\section{Preliminaries}
\label{sec:pre}

Let $m$ be a positive integer and set $E = \{1, 2,\dots,m\}$. An
(abstract) \emph{simplicial complex} on the vertex set $E$ is a
collection $ \Delta$ of subsets of $E$ such that (i) all singletons
$\{i\}$ with $i \in E$ belong to $ \Delta$ and (ii) $\sigma \subset
\tau \in  \Delta$ implies $\sigma \in  \Delta$. The elements of $ \Delta$ are
called \emph{faces} and those maximal with respect to inclusion
are called \emph{facets}. The dimension of a face $\sigma$ is
defined as one less than the cardinality of $\sigma$. The
dimension of $ \Delta$ is the maximum dimension of a face. The complex
$ \Delta$ is called \emph{pure} if all facets of $ \Delta$ have the same
dimension. Any abstract simplicial complex $\Delta$ has a
geometric realization, which is unique up to linear homeomorphism.
When we refer to a topological property of $ \Delta$, we mean the
corresponding property of the geometric realization of $ \Delta$.

For any subset $\rho$ of $E$, we denote by $x_\rho$ the
square-free monomial in the polynomial ring $k[x_1,\dots,x_m]$
with support $\rho$. The ideal $I_ \Delta$ of $k[x_1,\dots,x_m]$ which
is generated by the square-free monomials $x_\rho$ with $\rho
\notin  \Delta$ is called the \emph{Stanley-Reisner ideal} of $ \Delta$. The
\emph{face ring}, or \emph{Stanley-Reisner ring}, of $ \Delta$ over $k$, 
denoted $k[ \Delta]$, is
defined as the quotient ring of $k[x_1,\dots,x_m]$ by the ideal
$I_ \Delta$. Given a face $\sigma$ of $ \Delta$ of dimension at least $1$,
the \emph{stellar subdivision} of $ \Delta$ on $\sigma$ is the
simplicial complex $ \Delta_\sigma$ on the vertex set $E \cup \{m+1\}$
obtained from $ \Delta$ by removing all faces containing $\sigma$ and
adding all sets of the form $\tau \cup \{m+1\}$, where $\tau \in
 \Delta$ does not contain $\sigma$ and $\tau \cup \sigma \in  \Delta$. The
complex $ \Delta_\sigma$ is homeomorphic to $ \Delta$. We denote by $J{_\sigma}$
the ideal $(0:(x_{\sigma}))$ of $k[ \Delta]$, in other words
\[
    J{_\sigma} = \{ y \in k[\Delta] \colon yx_{\sigma}= 0  \}.
\]

The complex $ \Delta$ is said to be Gorenstein* (over $k$) if $k[ \Delta]$
is a Gorenstein ring and $ \Delta$ is non-acyclic (over $k$). It is
known \cite[Section II.5]{Sta} that $ \Delta$ is Gorenstein* if and
only if for any $\sigma \in  \Delta$ (including the empty face) we
have
\begin{equation}
\widetilde{H}_i ({\rm lk}_\Delta (\sigma), k) \ \cong \
\begin{cases} k, & \text{if $i = \dim ({\rm lk}_\Delta (\sigma))$} \\
0, & \text{otherwise,} \end{cases}
\label{eq:gorenstein}
\end{equation}
where ${\rm lk}_\Delta (\sigma) = \{\tau \smallsetminus \sigma: \tau
\in \Delta, \, \sigma \subset \tau\}$ is the link of $\sigma$ in
$ \Delta$ and $\widetilde{H}_* ({\rm lk}_\Delta (\sigma), k)$  denotes
simplicial homology of ${\rm lk}_\Delta (\sigma)$ with coefficients in 
the field $k$. By \cite[Corollary~5.1.5]{BH}, any Gorenstein* 
complex $ \Delta$ is pure. It follows from (\ref{eq:gorenstein}) that the
Gorenstein* property is inherited by links. In particular, any
codimension $1$ face of $ \Delta$ is contained in exactly $2$ facets of
$\Delta$. The class of Gorenstein* complexes includes all
triangulations of spheres.


Assume $R$ is a polynomial ring over a field $k$
with the degrees of all variables positive, and $M$ is a finitely 
generated graded $R$-module.  Let  
\[
      0 \to F_{g} \to F_{g-1} \to \dots  \to F_1 \to F_0 \to M\to 0
\] 
be the minimal graded free resolution of $M$ as $R$-module.   Write 
\[   
         F_i = \oplus_{j} R(-j)^{b_{ij}},
\]
then $b_{ij}$  is called the \emph{$ij$-th graded Betti number} of $M$, 
and we also denote it by $b_{ij}(M)$. For more details 
about free resolutions and Betti numbers 
see, for example, \cite[Sections~19, 20] {Ei}.

Assume $R$ is a ring. An element $r \in R$ will be called \emph{$R$-regular} if
the multiplication by $r$ map $R \to R, u \mapsto ru$ is injective. A sequence 
$r_1, \dots ,r_n$ of elements of
$R$ will be called a \emph{regular $R$-sequence} if $r_1$ is $R$-regular,   and,
for $2 \leq i \leq n$, we have that  $r_i$ is $R/(r_1, \dots ,r_{i-1})$-regular.

