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\title[Multigraded strong Lefschetz property for balanced simplicial complexes]{Multigraded strong Lefschetz property for balanced simplicial complexes}

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\author[Ryoshun Oba]{\texorpdfstring{Ryoshun Oba\thanks{\href{mailto:ryoshun\_oba@mist.i.u-tokyo.ac.jp}{ryoshun\_oba@mist.i.u-tokyo.ac.jp}.}\addressmark{1}}{Ryoshun Oba}}

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\address{\addressmark{1}Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-8656, Tokyo Japan}

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\abstract{Generalizing the strong Lefschetz property for an $\mathbb{N}$-graded algebra, we introduce multigraded strong Lefschetz property for an $\mathbb{N}^m$-graded algebra.
We show that, for $\bm{a} \in \mathbb{N}^m_+$, the generic $\mathbb{N}^m$-graded Artinian reduction of the Stanley--Reisner ring of an $\bm{a}$-balanced homology sphere over a field of characteristic $2$ satisfies the multigraded strong Lefschetz property.
A corollary is the inequality $h_{\bm{b}} \leq h_{\bm{c}}$ for $\bm{b} \leq \bm{c} \leq \bm{a}-\bm{b}$ for the flag $h$-vector of an $\bm{a}$-balanced simplicial sphere. This can be seen as a common generalization of the unimodality of the $h$-vector of a simplicial sphere by Adiprasito and balanced generalized lower bound inequality by Juhnke-Kubitzke and Murai.
Another combinatorial consequence is that a $k$-dimensional completely balanced simplicial complex which is a subcomplex of a simplicial $2k$-sphere satisfies $f_k\leq 2f_{k-1}$.}

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\keywords{Lefschetz property, Stanley--Reisner ring, balancedness, multigraded algebra, unimodality}

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\begin{document}

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\section{Introduction}
The $f$-vector and $h$-vector of simplicial complexes have been extensively studied in algebraic and topological combinatorics in the last decades.
Here, for a $(d-1)$-dimensional simplicial complex $\Delta$, the \emph{$f$-vector} of $\Delta$ is a sequence $(f_{-1},\ldots,f_{d-1})$, where $f_i$ is the number of $i$-dimensional faces of $\Delta$, and the \emph{$h$-vector} $(h_0,\ldots,h_d)$ of $\Delta$ is defined by $\sum_{i=0}^{d} h_i t^i = \sum_{i=0}^{d} f_{i-1} t^i(1-t)^{d-i}$ using a variable $t$.
For the study of $f$-vector and $h$-vector of simplicial complexes, Stanley--Reisner ring (or face ring) has been used.
A recent breakthrough announced by Adiprasito~\cite{Adi} (see also~\cite{APP,KX,PP}) is the hard Lefschetz theorem for the Stanley--Reisner ring of a simplicial (or homology) sphere, generalizing the work of Stanley~\cite{Sta80} for the boundary complex of simplicial polytopes.
Among many combinatorial consequences, this implies the celebrated $g$-conjecture, in particular generalized lower bound inequality (GLBI) asserting that the $h$-vector of a simplicial sphere is unimodal.

Adding extra combinatorial constraints to topological (or homological) ones and then studying behaviors of $f$- and $h$-vectors have been of interest (see \cite{Gal05,JM,Sta87} for example).
Juhnke-Kubitzke and Murai~\cite{JM} investigated completely balanced (or $(1,\ldots,1)$-balanced in the definition below) simplicial spheres, and showed (assuming Hard Lefschetz theorem~\cite{Adi}) that the Stanley--Reisner ring of its rank selected subcomplex possesses the top-heavy strong Lefschetz property (or dual-weak Lefschetz property).
A corollary is a balanced GLBI asserting that the $h$-vector of a completely balanced simplicial $(d-1)$-sphere satisfies 
\begin{equation} \label{eq:balancedGLBI}
\frac{h_i}{\binom{d}{i}} \leq \frac{h_{i+1}}{\binom{d}{i+1}} \text{ for } i < \frac{d}{2}.
\end{equation}

In this extended abstract of the full preprint~\cite{Oba24}, we investigate simplicial complexes with a combinatorial constraint called $\bm{a}$-balancedness. 
For a positive integer vector $\bm{a}=(a_1,\ldots,a_m)$ with $|\bm{a}|:=a_1+\cdots+a_m=d$, a pair $(\Delta,\kappa)$ of a $(d-1)$-dimensional simplicial complex $\Delta$ and a map $\kappa:V(\Delta) \rightarrow [m]:=\{1,\ldots,m\}$ is \emph{$\bm{a}$-balanced} if each face of $\Delta$ contains at most $a_j$ vertices of color $j$ for each $j=1,\ldots,m$.
Stanley~\cite{Sta79} initiated a research of $\bm{a}$-balanced simplicial complexes and showed that
the Stanley--Reisner ring of an $\bm{a}$-balanced simplicial complex admits a system of parameters homogeneous in the fine $\mathbb{N}^m$-grading induced by the coloring $\kappa$. 

With this in mind, we introduce multigraded strong Lefschetz property for an $\mathbb{N}^m$-graded algebra, generalizing strong Lefschetz property of an $\mathbb{N}$-graded algebra of \cite{HMMNWW}. 
We then show that the generic $\mathbb{N}^m$-graded Artinian reduction of the Stanley--Reisner ring of an $\bm{a}$-balanced homology sphere over a field of characteristic $2$ satisfies the multigraded strong Lefschetz property (\Cref{thm:main multiHL}).
Note that \Cref{thm:main multiHL} is a common generalization of the above mentioned two algebraic results.
Our proof of \Cref{thm:main multiHL} relies on an anisotropy technique, in particular differential identity, over a field of characteristic $2$ in~\cite{APP,APPS,KX,PP}.

A corollary of \Cref{thm:main multiHL} is a combinatorial result on flag $h$-vector.
The \emph{flag $f$-vector} of $\bm{a}$-balanced simplicial complex $(\Delta,\kappa)$ is an $m$-dimensional array $(f_{\bm{b}})_{\bm{0}\leq\bm{b}\leq\bm{a}}$ where $f_{\bm{b}}$ is the number of faces $\sigma \in \Delta$ with $|\sigma \cap \kappa^{-1}(j)|=b_j$ for $j=1,\ldots,m$. Here $\bm{c} \leq \bm{d}$ denotes the component-wise inequality $c_i \leq d_i$ for all $i$.
The \emph{flag $h$-vector} of $(\Delta,\kappa)$ is an $m$-dimensional array $(h_{\bm{b}})_{\bm{0}\leq\bm{b}\leq\bm{a}}$ defined by
\[
    \sum_{\bm{0} \leq \bm{b} \leq \bm{a}} h_{\bm{b}} \bm{t}^{\bm{b}} 
    = \sum_{\bm{0} \leq \bm{b} \leq \bm{a}} f_{\bm{b}} \bm{t}^{\bm{b}}(\bm{1}-\bm{t})^{\bm{a}-\bm{b}},
\]
where $\bm{t}=(t_1,\ldots,t_m)$ is a vector of variables. Here, we denote $\bm{t}^{\bm{c}}=t_1^{c_1}\cdots t_m^{c_m}$ for $\bm{t}=(t_1,\ldots,t_m)$ and $\bm{c}=(c_1,\ldots,c_m) \in \mathbb{N}$.
\begin{theorem} \label{thm:flag-h}
For an $\bm{a}$-balanced homology sphere $(\Delta,\kappa)$ over $\mathbb{F}_2$, we have $h_{\bm{b}} \leq h_{\bm{c}}$ for any $\bm{b},\bm{c} \in \mathbb{N}^m$ with $\bm{b} \leq \bm{c} \leq \bm{a}-\bm{b}$.
\end{theorem}
\noindent Note that \Cref{thm:flag-h} can be seen as a common generalization of GLBI and balanced GLBI. GLBI is the case of $m=1$ in \Cref{thm:flag-h}. Balanced GLBI (\ref{eq:balancedGLBI}) follows from the inequality  $h_{i\bm{e}_j}\leq h_{(i+1)\bm{e}_j}$ for $\bm{a}=\bm{1}+2i\bm{e}_j$ in \Cref{thm:main multiHL} together with the averaging argument of Goff, Klee and Novik~\cite{GKN}. See~\cite{JM} for details.

