystyle \pm \delta \bar x_{v_{F'}} \bar x_G
&=&
\displaystyle \pm \delta \bar x_{F'},
\end{array}
$$
where the third equality holds
because $x_v x_G = 0$ whenever $v$ does not lie in $F \cup F'$.
\end{proof}
\begin{cor} \label{pm}
Assume $\Gamma$ is pure and $F, F'$ are facets with
the same set of vertices that are connected by
a sequence $F = F_1,\ldots,F_r = F'$ in which $F_j$ and $F_{j-1}$
share a thin ridge for \mbox{$2 \leq j \leq r$}. Then
$\bar x_F = \pm \bar x_{F'}$.
\end{cor}
\begin{proof}
Use Proposition~\ref{ridge}: the product of ratios of determinants
telescopes to~$\pm 1$.%
\end{proof}
\begin{example}
Let $\Gamma$ be obtained by slitting a hollow tetrahedron along a
single edge, and attaching a ``pita pocket'' of two triangles sewn
together along two of their common edges. Suppose that the two pita
triangles have vertices $123$, while the slit tetrahedron has vertices
$2345$. Take a path from from the top pita triangle to the bottom
pita triangle by traversing the facets with vertices $234$, then
$345$, then $235$, and finally back to the bottom pita triangle.
Writing $[ijk] = \det(\Theta_F)$ when $F$ has vertex set $ijk$ (these
are {\em Pl\"ucker coordinates}\/), the sequence of ratios of
determinants is $[234]/[123]$, then $[345]/[234]$, then
$-[235]/[345]$, and finally $[123]/[235]$. Note that the product of
all these is $-1$, which is also the sign obtained by flipping from
the top pita triangle to the bottom one along one of the two
codimension~$1$ faces they share.
\end{example}
%\bibliographystyle{amsalpha}\bibliography{biblio}\end{document}%%%
\begin{thebibliography}{Mas03}
\bibitem[Ful93]{fulton}
W. Fulton,
Introduction to toric varieties.
\emph{Annals of Mathematics Studies} \textbf{131}.
Princeton University Press, Princeton, NJ, 1993.
\bibitem[Mas03]{masuda}
Mikiya Masuda, \emph{$h$-vectors of Gorenstein$\!$* simplicial
posets}, 2003. \textsf{arXiv:math.CO/0305203}
\bibitem[Sta91]{Sta91}
Richard~P. Stanley, \emph{{$f$}-vectors and {$h$}-vectors of
simplicial posets}, J. Pure Appl. Algebra \textbf{71} (1991),
no.~2-3, 319--331.
% \bibitem[Sta96]{Sta}
% Richard~P. Stanley, \emph{Combinatorics and commutative algebra},
% second ed., Progress in Mathematics, vol.~41, Birkh{\"{a}}user
% Boston Inc., Boston, MA, 1996.
\end{thebibliography}
\end{document}