\section {Proof of Theorem \ref{thm!mainalgebrathm} }
\label{sec!proof}

In this section, $ \Delta$ denotes an $(n-1)$-dimensional simplicial complex
on the vertex set $\{1, 2,\dots,m\}$.

\begin{remark}    \label {rem!idealquotientismonomial}
{\rm We will use the fact that $k[ \Delta]$ has no nonzero nilpotent elements
and that if $I_1, I_2$ are monomial ideals of $k[ \Delta]$, then so is the
ideal quotient
\[ (I_1 : I_2) \ = \ \{ y \in k[ \Delta] \colon  y I_2 \subset I_1 \}. \]
}
\end{remark}

\begin{remark}  \label {rem!separatednessproperty}
{\rm Assume that $ \Delta$ is Gorenstein*. If $e$ is a vertex of $ \Delta$ and $\sigma
\in  \Delta$ is a face that does not contain $e$, then there exists a facet
of $ \Delta$ that contains $\sigma$ but not $e$.
Indeed, let $\tau_1$ be a facet of $ \Delta$ containing $\sigma$. If $\tau_1$
contains $e$, then there exists a facet $\tau_2$ distinct from $\tau_1$
containing $\tau_1 \setminus  \{e\}$. This facet contains $\sigma$ and
does not contain $e$. }
\end {remark}



\begin{proposition}  \label {prop!technicalheart}
Let $ \Delta$ be a Gorenstein* simplicial complex on the vertex set $\{1,
2,\dots,m\}$ and let $\sigma$ be a face of $ \Delta$ of dimension at least $1$.
The ideal $J_{\sigma}$ is a
codimension $0$ ideal of $k[ \Delta]$ and the quotient $k[ \Delta]/J_{\sigma}$ is Gorenstein.
Moreover,
    \[ (0  : J_{\sigma} ) =  (x_\sigma). \]
\end{proposition}
\begin{proof}
The first claim is well-known, cf. \cite[Theorem 21.23]{Ei}, and
the second follows from the observation that $k[ \Delta] / J_{\sigma}  =  k[x_1,\dots,x_m]
/ I$, where $I$ is the Stanley-Reisner ideal of ${\rm lk}_ \Delta (\sigma)$, and the
fact that ${\rm lk}_ \Delta (\sigma)$ is also Gorenstein*.

We now prove that $(0 : J_{\sigma})
=  (x_\sigma)$. It is clear that $(x_\sigma) \subset (0 :  J_{\sigma})$. Since $(0  :
J_{\sigma})$ is a monomial ideal (Remark \ref{rem!idealquotientismonomial}), it
suffices to show that for any nonzero monomial $u \in (0 : J_{\sigma})$ we have $u
\in (x_\sigma)$. Let $\rho \in  \Delta$ be the support of $u$. By the way of
contradiction, suppose that $u$ is not in $(x_\sigma)$, so we may choose $i
\in (\sigma
\setminus \rho)$. By Remark~\ref{rem!separatednessproperty}, there exists a
facet $\tau$ of $ \Delta$ which does not contain $i$ and contains $\rho$. Since
$i$ is not in $\tau$ and $\tau$ is a facet, we have $x_i x_\tau = 0$ in
$k[ \Delta]$ and hence $x_\tau \in J_{\sigma}$. This fact and the assumption $u \in
(0 : J_{\sigma})$ imply that $x_\tau u  = 0$ in $k[ \Delta]$.
Since each variable which appears in $u$ also appears in $x_\tau$, we
conclude that $x_\tau$ is a nonzero nilpotent element of $k[ \Delta]$. This
contradicts Remark~\ref{rem!idealquotientismonomial} and completes the
proof of the proposition. \medskip
\end {proof}


\begin {remark}
{\rm The conclusion of Proposition~\ref{prop!technicalheart} is not true
under the weaker hypothesis that $k[ \Delta]$ is Gorenstein. For a counterexample
consider
\[   \Delta =\{\{1,2\},\{1,3\}, \{1\}, \{2\}, \{3\} \} \]
and  $\sigma=\{1,2 \}$. We have $k[ \Delta] =  k[x_1,x_2,x_3] /
(x_2 x_3)$, $J_{\sigma} = (0:x_1x_2) =(x_3)$, but  $(0 : J_{\sigma}) = (x_2)$.   
We believe that this is also  a counterexample to 
the second claim of Part a) of \cite[Theorem 21.23]{Ei},  this is the reason
we did not use this claim in the proof of Proposition~\ref{prop!technicalheart}.
} 
\end {remark}


Let $\sigma \in  \Delta$ be a face of dimension $d-1$ for some $d \ge 2$. We
recall that the stellar subdivision $ \Delta_\sigma$ of $ \Delta$ on $\sigma$ is
a simplicial complex on the vertex set $\{1, 2,\dots,m+1\}$.
We will use the (easy) fact that
\begin {equation}  \label{eqn!aboutstellar}
        k[ \Delta_\sigma] \cong  \frac{ k[x_1,\dots,x_{m+1}] } {(I_ \Delta, x_\sigma,
               x_{m+1} u_1,\dots,x_{m+1}u_r)},
\end {equation}
where $ \{u_1,\dots,u_r\}$ is a generating set of monomials for the ideal
$J_{\sigma}$ of $k[ \Delta]$.

\bigskip
\noindent
\emph{Proof of Theorem \ref{thm!mainalgebrathm}.}
Clearly there exists a unique element $\phi_{\sigma}$ of $M$ satisfying  
$\phi_{\sigma}(z) = x_\sigma$ and $\phi_{\sigma}(u) = 0$ for $u \in J_{\sigma}$. 
Given $f \in M$, we write $f(z)
= w_1 z + w_2$ with $w_1 \in  k[ \Delta][z]$ and $w_2 \in k[ \Delta]$ and set $g
= f -w_1 i \in M$, so that $g(z) = w_2$. For $u \in J_{\sigma}$ we have
\[ z g(u) = g(zu) = u g(z) = uw_2 \in k[ \Delta]. \]
Hence $g(u) = 0$ for all $u \in J_{\sigma}$, which implies  $w_2 \in (0 :
J_{\sigma})$. By Proposition~\ref{prop!technicalheart} we have $(0 : J_{\sigma})  =
(x_\sigma)$. As a consequence, there exist $w \in k[ \Delta]$ such that
$w_2 = w x_\sigma$ and hence $g = w \phi_{\sigma}$. This proves part (a) of the
theorem.