Another combinatorial corollary is related to the following balanced version of Gr\"{u}n\-baum--Kalai--Sarkaria conjecture posed by Kalai--Nevo--Novik~\cite{KNN}.
\begin{conj}[{\cite[Conjecture 8.2, Proposition 8.3]{KNN}}] \label{conj:bGKS}
    Let $\Delta$ be a $k$-dimensional completely balanced simplicial complex embedabble in $\mathbb{S}^{2k}$.
    Then, $f_{k} \leq 2f_{k-1}$ holds.
\end{conj}
\noindent We prove the following significant partial result on \Cref{conj:bGKS}.
\begin{theorem} \label{thm:bGKS}
    Let $\Delta$ be a $k$-dimensional completely balanced simplicial complex such that there is a simplicial $2k$-sphere $\Gamma$ with $\Delta \subset \Gamma.$
    Then, $f_{k} \leq 2f_{k-1}$ holds.
\end{theorem}
\noindent The derivation of \Cref{thm:bGKS} from \Cref{thm:main multiHL} is analogous to~\cite{Adi}, and we omit it from the extended abstract because of the length limit. The detailed argument for \Cref{thm:bGKS} can be found in the updated version of full preprint \cite{Oba24}.

This extended abstract is organized as follows.
After preliminaries are given in Section 2, Lee's formula for the evaluation map is recalled in Section 3.
In Section 4, generic $\mathbb{N}^m$-graded Artinian reduction is defined, and differential formula for the evaluation map in multigraded setting is derived.
In Section 5, we derive the main result \Cref{thm:main multiHL} about multigraded strong Lefschetz property over a field of characteristic $2$.
In Section 6, we briefly discuss further results in the full preprint~\cite{Oba24}.

\section{Preliminaries}
We highlight some definitions and notations we use (see \cite{Sta-book} for general reference).
We denote the set of nonnegative (resp.~positive) integers by $\mathbb{N}$ (resp.~$\mathbb{N}_+$).
\subsection{Simplicial complex and Stanley--Reisner ring}
Throughout, by a simplicial complex, we always mean an abstract simplicial complex, i.e., a downward closed collection of subsets of a finite set.
The vertex set of a simplicial complex $\Delta$ is denoted by $V(\Delta)$.

For an $\mathbb{N}^m$-graded module $M$ and $\bm{b} \in \mathbb{N}^m$, we denote by $M_{\bm{b}}$ the submodule of all homogeneous elements of degree $\bm{b}$.

Let $\mathbbm{k}$ be a field and let $\Delta$ be a simplicial complex. Let us denote by $\mathbbm{k}[\bm{x}]$ the polynomial ring $\mathbbm{k}[x_v:v \in V(\Delta)]$.
The Stanley--Reisner ring of $\Delta$ over $\mathbbm{k}$ is  $\mathbbm{k}[\Delta]=\mathbbm{k}[\bm{x}]/I_\Delta$, where $I_\Delta$ is the ideal generated by $x_\tau:=\prod_{v \in \tau} x_v$ over all $\tau \not\in \Delta$.
It is known that the Stanley--Reisner ring of $\Delta$ has Krull dimension $\dim \Delta+1$.
For a $(d-1)$-dimensional simplicial complex $\Delta$, a length $d$ sequence of linear forms $\Theta=(\theta_1,\ldots,\theta_d)$ of $\mathbbm{k}[\Delta]$ is called a \emph{linear system of parameters} (\emph{l.s.o.p.}~for short) for $\mathbbm{k}[\Delta]$ if $\mathbbm{k}[\Delta]/(\Theta)=\mathbbm{k}[\Delta]/(\theta_1,\ldots,\theta_d)$ is a finite dimensional $\mathbbm{k}$-vector space.
The resulting quotient algebra $\mathbbm{k}[\Delta]/(\Theta)$ is called an \emph{Artinian reduction} of $\mathbbm{k}[\Delta]$ with respect to $\Theta$.

For an $\bm{a}$-balanced simplicial complex $(\Delta,\kappa)$ with $\bm{a} \in \mathbb{N}^m$, the polynomial ring $\mathbbm{k}[\bm{x}]$ naturally has the $\mathbb{N}^m$-grading, sometimes called the \emph{fine grading}, defined by $\deg x_v = \bm{e}_{\kappa(v)}$, where $\bm{e}_j \in \mathbb{N}^m$ denotes the $j$-th unit coordinate vector.
For an $\bm{a}$-balanced simplicial complex $(\Delta,\kappa)$, we say that a system of parameters $\Theta$ for $\mathbbm{k}[\Delta]$ is \emph{$\mathbb{N}^m$-graded} (or $\bm{a}$-colored) if each $\theta_i$ is homogeneous in the fine $\mathbb{N}^m$-grading of $\mathbbm{k}[\Delta]$.
Stanley~\cite[Theorem 4.1]{Sta79} showed that if $\mathbbm{k}$ is an infinite field, every $\bm{a}$-balanced simplcial complex $(\Delta,\kappa)$ has an $\mathbb{N}^m$-graded l.s.o.p.~$\Theta$ for $\mathbbm{k}[\Delta]$, and $(\mathbbm{k}[\Delta]/(\Theta))_{\bm{b}}=0$ unless $\bm{0} \leq \bm{b} \leq \bm{a}$.
Note that for an $\mathbb{N}^m$-graded l.s.o.p.~$\Theta$ for the Stanley--Reisner ring of an $\bm{a}$-balanced simplicial complex, $\Theta$ contains exactly $a_j$ elements of degree $\bm{e}_j$ for each $j$.

