 module of any homology ball (see for instance \cite[Theorem 5.7.1]{BH}), which says that $\omega_{{\mathbb F}[\Gamma]}$ is the image of $I_{\partial \Gamma}$ under the projection $\mathbb{F}[\mathbf{x},\mathbf{y}]\twoheadrightarrow \mathbb{F}[\Gamma]$. Recalling that $\partial\Delta$ consists of the codimension $1$ faces of $\Gamma$ contained in exactly one facet, one checks that $I_{\partial \Gamma} = I_{\Gamma} + (\mathbf{y}^{[n] \setminus F} \mid F \in \mathcal{F}(\Delta))$. This is precisely the definition of a Bier sphere, see for instance \cite[Corollary 5.3]{DFN}. Note that if $\Delta$ is the $(n-1)$-simplex, then $\Gamma$ is actually a sphere, ${\mathbb F}[\Gamma]$ is Gorenstein and $\omega_{{\mathbb F}[\Gamma]} \cong {\mathbb F}[\Gamma]$; since $\mathbf{y}^{\varnothing} = 1$, we are done also in this case.
	
	The simplicial complex $\Gamma$ is actually vertex decomposable \cite[Remark 1.8]{Mur}, a property which is stronger than shellability.  Moreover, any Bier $(n-1)$-ball or $(n-2)$-sphere is naturally a subcomplex of the boundary complex $\Diamond_n$ of the $n$-cross-polytope, with $I_{\Diamond_n} = (x_iy_i : i\in [n])$.
	
	\begin{exam}\label{ex: Bier}
		Let $n=3$ and $\Delta$ be the simplicial complex $\{\varnothing, x_1,x_2,x_3,x_1x_2\}$. Then $I_{\Gamma}= (x_1x_3, x_2x_3) + (x_1y_1,x_2y_2,x_3y_3)$, and $\Gamma$ is the $2$-dimensional pure shellable homology ball with shelling order \[y_1y_2y_3 \prec \underline{x_1}y_2y_3 \prec y_1\underline{x_2}y_3 \prec y_1y_2\underline{x_3} \prec \underline{x_1x_2}y_3.\] Observe that $h(\Gamma)=f(\Delta)=(1,3,1)$. The complex $\partial\Gamma = \langle x_1x_2, x_2y_1,y_1x_3,y_2x_3,x_1y_2\rangle$ is a $5$-cycle.
	\end{exam}
	\begin{figure}[h]
	    \centering
	    \includegraphics{delta_gamma2.pdf}
	    \caption{The simplicial complex $\Delta$ as in \Cref{ex: Bier} and the associated Bier ball $\Gamma$.}
	    \label{fig:my_label}
	\end{figure}
	
	\begin{rema}
		The Bier sphere  $\partial\Gamma$ in \Cref{ex: Bier} is flag, but this is usually not the case in general. In \cite{HK}, the authors prove that there are only $8$ possible complexes $\Delta$ which give rise to flag Bier spheres.
	\end{rema}
	
	
	
	
	
	\section{\texorpdfstring{From Bier balls to Gorenstein algebras: the ring $R_{\Delta}$}{From Bier balls to Gorenstein algebras: the ring RDelta}}
	For the rest of this article we will be concerned with idealizing the canonical module of the Stanley--Reisner ring of a Bier ball. We set some notation for the rest of the paper.
	