By Proposition~\ref{prop!technicalheart}, the ring $k[ \Delta]/J_{\sigma}$ is
Gorenstein of the same dimension as $k[ \Delta]$. Therefore $(J_{\sigma},z)$ is a
codimension $1$ homogeneous ideal of the graded Gorenstein ring
$k[ \Delta][z]$, so the general theory of \cite{PR} applies.
Using part (a) we get
\[
         S \cong  \frac{ k[x_1,\dots ,x_{m+1},z] }
        {(I_ \Delta, x_{m+1}z-x_\sigma, x_{m+1} u_1,\dots,x_{m+1} u_r)},
\]
where the new variable $x_{m+1}$ has degree equal to $1$. It follows from
(\ref{eqn!aboutstellar}) that $S/(z) \cong k[ \Delta_\sigma]$. By \cite[Theorem
1.5]{PR}, $S$ is Gorenstein of dimension equal to the dimension of
$k[ \Delta][z]$. As a consequence $\dim S/(z) = \dim S -1$ and therefore $z$
is an $S$-regular element. This completes the proof of the theorem.
\qed \medskip


\section {Toric face ring interpretation}  \label {sec!toric_face_ring_interpretion}

Is it clear that  Theorem~\ref{thm!mainalgebrathm}  is equivalent to the following  theorem.

\begin{theorem} \label {thm!mainthm_combinatorialformulation}
Suppose that $ \Delta$ is a Gorenstein* simplicial complex and that
$\sigma \in  \Delta$ is a face of dimension $d-1$ for some $d \ge 2$.
Let $z_1, \dots ,z_{d-1}$ be $d-1$ new variables of degree $1$ and set $M_1 =
\Hom_{k[ \Delta][z_1, \dots ,z_{d-1}]} ((J_{\sigma}, z_1z_2 \cdots z_{d-1}),$
$k[ \Delta][z_1, \dots ,z_{d-1}])$.
\begin{enumerate}
\itemsep=0pt
\item[{\rm (a)}] $M_1$ is generated as a $k[ \Delta][z_1, \dots ,z_{d-1}]$-module by the
elements $i$ and $\phi_{\sigma}$, where $i \colon  (J_{\sigma}, z_1z_2 \cdots z_{d-1}) \to k[ \Delta][z_1, \dots ,z_{d-1}]$ is the
natural inclusion morphism, and $\phi_{\sigma}$ is uniquely specified by $\phi_{\sigma}(z_1z_2 \cdots z_{d-1}) =
x_\sigma$ and  $\phi_{\sigma}(u) = 0 $ for $u \in J_{\sigma}$.
\item[{\rm (b)}] Denote by $S_1$ the Kustin--Miller unprojection ring
of the pair \newline  $(J_{\sigma}, z_1z_2 \cdots z_{d-1}) \subset k[ \Delta][z_1, \dots ,z_{d-1}]$. Then 
$z_1,z_2, \dots ,z_{d-1}$ is an $S_1$-regular sequence,
and $k[ \Delta_\sigma]$ is isomorphic to $S_1/(z_1,z_2, \dots ,z_{d-1})$ as a $k$-algebra.
\end{enumerate}
\end{theorem}

We remark that, unlike in Theorem~\ref{thm!mainalgebrathm}, in 
Theorem~\ref {thm!mainthm_combinatorialformulation}  all variables have degree $1$ which
is the usual grading in the theory of Stanley--Reisner rings. 
Compare also \cite[Section~4]{BPcyclic},  where a similar product  $z_1z_2$ appears in a natural way 
when relating unprojection and cyclic polytopes.


Consider the  Kustin--Miller unprojection ring
\[
    S_1  =      \frac{ k[x_1,\dots ,x_{m+1},z_1, \dots ,z_{d-1}] }
        {(I_ \Delta, x_{m+1}z_1 \cdots z_{d-1}-x_\sigma, x_{m+1} u_1,\dots,x_{m+1} u_r)}
\]
appearing in Theorem~\ref {thm!mainthm_combinatorialformulation}, where as in 
Section~\ref{sec!proof}   $ \{u_1,\dots,u_r\}$ denotes a generating set of monomials for the ideal
$J_{\sigma} = (0 :  x_{\sigma})$ of $k[ \Delta]$. 
We will now give a combinatorial interpretation of $S_1$ using the notion
of toric face rings as defined by Stanley  in  \cite[p.~202]{Sta2}, 
compare also  \cite [Section 4] {BR} and \cite{BKR}. 
Let $M$ be a free $\mathbb Z$-module of rank $m+d-1$, and consider the ${\mathbb R}$-vector space 
$M_{{\mathbb R}} = M \otimes_{{\mathbb Z}} {\mathbb R}$.
We will define a (finite, pointed) rational polyhedral fan ${\mathcal F}$ in $M_{{\mathbb R}}$, such that 
$S_1$ is isomorphic to the toric face ring $k[{\mathcal F}]$.   For simplicity of 
notation we assume in the following that $\sigma = \{1,2, \dots ,d \}$.