\subsection{Homological properties} 
A simplicial complex $\Delta$ is called \emph{Cohen--Macaulay} over $\mathbbm{k}$ if there is an l.s.o.p.~$(\theta_1,\ldots,\theta_d)$ for $\mathbbm{k}[\Delta]$ such that $\mathbbm{k}[\Delta]$ is a free $\mathbbm{k}[\theta_1,\ldots,\theta_d]$-module.
By Reisner's theorem, a simplcial complex $\Delta$ is Cohen--Macaulay over $\mathbbm{k}$ if and only if it is pure and, for every face $\sigma \in \Delta$, $\widetilde{H}_i(\lk_\sigma(\Delta);\mathbbm{k})=0$ for all $i \neq \dim\Delta - |\sigma|$ (see~\cite[Corollary II.4.2]{Sta-book}). 
Here $\widetilde{H}_*(\Delta;\mathbbm{k})$ denotes the reduced simplicial homology group of $\Delta$ with coefficients in $\mathbbm{k}$, and $\lk_\tau(\Delta)=\{\sigma \in \Delta:\sigma \cap \tau = \emptyset, \sigma \cup \tau \in \Delta\}$ denoted the \emph{link} with respect to $\tau \in \Delta$.
Note that for an $\bm{a}$-balanced Cohen--Macaulay complex $(\Delta,\kappa)$, the equality $\dim (\mathbbm{k}[\Delta]/(\Theta))_{\bm{b}} = h_{\bm{b}}$ holds for $\bm{0} \leq \bm{b} \leq \bm{a}$.

For an $\mathbb{N}$-graded $\mathbbm{k}[\bm{x}]$-module $M$, its \emph{socle} is the submodule $\Soc(M)=\{a \in M: \mathfrak{m}a=0\}$, where $\mathfrak{m}=(x_1,\ldots,x_n)$ is the maximal graded ideal of $\mathbbm{k}[\bm{x}]$.
An $\mathbb{N}$-graded $\mathbbm{k}$-algebra of Krull dimension zero is said to be \emph{Gorenstein} if its socle is a one-dimensional $\mathbbm{k}$-vector space.
Note that an $\mathbb{N}$-graded finitely generated standard $\mathbbm{k}$-algebra $A=A_0\oplus \cdots \oplus A_d$ with $A_d \neq 0$ is Gorenstein if and only if $\dim A_d =1$ and the multiplication map $A_i \times A_{d-i} \rightarrow A_d \overset{\cong}{\rightarrow} \mathbbm{k}$ is a nondegenerate bilinear pairing for $i=0,\ldots,d$~\cite[Lemma 36]{AY}.

We say that a simplicial complex $\Delta$ is a \emph{simplcial $(d-1)$-sphere} if its geometric realization is homeomorphic to $\mathbb{S}^{d-1}$. 
Let $\mathbbm{k}$ be a field.
A simplcial complex $\Delta$ is a \emph{homology $(d-1)$-sphere} over $\mathbbm{k}$ if $\widetilde{H}_*(\lk_{\tau}\Delta;\mathbbm{k}) \cong \widetilde{H}_*(\mathbb{S}^{d-|\tau|-1};\mathbbm{k})$ for every face $\tau \in \Delta$.
If $\Delta$ is a homology sphere over $\mathbbm{k}$, an Artinian reduction $A=\mathbbm{k}[\Delta]/(\Theta)$ is Gorenstein with respect to any l.s.o.p.~$\Theta$~\cite[Theorem II.5.1]{Sta-book}.

A pure $(d-1)$-dimensional simplicial complex is \emph{strongly connected} if for every pair of facets $\sigma$ and $\tau$ of $\Delta$, there is a sequence of facets $\sigma=\sigma_0,\sigma_1,\ldots, \sigma_m=\tau$ such that $|\sigma_{i-1} \cap \sigma_i|=d-1$ for $i=1,\ldots,m$.
A \emph{$(d-1)$-pseudomanifold} (without boundary) is a strongly connected pure $(d-1)$-dimensional simplicial complex such that every $(d-2)$-face is contained in exactly two facets.
A $(d-1)$-pseudomanifold is \emph{orientable} over $\mathbbm{k}$ if $\widetilde{H}_{d-1}(\Delta;\mathbbm{k})\cong \mathbbm{k}$, and such a pseudomanifold is said to be \emph{oriented} if the facets are given an ordering such that the coefficients of nonzero $\mu \in \widetilde{H}_{d-1}(\Delta;\mathbbm{k})$ is constant over all oriented facets.

\section{Lee's formula for the evaluation map}
Let $\mathbbm{k}$ be a field of arbitrary characteristic, and let $\Delta$ be a $(d-1)$-dimensional simplicial complex.
Let $A=\mathbbm{k}[\Delta]/(\Theta)=A_0\oplus\cdots\oplus A_d$ be an Artinian reduction of $\mathbbm{k}[\Delta]$ with respect to an l.s.o.p.~$\Theta$.
Then, by \cite[Corollary 3.2]{TW}, $A_d$ is linearly isomorphic to $\widetilde{H}_{d-1}(\Delta;\mathbbm{k})$.
Thus, for a $(d-1)$-pseudomanifold $\Delta$ (without boundary) orientable over $\mathbbm{k}$, $A_d$ is a one-dimensional linear space. 
The linear isomorphism $\Psi:A_d \overset{\cong}{\rightarrow} \mathbbm{k}$ which is determined unique up to the scaling is called the \emph{evaluation map} (or \emph{degree map, volume map, Brion's isomorphism}).
Lee~\cite{Lee} gave an explicit description of the evaluation map $\Psi$ with the appropriate scaling. We shall recall this formula below.

Let us prepare some conventions and notations used throughout.
We always assume that $V(\Delta)=[n]:=\{1,\ldots,n\}$ and let $\mathbbm{k}[\bm{x}]=\mathbbm{k}[x_1,\ldots,x_n]$. For a sequence $J=(v_1,\ldots,v_k)$ of vertices, we denote $x_J=x_{v_1}\cdots x_{v_k}$.
We abbreviate the projection from $\mathbbm{k}[\bm{x}]$ to an Artinian reduction $A$ of $\mathbbm{k}[\Delta]$ as long as it is not confusing.
So, for example, the composite $\mathbbm{k}[\bm{x}]_d \twoheadrightarrow A_d \overset{\cong}{\rightarrow} \mathbbm{k}$ is also denoted as $\Psi$.
An l.s.o.p.~$\Theta=(\theta_1,\ldots,\theta_d)$ for $\mathbbm{k}[\Delta]$ is identified with a map $p:V(\Delta) \rightarrow \mathbbm{k}^d$ through the relation $\theta_k=\sum_{v\in V(\Delta)} p(v)_k x_v$ for $k=1,\ldots,d$. The map $p$ is called a \emph{point configuration}.
For an (oriented) facet $\sigma=[v_1,\ldots,v_d]$ of $\Delta$, let $[\sigma]=\det \begin{pmatrix} p(v_1) & \cdots & p(v_d) \end{pmatrix}$.

We also need the following notations to state Lee's formula.
Let $v^*$ be a new vertex not in $V(\Delta)$ with an associated position\footnote{Here, $p'(v^*)$ has to be in sufficiently general position so that none of $[\sigma-v_i+v^*]$ vanishes. One may need to extend the field to choose such a vector.} $p'(v^*) \in \mathbbm{k}^d$, and for an (oriented) facet $\sigma=v_1,\ldots,v_d$, let $[\sigma-v_i+v^*]$ be the determinant of the matrix obtained by replacing the $i$-th column of the matrix $\begin{pmatrix} p(v_1) & \cdots & p(v_d) \end{pmatrix}$ with $p'(v^*)$.