	\begin{nota} \label{notat:main}
	From now on, if not specified otherwise:
	\begin{itemize}
	    \item $\Delta$ is a pure $(d-1)$-dimensional simplicial complex on $n$ vertices, and its Stanley--Reisner ideal $I_{\Delta}$ lives in ${\mathbb F}[x_1, \ldots, x_n]$;
	    \item $S$ is the polynomial ring ${\mathbb F}[\mathbf{x}, \mathbf{y}]={\mathbb F}[x_1, \ldots, x_n, y_1, \ldots, y_n]$;
	    \item $\Gamma$ is the $(n-1)$-dimensional Bier ball obtained from $\Delta$ as in \Cref{def: whiskers} (with a slight abuse of notation, we will use the name ``Bier ball'' also when $\Delta$ is the full simplex);
	    \item $R_{\Delta}$ is the $n$-dimensional standard graded Gorenstein ${\mathbb F}$-algebra $\widetilde{{\mathbb F}[\Gamma]} = {\mathbb F}[\Gamma] \ltimes \omega_{{\mathbb F}[\Gamma]}(n-d-1)$, as in \Cref{thm:idealization} (note that, by \Cref{prop: Bier dual}.iv, ${\mathbb F}[\Gamma]$ is level and $a({\mathbb F}[\Gamma]) = -(n-d) = d-n$);
	    \item given two facets $F_1$ and $F_2$ of $\Delta$, we will denote by $b_{F_1, F_2}$ the binomial $\mathbf{y}^{F_1 \setminus F_2}z_{F_1} - \mathbf{y}^{F_2 \setminus F_1}z_{F_2}$ inside $S[z_F \mid F \in \mathcal{F}(\Delta)]$. Notice that $b_{F_2, F_1} = -b_{F_1, F_2}$, and the case where $F_1$ and $F_2$ coincide yields the zero binomial.
	\end{itemize}
	\end{nota}
	
	\begin{rema} \label{rem:hvector_RDelta}
		The construction of $R_{\Delta}$ shows that if $f=(1,f_0,\dots,f_{d-1})$ is the $f$-vector of a pure simplicial complex, then $(1, f_0+f_{d-1},f_1+f_{d-2},\dots,f_{d-1}+f_0,1)$ is the $h$-vector of a Gorenstein standard graded algebra. In particular, it is an $M$-sequence (see \cite{St_green}). We will see a stronger condition on $\Delta$ for this vector to be the the $h$-vector of a quadratic -- or even Koszul -- Gorenstein algebra.
	\end{rema}

	\begin{prop} \label{prop:RDelta_presentation}
	Let $\Delta$ be a pure simplicial complex of dimension $d-1$. Then \[R_{\Delta} = \frac{S[z_F \mid F \in \mathcal{F}(\Delta)]}{I_{\Gamma} + \mathcal{L} + \left(z_F \mid F \in \mathcal{F}(\Delta)\right)^2},\] where \[\mathcal{L} = ( x_iz_F \mid F \in \mathcal{F}(\Delta),\  i \notin F) + (b_{F_1, F_2} \mid F_1, F_2 \in \mathcal{F}(\Delta), F_1 \neq F_2).\]
	In particular, the defining ideal of $R_{\Delta}$ is generated by
	\begin{itemize}
	    \item $\mathbf{x}^{N_i}$, where $N_i$ is a minimal nonface of $\Delta$,
	    \item $x_iy_i$, where $i$ ranges between $1$ and $n$,
	    \item $z_{F_1}z_{F_2}$, where $F_1$ and $F_2$ are (possibly coincident) facets of $\Delta$,
	    \item $x_iz_F$, where $F$ is a facet of $\Delta$ and $i \notin F$,
	    \item $b_{F_1, F_2}$, where $F_1$ and $F_2$ are distinct facets of $\Delta$ and $b_{F_1, F_2}$ is the binomial defined in \Cref{notat:main},
	\end{itemize}
	and such a presentation does not depend on the characteristic of the field ${\mathbb F}$.
	\end{prop}
	\begin{proof}
	    Since the defining ideal of the claimed presentation of $R_{\Delta}$ involves only monomials and binomials of the form $\mathbf{m} - \mathbf{m'}$, such a presentation must hold in every characteristic, see e.g.~\cite[Lemma A.1.(iii)]{DKoszul}.
	    
	    By Theorem \ref{thm:idealization}.ii, in order to find a presentation for $R_{\Delta}$ it is enough to compute the syzygies of $\omega_{\mathbb{F}[\Gamma]}$ as an