Denote by $
    e_{x,1}, \dots ,e_{x,m},  e_{z,1}, \dots , e_{z,d-1} $
a fixed ${\mathbb Z}$-basis of $M$,  and set
\[ 
  e_{a} = (e_{x,1}+ \dots + e_{x,d}) - (e_{z,1}+ \dots + e_{z,d-1}) \in M.
\]
Assume $\tau=\{a_1, \dots ,a_p\}$ is
a face of $\Delta$. If  $\sigma$ is not a face of 
$\tau$ we set 
$c_{\tau}$ to be the cone in $M_{{\mathbb R}}$ spanned by the basis vectors
\[
    e_{x,a_1}, \dots ,   e_{x,a_p},   e_{z,1},  \dots , e_{z,d-1},
\]
while if $\sigma$ is a face of $\tau$ we set 
$c_{\tau}$ to be the cone in $M_{{\mathbb R}}$ spanned by the (non-affinely independent) vectors
\[
    e_{x,a_1}, \dots ,   e_{x,a_p},   e_{z,1},  \dots , e_{z,d-1},  e_{a}.
\]
It is  easy to see that the collection of cones  $\{ c_{\tau}  \bigm|  \tau \text { face of } \Delta \}$
together with their faces form a fan  ${\mathcal F}$ in ${M_{{\mathbb R}}}$ and that the toric face ring $k[{\mathcal{F}}]$
is isomorphic as a $k$-algebra to  $S_1$.




\begin{example}  \label {example_with_picture}
Consider the example given after the statement of
Theorem~\ref{thm!mainalgebrathm}. That is, let $\Delta$ be the boundary of a
triangle with vertices corresponding to the variables $x_{1},x_{2},x_{3}$, and
denote by $\Delta_{\sigma}$ the stellar subdivision of $\Delta$ with respect
to the face  $x_{1}x_{2}$. We embed the fan $\mathcal{F}$
into $\mathbb{R}^{3}$ by assigning to the variables $x_{1},x_{2},x_{3},x_{4}$
the rays generated by $\left(  1,0,0\right)  $, $\left(  0,1,0\right)  $,
$\left(  -1,-1,-1\right)  $, $\left(  0,0,1\right)  \in\mathbb{Z}^{3}$, i.e.,
those of the standard fan of $\mathbb{P}^{3}$ as a toric variety. Then the ray
associated to $z$ is generated by $\left(  1,1,-1\right)  $. The right hand side
of Figure~\ref{elliptic curve unprojection} visualizes the Kustin-Miller
unprojection ring $S\cong k\left[  \mathcal{F}\right]  $ via representing each
cone of the embedded fan $\mathcal{F}$ by a polytope spanning it.
There are $3$ polytopes of maximal dimension, spanned by 
$\{x_1,x_3,z\}, \{x_2,x_3,z\}$ and $\{x_1,x_4,x_2,z\}$. Notice that
subdividing the cone corresponding to $x_{1},x_{4},x_{2},z$ into $x_{1}%
,x_{4},z$ and $x_{4},x_{2},z$ amounts to passing from $S$ to the polynomial
ring in the variable $z$ over $k\left[  \Delta_{\sigma}\right]  $.
\end{example}



\begin{figure}
[h]
\begin{center}
\includegraphics[
height=2.495in,
width=4.1597in
]%
{fans.eps}%
\caption{Unprojection via toric face rings}%
\label{elliptic curve unprojection}%
\end{center}
\end{figure}



\section {Application to stacked polytopes}  \label {sec!application_to_stacked_polytopes}

\subsection {The Kustin--Miller complex construction }  \label{subs!generalities_about_KM_complexes}

The following construction, which is due to Kustin and Miller  \cite {KM},
will be important in the applications to stacked polytopes
contained in Subsection~\ref {subs!minimal_resol_for_stacked}.

Assume $R$ is a polynomial ring over a field  with the degrees of all variables positive,
 and $I \subset J \subset R$ are two 
homogeneous ideals of $R$ such
that both  quotient rings $R/I$ and $R/J$ are Gorenstein and $\dim R/J = \dim R/I -1$. 
We define $k_1,k_2 \in \mathbb Z$ such that $\omega_{R/I} = R/I(k_1)$   and  
$\omega_{R/J} = R/J(k_2)$, compare~\cite[Proposition~3.6.11]{BH}, and assume that $k_1 > k_2$.  Moreover, let 
\[
      0 \to  A_{g} \to A_{g-1}  \to  \dots  \to A_1 \to A_0 \to R/J \to 0
\] 
and
\[ 
    0 \to B_{g-1} \to  \dots  \to B_1 \to B_0 \to R/I \to 0
\]
be the minimal graded free resolutions of $R/J$ and $R/I$ respectively as 
$R$-modules.     Denote by $S = R[T]/Q$  the Kustin--Miller unprojection  ring of the pair 
$J \subset R/I$, where $T$ is a new variable of degree $k_1-k_2$.  
Kustin and Miller constructed in \cite {KM}
a graded free resolution of 
$S$ as $R[T]$-module of the form 
\[
      0 \to F_{g} \to F_{g-1} \to \dots  \to F_1 \to F_0 \to S  \to 0,
\] 
where,  when $g \geq 3$,   
\begin {eqnarray*}
  &  &  F_0 =  B_0', \quad \quad  F_1 = B_1' \oplus A_1'(k_2-k_1),  \\\ 
  &  &   F_i = B_i' \oplus A_i'(k_2-k_1) \oplus  B_{i-1}'(k_2-k_1), \quad \quad   \text{for} \; \; 2 \leq i \leq g-2, \\
  &  &  F_{g-1} = A_{g-1}'(k_2-k_1) \oplus  B_{g-2}'(k_2-k_1), \quad \quad F_{g} =  B_{g-1}'(k_2-k_1), 
\end {eqnarray*}
cf. \cite [p.~307, Equation (3)]{KM}.  When $g=2$ we have 
\[ 
     F_0 =  B_0', \quad   F_1 =  A_1'(k_2-k_1), \quad  F_2 =  B_1'(k_2-k_1).
\]
In the above expressions, for an $R$-module $M$ we denoted 
by $M'$ the $R[T]$-module $M \otimes_R R[T]$.  This resolution is, in general,
not minimal, see Example~\ref{example!nonminal_km_complex_construction} below. However, 
in the case of  stacked  and cyclic polytopes  it
is minimal, see Theorem~\ref{thm!applicationtostackedpolytopes} and \cite{BPcyclic}.
  We call the complex consisting of the $F_i$  the \emph{Kustin--Miller complex construction}.