Now we are ready to state Lee's formula.
\begin{lemma} \label{lem:Lee formula}
Let $\Delta$ be an orientable $(d-1)$-pseudomanifold over a field $\mathbbm{k}$.
Let $A$ be an Artinian reduction of $\mathbbm{k}[\Delta]$ with respect to $\Theta$, and let $\Psi:A_d \rightarrow \mathbbm{k}$ be the evaluation map.
Then, under suitable normalization, for any length $d$ sequence of vertices $J=(v_1,\ldots,v_d)$,
\begin{align} 
\Psi(x_J)&=\sum_{\sigma \in \Delta:\text{ facet containing }\{J\}} \Psi(x_\sigma) \frac{\prod_{k=1}^{d}[\sigma + v^* - v_k]}{\prod_{v \in \sigma} [\sigma+v^*-v]} \notag\\
&=\sum_{\sigma \in \Delta:\text{ facet containing }\{J\}} \frac{1}{[\sigma]} \frac{\prod_{k=1}^{d}[\sigma + v^* - v_k]}{\prod_{v \in \sigma} [\sigma+v^*-v]} \label{eq:eval}
\end{align}
holds.
Here $\{J\}$ denotes the set $\{v_1,\ldots,v_d\}$, and the sum is taken over all oriented facets of $\Delta$ containing $\{J\}$.
\end{lemma}
Throughout, we assume that the evaluation map $\Psi$ is normalized so that (\ref{eq:eval}) holds.

\section{Generic Artinian reduction}
\subsection{Generic \texorpdfstring{$\mathbb{N}^m$}{Nm}-graded Artinian reduction}
For an $\bm{a}$-balanced simplicial complex $(\Delta,\kappa)$ with $\bm{a} \in \mathbb{N}^m_+$ and $|\bm{a}|=d$, we define the generic $\mathbb{N}^m$-graded Artinian reduction of $\mathbbm{k}[\Delta]$ as follows.
Fix a partition $\mathcal{I}_1 \sqcup \cdots \sqcup \mathcal{I}_m$ of $[d]$ with $|\mathcal{I}_j|=a_j$ for $j=1,\ldots,m$.
Consider the set of new auxiliary indeterminates
\begin{equation*}
\{p_{k,v}: k \in [d], v \in V(\Delta), k \in \mathcal{I}_{\kappa(v)}\}
\end{equation*}
and let $\widetilde{\mathbbm{k}}=\mathbbm{k}(p_{k,v})$ be the rational function field of these indeterminates with coefficients in $\mathbbm{k}$. 
Define the $\mathbb{N}^m$-graded l.s.o.p.~$\Theta=(\theta_1,\ldots,\theta_d)$ by 
\[
\begin{pmatrix}
\theta_1 \\
\vdots \\
\theta_d
\end{pmatrix}
=
\bm{P}
\begin{pmatrix}
x_1 \\
\vdots \\
x_n
\end{pmatrix},
\]
where the $(k,v)$-th entry of the coefficient matrix $\bm{P}$ is $p_{k,v}$ if $k \in \mathcal{I}_{\kappa(v)}$ and $0$ otherwise.
The quotient $\mathbb{N}^m$-graded algebra $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ is called the \emph{generic $\mathbb{N}^m$-graded Artinian reduction} of $\mathbbm{k}[\Delta]$ (with respect to a coloring $\kappa$).
Note that, when $m=1$, the generic $\mathbb{N}$-graded Artinian reduction coincides with the generic Artinian reduction in the sense of~\cite{PP}. We remark that to be consistent with the definition of~\cite{PP}, $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ is called the generic $\mathbb{N}^m$-graded Artinian reduction of $\mathbbm{k}[\Delta]$, not of $\widetilde{\mathbbm{k}}[\Delta]$, though $A$ is the Artinian reduction of $\widetilde{\mathbbm{k}}[\Delta]$ in the usual sense.
By~\cite[Theorem 4.1]{Sta79}, as an $\mathbb{N}^m$-graded algebra, $A$ is decomposed into $\mathbb{N}^m$-homogeneous components as $A=\bigoplus_{\bm{0} \leq \bm{b} \leq \bm{a}} A_{\bm{b}}$.
The homogeneous decomposition as $\mathbb{N}$-graded algebra is denoted as $A=\bigoplus_{i=0}^d A_i$.

\subsection{Differential formula in characteristic \texorpdfstring{$2$}{2}}
In the generic $\mathbb{N}$-graded Artinian reduction, the right-hand-side of (\ref{eq:eval}) in Lee's formula is a rational function of new indeterminates $p_{k,v}$.
Papadakis--Petrotou~\cite{PP} considered taking partial derivative of (\ref{eq:eval}) with respect to new indeterminates $p_{k,v}$, and they prove a remarkable formula in characteristic $2$.
This formula is later generalized by Karu--Xiao~\cite[Theorem 4.1]{KX}.
We recall this formula.

In this subsection, we assume that the field $\mathbbm{k}$ is of characteristic $2$. Then automatically every pseudomanifold is orientable.
For a pseudomanifold $\Delta$, let $A=\mathbbm{k}[\Delta]/(\Theta)$ be the generic ($\mathbb{N}$-graded) Artinian reduction of $\mathbbm{k}[\Delta]$, where $\widetilde{\mathbbm{k}}=\mathbbm{k}(p_{kv}:k \in [d], v \in V(\Delta))$.
For a length $d$ sequence $I=(v_1,\ldots,v_d)$ of vertices, define the differential operator $\partial_I$ by $\partial_{p_{1,v_1}}\circ \cdots \circ \partial_{p_{d,v_d}}$, where $\partial_{p_{k,v}}$ denoted the (formal) partial derivative with respect to $p_{k,v}$. Under these notations, the following holds.
\begin{theorem}{\cite[Theorem 4.1]{KX}} \label{thm:KX}
Let $\Delta$ be a $(d-1)$-pseudomanifold, and let $\mathbbm{k}$ be a field of characteristic $2$.
Let $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ be the generic $\mathbb{N}$-graded Artinian reduction of $\mathbbm{k}[\Delta]$, where $\widetilde{\mathbbm{k}}=\mathbbm{k}(p_{kv}:\, 1\leq k\leq d, \, v\in V(\Delta))$.
Let $\Psi:A_{d} \rightarrow \widetilde{\mathbbm{k}}$ be the evaluation map normalized as in~\Cref{lem:Lee formula}.
Then, for any length $d$ sequences $I$ and $J$ of vertices,
\[
\partial_I \Psi(x_J) =\Psi(\sqrt{x_Ix_J})^2
\]
holds.
Here, for a monomial $x_L$, define its square root $\sqrt{x_L}$ by $x_K$ if there is a monomial $x_K$ with $x_K^2=x_L$ and $0$ otherwise.
\end{theorem}


We generalize the formula in \Cref{thm:KX} in the setting of generic $\mathbb{N}^m$-graded Artinian reduction by a simple trick of substitution.
Let $(\Delta,\kappa)$ be an $\bm{a}$-balanced pseudomanifold and let $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ be the generic $\mathbb{N}^m$-graded Artinian reduction of $\mathbbm{k}[\Delta]$, where $\widetilde{\mathbbm{k}}=\mathbbm{k}(p_{kv})$.
We call a length $d$ sequence of vertices $I=(v_1,\ldots,v_d)$ (possibly with repetition) \emph{$\kappa$-transversal} if $k \in \mathcal{I}_{\kappa(v_k)}$ for $k=1,\ldots,d$.
Note that $I=(v_1,\ldots,v_d)$ is a $\kappa$-transversal sequence if and only if there exist corresponding variables $p_{1,v_1}, \ldots, p_{d,v_d}$.
Note also that for every degree $\bm{a}$ monomial $x_J$ in $\mathbbm{k}[\bm{x}]$, $J$ can be reordered into a $\kappa$-transversal sequence.
For a $\kappa$-transversal sequence $I=(v_1,\ldots,v_d)$, define the differential operator $\partial_I$ by $\partial_{p_{1,v_1}}\circ \cdots \circ \partial_{p_{d,v_d}}$.
The following differential formula for the map $\Psi$ holds.
\begin{lemma} \label{lem:diff}
Let $(\Delta,\kappa)$ be an $\bm{a}$-balanced $(d-1)$-pseudomanifold for $\bm{a}\in\mathbb{N}^m_+$ with $|\bm{a}|=d$ and let $\mathbbm{k}$ be a field of characteristic $2$.
Let $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ be the generic $\mathbb{N}^m$-graded Artinian reduction of $\mathbbm{k}[\Delta]$.
Let $\Psi:A_{\bm{a}} \rightarrow \widetilde{\mathbbm{k}}$ be the evaluation map normalized as in~\Cref{lem:Lee formula}.
Then, for any $\kappa$-transversal sequence $I$ and any length $d$ sequence $J$ of vertices,
\[
\partial_I \Psi(x_J) =\Psi(\sqrt{x_Ix_J})^2
\]
holds.
Here, for a monomial $x_L$, define its square root $\sqrt{x_L}$ by $x_K$ if there is a monomial $x_K$ with $x_K^2=x_L$ and $0$ otherwise.
\end{lemma}
\begin{proof}
When $m=1$, this formula coincides with the formula in \Cref{thm:KX}.
For general case, the identity is obtained by substituting $p_{k,v}$ to $0$ in the formula in \Cref{thm:KX} for all indeterminates corresponding to pairs $(k,v)$ with $k \not\in \mathcal{I}_{\kappa(v)}$.
Note that this substitution is a valid one since the denominator of (\ref{eq:eval}) does not vanish after the substitution by the Kind--Kleinschmidt's criterion on l.s.o.p.~for Stanley--Reisner ring~\cite[Lemma III.2.4]{Sta-book}.
\end{proof}
\Cref{lem:diff} can be readily strengthened as below.
\begin{corollary} \label{cor:diff general}
Let $(\Delta,\kappa)$, $d$, $A$, $\Psi$ be as in \Cref{lem:diff}. 
For a $\kappa$-transversal sequence $I$, an element $g \in A_i$ with $i \leq \frac{d}{2}$, and a length $d-2i$ sequence $J$ of vertices,
\[
\partial_I \Psi(g^2x_J) =\Psi(g\sqrt{x_Ix_J})^2
\]
holds.
\end{corollary}
\begin{proof}
Writing $g=\sum_{K} \lambda_K x_K$ ($\lambda_K \in \widetilde{\mathbbm{k}}$)\footnote{Recall that we are abbreviating the projection from the polynomial ring to $A$.}, we have
\begin{align*}
\partial_I \Psi(g^2x_J) &= \partial_I \Psi\left(\sum_{K} \lambda_K^2 x_K^2 x_J\right) & (\text{by characteristic $2$}) \\
&=\sum_{K} \partial_I (\lambda_K^2 \Psi(x_K^2x_J)) & (\text{by linearlity of $\Psi, \partial_I$}) \\
&=\sum_{K} \lambda_K^2 \partial_I  \Psi(x_K^2x_J) & (\text{by } \partial_{p_{k,v}}(f^2g)=f^2\partial_{p_{k,v}}(g) \text{ for $f,g \in \widetilde{\mathbbm{k}}$ in char.\ $2$}) \\
&=\sum_{K} \lambda_K^2 \Psi(x_K \sqrt{x_Ix_J})^2 & (\text{by \Cref{lem:diff}}) \\
&=\Psi(g \sqrt{x_Ix_J})^2.
\end{align*}
\end{proof}

\section{Multigraded strong Lefschetz property via anisotropy}
Throughout this section, we assume that $\mathbbm{k}$ is a field of characteristic $2$ and $(\Delta,\kappa)$ is an $\bm{a}$-balanced homology sphere over $\mathbb{F}_2$ for $\bm{a} \in \mathbb{N}^m_+$ with $|\bm{a}|=d$. 
Let $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ be the generic $\mathbb{N}^m$-graded Artinian reduction of $\mathbbm{k}[\Delta]$, where $\widetilde{\mathbbm{k}}=\mathbbm{k}(p_{kv})$.
By Gorensteiness, the multiplication map $A_i \times A_{d-i} \rightarrow A_d \overset{\Psi}{\rightarrow} \widetilde{\mathbbm{k}}$ is a nondegenerate for each $0 \leq i\leq d$.
Hence, the multiplication map $A_{\bm{b}} \times A_{\bm{a}-\bm{b}} \rightarrow A_{\bm{a}} \overset{\Psi}{\rightarrow} \widetilde{\mathbbm{k}}$ is nondegenerate for each $\bm{b}\in\mathbb{N}^m$ with $\bm{b} \leq \bm{a}$.
We call this property as \emph{multigraded Poincar\'{e} duality}.

Let $\mathbbm{k}$ be a field and let $A=\bigoplus_{\bm{0} \leq \bm{b} \leq \bm{a}} A_{\bm{b}}$ be an Artinian Gorenstein standard $\mathbb{N}^m$-graded $\mathbbm{k}$-algebra\footnote{An $\mathbb{N}^m$-graded algebra is \emph{standard} if it is generated by $A_{\bm{e}_1} \oplus \cdots \oplus A_{\bm{e}_m}$.} with $A_{\bm{0}}\cong A_{\bm{a}} \cong \mathbbm{k}$.
We say that $A$ has the \emph{multigraded strong Lefschetz property} (as an $\mathbb{N}^m$-graded algebra) if there is an element $\ell_j \in A_{\bm{e}_j}$ for each $j=1,\ldots,m$ such that the multiplication map
\[
\times\bm{\ell}^{\bm{a}-2\bm{b}} : A_{\bm{b}} \rightarrow A_{\bm{a}-\bm{b}}
\]
is an isomorphism for all $\bm{b} \in \mathbb{N}^m$ with $\bm{b} \leq \frac{\bm{a}}{2}$ , where $\bm{e}_j \in \mathbb{N}^m$ is the $j$-th unit coordinate vector. 
We prove the following.
\begin{theorem} \label{thm:main multiHL}
Let $\mathbbm{k}$ be a field of characteristic $0$ or $2$ and let $(\Delta,\kappa)$ be an $\bm{a}$-balanced homology sphere over $\mathbb{F}_2$. 
Then the generic $\mathbb{N}^m$-graded Artinian reduction $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ of the Stanley--Reisner ring $\mathbbm{k}[\Delta]$ has the multigraded strong Lefschetz property.
\end{theorem}
\noindent
Here $\widetilde{\mathbbm{k}}$ is a purely transcendental field extension of $\mathbbm{k}$ resulted in the generic $\mathbb{N}^m$-graded Artinian reduction.