\begin {example} {\rm  Assume $k_2-k_1 = -1$,   and that the $2$ complexes are
\[
     0 \to A_4 \to A_3 \to  A_2 \to A_1 \to A_0 \to  0
\]
and
\[
      0 \to B_3 \to B_2 \to  B_1 \to B_0 \to 0 
\]
Then, the Kustin--Miller complex construction is of the form
{ \small
\[  
  0 \to  B_3'(-1)  \to B_2'(-1) \oplus A_3'(-1) \to  B_1'(-1)  \oplus A_2'(-1) \oplus B_2' 
        \to A_1'(-1) \oplus B_1' \to B_0' \to 0
\] }  }
\end {example}

\begin {example} {\rm  \label{example!nonminal_km_complex_construction}
Let $\Delta$ be the simplicial complex with Stanley--Reisner ideal
$(x_1x_2x_3, x_4x_5)$,  $\Delta$ is just the stellar subdivision of a  facet  
of the boundary complex of the $3$-simplex.
 Then $\sigma = \{1,2\}$ is a face of $\Delta$, and one can easily check
that the Kustin--Miller complex construction gives a  graded resolution of $k[\Delta_{\sigma}]$
which is not minimal, the reason being that the Stanley-Reisner ideal of $\Delta_{\sigma}$
needs $3$ generators and not $5$.  }
\end {example}



\subsection { The minimal resolution for stacked polytopes}   \label{subs!minimal_resol_for_stacked}

Assume  $d \geq 2$ is a fixed integer. 
Recall from \cite [p.~448]{TH2}, that starting from a $d$-simplex
one can add new vertices by building shallow pyramids over facets 
to obtain a simplicial convex $d$-polytope with $m$ vertices,
called a \emph{stacked polytope} $P_d(m)$.  
We denote by  $\Delta P_d (m)$ the boundary simplicial complex of the
simplicial polytope $P_d (m)$.
By definition,  $\Delta P_d(m)$ has 
as  elements the empty set and the sets of  vertices of the proper faces of
$P_d(m)$, 
cf.~\cite[Corollary~5.2.7]{BH}.
There is a  slight abuse of notation here,  since the combinatorial type 
of $\Delta P_d(m)$  does not depend only on $d$ and $m$ but also
on the specific choices of the sequence of facets we used when
building the shallow pyramids.
The graded Betti numbers $b_{ij}$ of the Stanley-Reisner ring
$k[\Delta P_d(m)]$ have been
calculated in \cite[Theorem~1.1]{TH2}, and it turns out that they
only depend on $d$ and $m$. 

It is clear that, for $d < m$,  the simplicial complex 
$\Delta P_d(m+1)$ can be considered as the stellar subdivision of 
the boundary simplicial complex $\Delta P_d(m)$ of a 
stacked polytope $P_d(m)$ with respect to a facet $\sigma$
of $\Delta P_d(m)$.  Since $\sigma$ is a facet, the ideal
$(J_\sigma, z)$   is generated by the regular sequence
$  x_\rho,z$, where $\rho$ takes values in the set of vertices
of $\Delta P_d(m)$ which are not vertices of $\sigma$. Hence,
the minimal graded free resolution of $(J_\sigma,z)$ is a Koszul complex.
Combining Theorem~\ref{thm!mainalgebrathm}
with the Kustin--Miller complex construction described in 
Subsection~\ref{subs!generalities_about_KM_complexes}
we can get, starting with the Koszul complex and the
minimal graded free resolution of $k[\Delta P_d(m)]$, 
a  graded free resolution of $k[\Delta P_d(m+1)]$.
The following theorem, whose proof will be given in 
Subsection~\ref{subs!proofofapplictostacked}, 
 states that in this way we get a minimal graded free resolution of
$k[\Delta P_d(m+1)]$.     We remark that, when  $d=2$ or $d \geq 4$,
we do not  use  in the proof of the theorem  the calculation
of the graded Betti numbers of $k[\Delta P_d(m)]$ obtained in \cite{TH2},
and, moreover,  we recover these numbers in 
Proposition~\ref{prop!keytechnicalfor_stacked_application}.




\begin {theorem}  \label{thm!applicationtostackedpolytopes}
Assume $d \geq 2$ and $d+1  < m$. The resolution  of  $k[\Delta P_d(m+1)]$,
obtained using the Kustin--Miller complex construction starting
from the minimal graded free resolution of $k[\Delta P_d(m)]$ and the Koszul complex
resolving $(J_\sigma , z)$ is minimal.
\end {theorem}



\subsection {Proof of Theorem~\ref{thm!applicationtostackedpolytopes} }  \label{subs!proofofapplictostacked}

We need the following combinatorial definition.
Assume $d \geq 2$ and $d < m$. For  $1 \leq i \leq m-d-1$ we define 
\[
    \theta (d,m,i) = i \binom {m-d}{i+1},
\]
compare~\cite [p.~448] {TH2}.  Moreover we set $ \theta (d,m,0)  =  \theta (d,m,m-d)  = 0$.