Our proof of \Cref{thm:main multiHL} relies on anisotropy technique used in~\cite{APP,APPS,KX,PP}.
For a vector space $W$ over a field $\mathbbm{k}$, a bilinear form $\varphi:W \times W \rightarrow \mathbbm{k}$ is \emph{anisotropic} if $\varphi(u,u) \neq 0$ holds for any nonzero $u \in W$.
Note that a bilinear form is anisotropic if and only if the restriction $\varphi|_{W' \times W'}$ is nondegenerate for any nonzero subspace $W'$ of $W$.
We prove the following combination of anisotropy and multigraded strong Lefschetz property in a field of characteristic $2$ with the explicit Lefschetz elements.
\begin{theorem} \label{thm:main anis}
Let $(\Delta,\kappa)$ be an $\bm{a}$-balanced homology sphere over $\mathbb{F}_2$, and let $\mathbbm{k}$ be a field of characteristic $2$. 
Let $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ be the generic $\mathbb{N}^m$-graded Artinian reduction of $\mathbbm{k}[\Delta]$.
Define $\ell_j=\sum_{v \in \kappa^{-1}(j)} x_v \in A_{\bm{e}_j}$ for $j=1,\ldots,m$.
Then, for any $\bm{b} \in \mathbb{N}^m$ with $\bm{b} \leq \frac{\bm{a}}{2}$, the bilinear form $\mathcal{Q}:A_{\bm{b}}\times A_{\bm{b}} \rightarrow \widetilde{\mathbbm{k}}$ defined by
\[
\mathcal{Q}(g,h)=\Psi(gh \bm{\ell}^{\bm{a}-2\bm{b}})
\]
is anisotropic, where $\Psi:A_{\bm{a}} \rightarrow \widetilde{\mathbbm{k}}$ is the evaluation map.
\end{theorem}

Toward the proof of \Cref{thm:main anis}, we first prove an auxiliary lemma, which can be seen as the combination of a multigraded version of weak Lefschetz property and anisotropy.
\begin{lemma} \label{lemma:WLP anis}
Let $(\Delta,\kappa)$, $\bm{a}$, $\mathbbm{k}$, $A$, $\ell_j$ be as in \Cref{thm:main anis}.
Let $S$ be a (possibly empty) subset of $[m]$ and let $\bm{e}_S=\sum_{j \in S} \bm{e}_j \in \mathbb{N}^m$ be the characteristic vector of $S$.
For $\bm{b} \in \mathbb{N}^m$ with $2\bm{b}+\bm{e}_S \leq \bm{a}$, define the bilinear form $\mathcal{Q}' : A_{\bm{b}}\times A_{\bm{b}} \rightarrow A_{2\bm{b}+\bm{e}_S}$ by
\[
\mathcal{Q}'(g,h)=gh \bm{\ell}^{\bm{e}_S}.
\]
Then $\mathcal{Q}'(g,g)\neq 0$ for any nonzero $g \in A_{\bm{b}}$,.
\end{lemma}
\begin{proof}
Suppose that $g$ is a nonzero element of $A_{\bm{b}}$.
As $A_{\bm{a}-\bm{b}}$ is generated by monomials, by multigraded Poincar\'{e} duality of $A$, there is a monomial $x_K$ of degree $\bm{a}-\bm{b}$ such that $gx_K \neq 0$ in $A_{\bm{a}}$. Its square $x_K^2$ is of degree $2\bm{a}-2\bm{b}$, where $2\bm{a}-2\bm{b} \geq \bm{a}+\bm{e}_S$ by assumption.
Hence there is a $\kappa$-transversal sequence $I$ and a set of vertices $U^* \in V_S:=\prod_{j \in S} \kappa^{-1}(j)$ and a length $d-2|\bm{b}|-|S|$ sequence of vertices $J$ satisfying $x_K^2=x_Ix_{U^*}x_J$\footnote{Since, for any degree $\bm{a}$ monomial $x_L$, $L$ can be reordered into $\kappa$-transversal sequence, the desired decomposition $x_K^2=x_Ix_Ux_J$ is obtained by assigning variables in greedy way.}.

Now we have the following identity:
\begin{align}
\partial_I\Psi(\mathcal{Q}'(g,g)x_J) &= \sum_{U \in V_S} \partial_I \Psi(g^2x_Ux_J) & \text{(by linearity of $\Psi$, $\partial_I$ and $\bm{\ell}^{\bm{e}_S}=\sum_{U \in V_S} x_U$)} \notag \\
&=\sum_{U \in V_S} \Psi(g\sqrt{x_Ix_Ux_J})^2 &\text{(by \Cref{cor:diff general})} \notag \\
&\overset{(*)}{=}\Psi(g\sqrt{x_Ix_{U^*}x_J})^2 = \Psi(g x_K)^2 \label{eq:WLP}
\end{align}
Here, in $(*)$, we are using the fact that, by the definition of square root, for a fixed monomial $x_Ix_J$, there is a unique squarefree monomial $x_{U'}$ with $\sqrt{x_Ix_{U'}x_J} \neq 0$.
By our choice of $U^*$, this is achieved by $x_{U'}=x_{U^*}$. 
As monomials $x_U$ for $U \in V_S$ are all distinct and squarefree, the equality $(*)$ holds.
Now, $g x_K$ is a nonzero element in $A_{\bm{a}}$ and $\Psi$ is an isomorphism, so we have $\Psi(g x_K)^2 \neq 0$.
Hence, by the identity (\ref{eq:WLP}), $\partial_I\Psi(\mathcal{Q}'(g,g) x_J)$ must be nonzero. Therefore $\mathcal{Q}'(g,g)$ is nonzero.
\end{proof}

Now we are ready to prove \Cref{thm:main anis}.
\begin{proof}[Proof of \Cref{thm:main anis}]
Suppose that $\mathcal{Q}(g,g)=0$ holds for $g \in A_{\bm{b}}$. As $\Psi$ is an isomorphism, we have $g^2\bm{\ell}^{\bm{a}-2\bm{b}}=0$.
By applying \Cref{lemma:WLP anis} for 
\[
g\prod_{j \in [m]} \ell_j^{\left\lfloor \frac{a_j-2b_j}{2}\right\rfloor}
\]
and $S=\{j \in [m]:a_j-2b_j \text{ is odd}\}$, we have
\begin{equation}
g\prod_{j \in [m]} \ell_j^{\left\lfloor \frac{a_j-2b_j}{2}\right\rfloor}=0. \label{eq:main proof}
\end{equation}
By multiplying $g$ to both sides of (\ref{eq:main proof}), we obtain  
\[
g^2\prod_{j \in [m]} \ell_j^{\left\lfloor \frac{a_j-2b_j}{2}\right\rfloor}=0.
\]
By repeating in this way, we can reduce the power of $\ell_j$s' and we eventually obtain $g=0$.
\end{proof}
Now \Cref{thm:main multiHL} for characteristic $2$ is immediate.
\begin{proof}[Proof of \Cref{thm:main multiHL} for characteristic $2$]
Suppose that the field $\mathbbm{k}$ is of characteristic $2$.
Define the Lefschetz elements $\ell_j$ for $j=1,\ldots,m$ as in \Cref{thm:main anis}.
Then, \Cref{thm:main anis} implies that the linear map $\times \bm{\ell}^{\bm{a}-2\bm{b}}:A_{\bm{b}} \rightarrow A_{\bm{a}-\bm{b}}$ is injective for every $\bm{b} \leq \frac{\bm{a}}{2}$.
By multigraded Poincar\'{e} duality of $A$, we have $\dim A_{\bm{b}}=\dim A_{\bm{a}-\bm{b}}$, and thus the map is an isomorphism.
\end{proof}
\Cref{thm:flag-h} is readily obtained as a corollary of \Cref{thm:main multiHL}.
\begin{proof}[Proof of \Cref{thm:flag-h}]
By \Cref{thm:main multiHL} over a field $\mathbbm{k}$ of characteristic $2$, the composite
\[
A_{\bm{b}} \overset{\times \bm{\ell}^{\bm{c}-\bm{b}}}{\longrightarrow} A_{\bm{c}} \overset{\times \bm{\ell}^{\bm{a}-\bm{b}-\bm{c}}}{\longrightarrow} A_{\bm{a}-\bm{b}}
\]
is an isomorphism. So, the linear map $\times \bm{\ell}^{\bm{c}-\bm{b}}:A_{\bm{b}}\rightarrow A_{\bm{c}}$ is injective.
Thus, $h_{\bm{b}} = \dim A_{\bm{b}} \leq \dim A_{\bm{c}} = h_{\bm{c}}$ holds.
\end{proof}

\section{Further results in the full preprint}
We end this extended abstract by listing the further contents of the full paper~\cite{Oba24}.
We give a proof of \Cref{thm:main multiHL} for a field of characteristic $0$.
We further generalize \Cref{thm:main multiHL} to manifolds and simplicial cycles and doubly Cohen--Macaulay complexes. 
This is a multigraded generalization of the almost strong Lefschetz property of manifolds~\cite[Section 8]{AY} and the strong Lefschetz property of simplicial cycles (after Gorensteinification)~\cite[Theorem I]{APP} and the top-heavy strong Lefschetz property for doubly Cohen--Macaulay complexes~\cite[Corollary 3.2]{APP}.
A combinatorial corollary, we obtain a generalization of \Cref{thm:flag-h} to the flag $h''$-vector of manifolds (without boundary), which is a common generalization of manifold GLBI~\cite{NS} and balanced manifold GLBI~\cite{JMNS}.


\acknowledgements{I thank Satoshi Murai for his comments. I have also benefited a lot from his unpublished manuscript. 
I also thank Shin-ichi Tanigawa for his supervision and helpful comments.}

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\end{document}


\section{From characteristic $2$ to characteristic $0$}
In this section, we prove the multigraded strong Lefschetz property (\Cref{thm:main multiHL}) over a field of characteristic $0$ based on the result over a field of characteristic $2$.
Though the argument in this section may be well-known, we include it for completeness.

We start with a lemma about the basis (see also~\cite[Lemma 5.1]{KX}).
\begin{lemma} \label{lem:basis}
Let $(\Delta,\kappa)$ be an $\bm{a}$-balanced homology sphere over $\mathbb{F}_2$ and let $\mathbbm{k}$ be a field of characteristic $0$.
Let $\mathcal{B}$ be a set of monomials that forms a basis of the generic $\mathbb{N}^m$-graded Artinian reduction $\widetilde{\mathbb{F}_2}[\Delta]/(\Theta)$ of $\mathbb{F}_2[\Delta]$.
Then, $\mathcal{B}$ also forms a basis of the generic $\mathbb{N}^m$-graded Artinian reduction $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ of $\mathbbm{k}[\Delta]$.
\end{lemma}
\begin{proof}
Let $d=|\bm{a}|$ and let $\widetilde{\mathbbm{k}}=\mathbbm{k}(p_{k,v})$.
By~\cite[Lemma 2.1 (2)]{KX} and the fact that whether $\Delta$ is a homology sphere over a given field only depends on its characteristic, $\Delta$ is also a homology sphere over $\widetilde{\mathbbm{k}}$.
Hence, by Reisner's theorem, $\widetilde{\mathbb{F}_2}[\Delta]/(\Theta)$ and $A$ has the same dimension $h_0+\cdots+h_d$ as a $\widetilde{\mathbb{F}_2}$-vector space and a $\widetilde{\mathbbm{k}}$-vector space, respectively.
Thus, it suffices to prove that $\mathcal{B}$ is linearly independent in $A$.

Suppose on the contrary, $\mathcal{B}$ is linearly dependent in $A$.
Then, in $\widetilde{\mathbbm{k}}[\Delta]$, there are linearly dependent elements $b_1 \bm{\theta}^{\bm{\alpha}^1}, \ldots, b_k \bm{\theta}^{\bm{\alpha}^k}$ ($b_i \in \mathcal{B}, \bm{\alpha}^i \in \mathbb{N}^m$), where $\bm{\theta}^{\bm{\alpha}} = \theta_1^{\alpha_1} \cdots \theta_d^{\alpha_d}$.
Using the standard basis $\{m_j\}$ of $\widetilde{\mathbbm{k}}[\Delta]$ consisting of all monomials whose support is contained in $\Delta$, write these elements as 
$b_l\bm{\theta}^{\bm{\alpha}^l}=\sum_{j \in J} t_{lj}m_j$ for some finite subset $J$.
Then the set of vectors $\{(t_{lj})_{j\in J}:l=1,\ldots,k\}$ is linearly dependent over a field of characteristic $0$.
Note that each $t_{lj}$ is a $\mathbb{Z}$-coefficient polynomial of $p_{k,v}$, so all the subdeterminants of the matrix $(t_{lj})_{l \in [k],j\in J}$ are in $\mathbb{Z}[p_{k,v}]$.
Since a polynomial in $\mathbb{Z}[p_{k,v}]$ vanishes in characteristic $2$ if it vanishes in characteristic $0$.
Hence the set of vectors $\{(t_{lj})_{j\in J}:l=1,\ldots,k\}$ is also linearly dependent over a field of characteristic $2$, and thus the same set of elements $b_1 \bm{\theta}^{\bm{\alpha}^1}, \ldots, b_k \bm{\theta}^{\bm{\alpha}^k}$ is linearly dependent in $\widetilde{\mathbb{F}_2}[\Delta]$.
This is a contradiction since $\widetilde{\mathbb{F}_2}[\Delta]$ is a free $\widetilde{\mathbb{F}_2}[\theta_1,\ldots,\theta_d]$-module.
\end{proof}
We now prove \Cref{thm:main multiHL} for a field of characteristic $2$.
\begin{proof}[Proof of \Cref{thm:main multiHL} for a characteristic $0$ field]
Let $\mathbbm{k}$ be a field of characteristic $0$ and let $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$.
Define Lefschetz elements as in \Cref{thm:main anis}.
As discussed in the proof \Cref{lem:basis}, $\Delta$ is also a homology sphere over $\widetilde{\mathbbm{k}}$.
Hence $\mathbbm{k}[\Delta]$ is a free $\mathbbm{k}[\theta_1,\ldots,\theta_d]$-module and $\dim A_{\bm{b}}=\dim A_{\bm{a}-\bm{b}}$ holds.
So it suffices to prove the injectivity of the linear map $\times\bm{\ell}^{\bm{a}-2\bm{b}}:A_{\bm{b}} \rightarrow A_{\bm{a}-\bm{b}}$.