\begin {lemma}   \label{lem!combinatorial_induction_for_stacked} {\rm (Compare 
\cite [p.~451] {TH2}).}
Assume  $1 \leq i \leq m-d$. Then
\[
     \theta (d,m+1,i) = \theta (d,m,i) +  \binom {m-d}{i}   +\theta (d,m,i-1).
\]
(By our conventions, for $i=1$ the equality becomes  $\theta (d,m+1,1) =  \theta (d,m,1) + (m-d)$,
while for $i = m-d$  it becomes  $\theta (d,m+1,m-d) =  \theta (d,m,m-d-1) + 1$).
\end {lemma}

\begin {proof} Assume first  $2 \leq i \leq m-d-1$. Using the Pascal triangle identity
$\binom {m}{d} = \binom{m-1}{d} + \binom {m-1}{d-1}$ we have
{ \small
\begin{eqnarray*}
    \theta (d,m+1,i)   &  =  &  i \binom {m+1-d}{i+1} =  i (\binom{m-d}{i+1} + \binom{m-d}{i})  \\
       &  = &    i \binom{m-d}{i+1} +  \binom{m-d}{i} + (i-1)  \binom{m-d}{i}  \\
       & = &   \theta (d,m,i) +   \binom {m-d}{i}  + \theta (d,m,i-1).
\end{eqnarray*}   }
The special cases $i=1$ and $i=m-d$ are proven by the same argument. \end {proof}
The following proposition is well-known.


\begin {prop}  \label{prop!dividing_resolutions_by_graded_elements}
{\rm (\cite [Proposition~1.1.5] {BH}).}
Assume   $R= k[x_1, \dots ,x_n]$ is a polynomial
ring over a field $k$ with the degrees of all variables positive, and $I \subset R$ a homogeneous ideal.
Moreover, assume that $x_n$ is $R/I$-regular.  Denote by $cF$ the minimal graded free resolution
of $R/I$ as $R$-module. We then have that $cF \otimes_R R/(x_n)$ is the minimal graded
free resolution of   $R/(I,x_n)$  as  $k[x_1, \dots ,x_{n-1}]$-module, where we
used the natural isomorphisms \newline  $ R \otimes_R R/(x_n) \cong R/(x_n) \cong k[x_1, \dots ,x_{n-1}]$.
\end {prop}


The proof of the following proposition is an immediate corollary of the construction
of the Koszul complex in \cite[Section~1.6]{BH}.


\begin {prop}  \label{prop!Koszul_complex_with_different_degrees}
   Assume $R= k[x_1, \dots ,x_n]$ is a polynomial ring
over a field $k$ with the degrees of all variables positive, 
$p \leq n$ a fixed integer, and $g_1, \dots ,g_p$, an $R$-regular
sequence consisting of homogeneous elements of $R$, with 
$\deg g_i = 1$, for $1 \leq i \leq p-1$, and $\deg g_p = q \geq 1$.
Then, the minimal resolution of $R/(g_1, \dots ,g_p)$ is of the form
\[
     0 \to F_p \to F_{p-1}  \dots  \to F_1 \to F_0,
\]
with  $ F_0 \cong R$, $F_p \cong R(-p-q+1)$, and 
\[
    F_i \cong  R(-i)^{b_i} \oplus R(-q-i+1)^{b_{p-i}}
\]
for $1 \leq i \leq p-1$, where   $b_i = \binom {p-1}{i}$.
\end {prop}


Theorem~\ref {thm!applicationtostackedpolytopes} will follow from the following 
more precise proposition. Notice that as we have already mentioned the statements about the 
graded Betti  numbers of $k[\Delta P_d(m)]$  have been proven before in \cite{TH2}.   For 
$d \not= 3$ we  do not  use in our proof the results of \cite{TH2}.



\begin {prop}  \label {prop!keytechnicalfor_stacked_application}
Assume $d \geq 2$ and $d+1 < m$.  Set  $b_{ij} = b_{ij}(k[\Delta P_d(m)])$.
Then the statement of Theorem~\ref {thm!applicationtostackedpolytopes} is true for $(d,m)$.
Moreover, we have that if   $d=2$ then 
$b_{ij}= 1$ for $(i,j) \in \{ (0,0), (m-d,m) \}$, 
\[
   b_{i,i+1} = \theta(d,m,i) +  \theta (d,m,m-d-i),
\]
for $1 \leq i \leq m-d-1$, and $b_{ij} = 0$ otherwise.  
If $d  \geq 3$, we have
$b_{ij}= 1$ for $(i,j) \in \{ (0,0),  (m-d,m) \}$, 
\[
     b_{i,i+1} = \theta(d,m,i), \quad   b_{i,d+i-1} = \theta(d,m,m-d-i),
\]
for $1 \leq i \leq m-d-1$, and $b_{ij} = 0$ otherwise.
\end {prop}



\begin {proof}   We fix $d \geq 2$ and  use induction on $m$. 
If $d \geq 2$ and $m = d+2$ then $k[\Delta P_d(m)]$ is a type  $(2,d)$ 
codimension $2$  complete intersection and everything is clear. 


Assume $d \not = 3$, and that Proposition~\ref{prop!keytechnicalfor_stacked_application}
is true for $(d,m)$.   By Theorem~\ref{thm!mainalgebrathm}, the
extension ring $S$ of $k[\Delta P_d(m+1)]$ is the Kustin--Miller unprojection
ring of the pair $(J_{\sigma},z) \subset k[\Delta P_d(m)][z]$. As we noticed above,
the ideal $(J_{\sigma},z)$ is generated by a regular sequence, 
so the Koszul complex described in 
Proposition~\ref{prop!Koszul_complex_with_different_degrees} 
is the minimal resolution of $k[\Delta P_d(m)][z]/(J_{\sigma}, z)$.
Combining Proposition~\ref{prop!dividing_resolutions_by_graded_elements}
and the discussion of  Subsection~\ref{subs!generalities_about_KM_complexes}, 
starting from the Koszul
complex and the minimal graded free resolution of $k[\Delta P_d(m)]$, the Kustin--Miller
complex construction gives a graded free resolution of $k[\Delta P_d(m+1)]$.
Using Lemma~\ref{lem!combinatorial_induction_for_stacked} 
this complex has the conjectured graded Betti numbers, and since there
are no degree $0$ morphisms it is necessarily minimal.

When  $d=3$  the above arguments work except for the minimality argument, 
since there are degree $0$ morphisms. But comparing the graded Betti number of the Kustin--Miller
complex construction with the graded Betti numbers of 
$k[\Delta P_d(m+1)]$ calculated
in \cite{TH2} we again obtain the minimality of the Kustin--Miller complex
construction.
\end {proof}



\section{Remarks and open questions}
\label{sec:rem}


In the following we use the notation  introduced
in Section \ref{intro} and Theorem \ref{thm!mainalgebrathm}.


\begin {remark}
{\rm It follows from Theorem~\ref{thm!mainalgebrathm} that $S$ is a $1$-parameter 
deformation ring of $k[ \Delta_\sigma]$,  compare  \cite[Exerc.~18.18]{Ei}.
 The fact that such a
deformation ring of $k[ \Delta_\sigma]$ exists is a special case of more
general results due to Altmann and Christophersen \cite{AC1,AC2}. }
\end {remark}


\begin {remark}
{\rm  Using the Kustin--Miller complex construction described in 
Subsection~\ref{subs!generalities_about_KM_complexes}, 
we can construct a graded free resolution of $S$, therefore 
using Proposition~\ref {prop!dividing_resolutions_by_graded_elements} also of $k[ \Delta_\sigma]$,
starting from graded free resolutions of $k[ \Delta]$ and $k[ \Delta]/J_{\sigma}$. In particular, it
follows that
\[ F (k[ \Delta_\sigma], t) =
F (k[ \Delta], t) + (t+t^2+ \cdots + t^{d-1}) \, F (k[ \Delta]/J_{\sigma}, t), \]
where $F (R, t)$ stands for the Hilbert series of $R$
and $d-1$ is the dimension of the face $\sigma$. 
This equality can be rewritten as
\begin{equation}
h ( \Delta_\sigma, t) = h ( \Delta, t) + (t+t^2+ \cdots + t^{d-1}) \,
h ({\rm lk}_ \Delta (\sigma), t),
\label{eq:h}
\end{equation}
where $h (\Gamma, t)$ stands for the $h$-polynomial \cite[Section II.2]{Sta}
of the simplicial complex $\Gamma$. It is not hard to see that (\ref{eq:h})
holds for any pure simplicial complex $ \Delta$. Indeed, one can check directly
that (\ref{eq:h}) is equivalent to the formula
\[ f_j ( \Delta_\sigma) =  f_j ( \Delta) - f_{j-d} ({\rm lk}_ \Delta (\sigma)) \, + \,
\sum_{i \ge 0} \  \binom{d}{j-1} f_{i-1} ({\rm lk}_ \Delta (\sigma)), \]
where $f_j (\Gamma)$ denotes the number of $j$-dimensional faces
of a complex $\Gamma$. That formula follows from the definition of
$ \Delta_\sigma$. }
\end {remark}



\begin{remark}
{\rm In \cite{NP}, Neves and the second author introduced the 
$\binom{n}{2}$ Pfaffians format, starting from a certain hypersurface ideal. 
We give a monomial interpretation
of the construction. Start with the boundary simplicial complex  $\Delta$ of the 
$(n-1)$-simplex.  Denote
by $\Delta_1$ the simplicial complex obtained by the stellar 
subdivisions of all facets of $\Delta$. 
It is easy to check that the Stanley--Reisner ideal of $\Delta_1$ is equal to
$\widetilde{I_n}$, where $\widetilde{I_n}$  denotes the ideal
obtained by substituting  $z_i=0$, for $1 \leq i \leq n$,
and $r_{d_1, \dots , d_n} = 1$, for $(d_1, \dots , d_n) \in \{ 0,1 \}^n$,
to the ideal  $I_n$ defined in \cite [Definition~2.2]{NP}.

 
Similarly, in \cite[Section~4.3]{NP2}, Neves and the second author
constructed a codimension $11$ Gorenstein ideal 
starting from a certain codimension $2$ complete intersection ideal.
The monomial interpretation of the construction is as follows.
Denote by $\Delta$ the simplicial complex which is the join 
\cite [p.~221]{BH} of $2$ copies of the boundary simplicial complex 
of the $2$-simplex. $\Delta$ has  Stanley--Reisner ideal equal 
to  $(x_{11}x_{12}x_{13}, x_{21}x_{22}x_{23})$ and  exactly
$9$ facets. Denote by $\Delta_1$ the simplicial complex 
obtained by the stellar subdivisions of $\Delta$ on these $9$ facets.
Using the notations of  \cite[Section~2]{NP2}, denote by 
$I_{\mathcal L}$ the kernel of the surjection  $R[y_u \bigm| u \in {\mathcal L}] \to R_{\mathcal L}$.
It is easy to check that the Stanley--Reisner ideal of $\Delta_1$ is equal to
$\widetilde{I_{\mathcal L}}$, where $\widetilde{I_{\mathcal L}}$  
 denotes the ideal obtained by substituting  $x_{3i}=0$, for $1 \leq i \leq 3$, to 
${I_{\mathcal L}}$.  }
\end {remark}


\begin {remark}
{\rm Theorem~\ref{thm!mainalgebrathm} suggests that, given a
Gorenstein* simplicial complex $ \Delta$, the essential information
about the relationship between the Stanley--Reisner rings of $ \Delta$
and a stellar subdivision $ \Delta_\sigma$ of $ \Delta$ is contained in the
module $\Hom_{k[ \Delta][z]} ((J_{\sigma}, z), k[ \Delta][z])$. We believe that for
many other natural combinatorial operations on simplicial
complexes, the key information about the relationship between the
corresponding Stanley--Reisner rings is contained in suitable
$\Hom$ modules over the original ring. We give below two examples,
compare also \cite{BPcyclic} for the case of cyclic polytopes.

For the first example, consider the simplicial complex $ \Delta$ with vertex
set $\{1, 2,\dots,6\}$ and facets
\[ \{1,2,3\}, \{1,3,4\}, \{1,4,5\}, \{1,5,6\}, \{1,2,6\}, \{2,3,4\},
\{2,4,5\}, \{2,5,6\} \]
and the simplicial complex $ \Delta'$ with vertex set $\{1, 2,\dots,7\}$
and facets

\medskip
\begin{tabular}{c}
\{1,2,3\}, \{1,2,6\}, \{1,3,4\}, \{1,4,5\}, \{1,5,6\}, \{2,3,7\},
\{2,6,7\}, \{3,4,7\}, \\ \{4,5,7\}, \{5,6,7\}.
\end{tabular}

\medskip
\noindent The complex $ \Delta$ is combinatorially isomorphic to the
boundary simplicial complex of a $3$-dimensional cyclic polytope with $6$
vertices. It can be obtained from $ \Delta'$ by `sliding' (moving) vertex
$7$ to vertex $2$. Set $R = k[ \Delta][z]$ and $I_1 = (x_1, z) \subset
R$, where $k[ \Delta] = k[x_1,\dots,x_6]/I_ \Delta$. It can be shown that
$\Hom_{R} (I_1,R)$ is generated as an $R$-module by the inclusion
morphism $I_1 \hookrightarrow R$ together with two elements $\phi_1, \phi_2$
defined by
\[ \phi_1(x_1)=\phi_2(x_1)=0, \quad \; \; \phi_1(z) = x_2x_4, \; \phi_2(z) =
x_2x_5. \]
Moreover, if we set
\[ I_2 = I_ \Delta + (x_1x_7, x_7z-\phi_1(z), x_1x_8, x_8z-\phi_2(z))
           \subset k[x_1, \dots ,x_8,z], \]
then we have
\[ k[ \Delta'] \cong \frac {k[x_1,\dots,x_8,z]} {(I_2, z, x_8-x_7)}. \]

\medskip
\noindent For an example in a different direction, consider the
boundary simplicial complex $ \Delta$ of the standard $3$-simplex on the vertex
set $\{1,2,3,4\}$ and the simplicial complex $ \Delta'$ with vertex set
$\{1, 2,\dots,7\}$ and facets

\medskip
\begin{tabular}{c}
\{1,2,5\}, \{1,3,5\}, \{1,3,6\}, \{1,4,6\}, \{1,4,7\}, \{1,2,7\}, 
\{2,5,7\}, \{3,5,6\}, \\ \{4,6,7\}, \{5,6,7\}.
\end{tabular}

\medskip
\noindent 
We note that $ \Delta'$ is a subdivision of $ \Delta$ which cannot 
be obtained from $ \Delta$ by a sequence of stellar subdivisions. We 
set $R=k[ \Delta][z]$, where $k[ \Delta] = k[x_1,x_2,x_3,x_4]/I_ \Delta$ and, 
for $2 \leq i \leq 4$, $I_i =
(x_1x_i,z) \subset R$. For $2 \leq i \leq 4$ we have that
$\Hom_{R} (I_i,R)$ is generated as an $R$-module by the inclusion
morphism $ I_i \hookrightarrow  R$ together with a single element
$\phi_i$ such that $\phi_i (x_0x_i) = 0$ and $\phi_i (z) = x_ax_b$, with
$\{i,a,b\}=\{2,3,4\}$. We denote by $K(R)$ the total ring of
fractions of $R$, defined as the localization of $R$ with respect
to its multiplicative subset of regular elements. Consider the
three elements $\tilde{s}_i = x_ax_b/z \in  K(R)$ (indices as
before) and denote by $A$ the $R$-subalgebra of $K(R)$ generated
by $R$ and $\tilde{s}_2, \tilde{s}_3, \tilde{s}_4$. It can be
shown that $k[ \Delta']$ is isomorphic to the quotient of the ring
$A/(z)$ by its radical.}
\end {remark}


\begin {remark}
{\rm   
It is plausible that our ideas also generalize to
non-Gorenstein simplicial complexes. To do this a
more detailed study of non-Gorenstein
unprojections would be necessary.}
\end {remark}



\begin {remark}
{\rm Combining the results of the present note with those of
\cite{KM3} we get a link between stellar subdivisions of
non-acyclic Gorenstein simplicial complexes and linkage theory 
\cite{Mi}. Is it possible to use this connection to define 
new combinatorial invariants of simplicial complexes?}
\end{remark}


\vspace{0.1 in}
\noindent
\emph{Acknowledgements}.  The authors are grateful to Christos
Athanasiadis for important discussions and suggestions. They
also thank  Jo\~{a}o Martins  for useful
discussions, and  Tim R\"omer for useful comments on an earlier 
version of this manuscript.