For this, pick a basis $\mathcal{B}$ of $\widetilde{\mathbb{F}_2}[\Delta]/(\Theta)$ consisting of monomials (such a basis can be obtained by ordering all the monomials of $\mathbb{F}_2[\bm{x}]$ of degree at most $d=|\bm{a}|$ and choosing linearly independent ones in a greedy manner).
By \Cref{lem:basis}, $\mathcal{B}$ is also a basis for $A$. 
Let $\{m_1,\ldots,m_k\}=\mathcal{B} \cap \widetilde{\mathbbm{k}}[\bm{x}]_{\bm{b}}$ be a basis of $A_{\bm{b}}$.
Then $\times\bm{\ell}^{\bm{a}-2\bm{b}}:A_{\bm{b}} \rightarrow A_{\bm{a}-\bm{b}}$ is injective if and only if the set of polynomials $\mathcal{S}=\{m_l\bm{\ell}^{\bm{a}-2\bm{b}} \bm{\theta}^{\bm{\alpha}}: l=1,\ldots,k, \bm{\alpha} \in \mathbb{N}^d\}$ is linearly independent in $\widetilde{\mathbbm{k}}[\Delta]$.
By \Cref{thm:main multiHL} for a field of characteristic $2$, we know that $\mathcal{S}$ is linearly independent in $\widetilde{\mathbb{F}_2}[\Delta]$. 
By the similar argument as in the proof of \Cref{lem:basis}, this implies the linear independence of $\mathcal{S}$ in $\widetilde{\mathbbm{k}}[\Delta]$.
Thus the map $\times\bm{\ell}^{\bm{a}-2\bm{b}}:A_{\bm{b}} \rightarrow A_{\bm{a}-\bm{b}}$ is injective.
\end{proof}

\section{Manifolds, simplicial cycles, 2-CM complexes}
\subsection{Manifolds}
For homology manifolds over $\mathbb{F}_2$, we have the following theorem, which can be seen as a multigraded version of almost Lefschetz property.
\begin{theorem} \label{thm:manifold almost HL}
For $\bm{a} \in \mathbb{N}^m_+$, let $(\Delta,\kappa)$ be an $\bm{a}$-balanced homology manifold over $\mathbb{F}_2$.
For a field $\mathbbm{k}$ of characteristic $0$ or $2$, let $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ be the generic $\mathbb{N}^m$-graded Artinian reduction of $\mathbbm{k}[\Delta]$, and let $\ell_j$ be a generic element in $A_{\bm{e}_j}$ for $j=1,\ldots,m$.
Then, for any $\bm{b} \in \mathbb{N}^m$ and $j \in [m]$ with $\bm{b} \leq \frac{\bm{a}-\bm{e}_j}{2}$, the multiplication map
\[
\times \bm{\ell}^{\bm{a}-2\bm{b}-\bm{e}_j} : A_{\bm{b}+\bm{e}_j} \to A_{\bm{a}-\bm{b}}
\]
is surjective.
\end{theorem}
To prove \Cref{thm:manifold almost HL}, we highlight useful conventions for star and link.
Let $\Theta$ be an l.s.o.p.~for $\mathbbm{k}[\Delta]$.
Recall that $\Theta$ is identified with a point configuration $p:V(\Delta) \rightarrow \mathbb{R}^{\dim \Delta}$.
The l.s.o.p.~for the star is obtained by the restriction of $p$ to $V(\st_\tau \Delta)$, and an l.s.o.p.~for the link is obtained by the orthogonal projection of $p|_{V(\st_\tau \Delta)}$ to $(\spa p(\tau))^\bot$.
Throughout, we always assume this convention for star and link, and when $A=\mathbbm{k}[\Delta]/(\Theta)$ is the Artinian reduction of $\mathbbm{k}[\Delta]$ with respect to $\Theta$, the corresponding Artinian reduction of $\mathbbm{k}[\st_\tau \Delta]$ and $\mathbbm{k}[\lk_\tau \Delta]$ are denoted by $A(\st_\tau \Delta)$ and $A(\lk_\tau \Delta)$, respectively.
Note also that if $(\Delta,\kappa)$ is $\bm{a}$-balanced and $\Theta$ is an $\mathbb{N}^m$-graded l.s.o.p., for a face $\tau \in \Delta^{(\bm{b})}$, $\lk_\tau(\Delta)$ is $(\bm{a}-\bm{b})$-balanced by the restriction of $\kappa$ and the corresponding l.s.o.p.~for $\mathbbm{k}[\lk_\tau \Delta]$ is also $\mathbb{N}^m$-graded.
Here, for an $\bm{a}$-balanced simplicial complex $(\Delta,\kappa)$, we denote $\Delta^{(\bm{b})}=\{\tau\in\Delta:|\tau \cap \kappa^{-1}(j)|=b_j \text{ for all }j\in [m]\}$.
We have the following lemmas.
\begin{lemma}[Cone lemma \cite{Lee}] \label{lem:cone}
Let $(\Delta,\kappa)$ be an $\bm{a}$-balanced simplicial complex with $\bm{a}\in\mathbb{N}^m$ and let $\Theta$ be an $\mathbb{N}^m$-graded l.s.o.p.~for $\mathbbm{k}[\Delta]$.
Let $A=\mathbbm{k}[\Delta]/(\Theta)$ be the Artinian reduction of $\mathbbm{k}[\Delta]$ with respect to $\Theta$.
Then for any vertex $v \in V(\Delta)$ and any $\bm{b} \in \mathbb{N}^m$, we have an isomorphism
\[
A(\lk_v \Delta)_{\bm{b}} \cong A(\st_v \Delta)_{\bm{b}}.
\]
\end{lemma}
\begin{lemma} \label{lem:partition}
Let $(\Delta,\kappa)$ be an $\bm{a}$-balanced simplicial complex with $\bm{a}\in\mathbb{N}^m$ and let $\Theta$ be an $\mathbb{N}^m$-graded l.s.o.p.~for $\mathbbm{k}[\Delta]$.
Let $A=\mathbbm{k}[\Delta]/(\Theta)$ be the Artinian reduction of $\mathbbm{k}[\Delta]$ with respect to $\Theta$.
Then, for each $j \in [m]$, there is a degree preserving surjection 
\[
\bigoplus_{v \in \Delta^{(\bm{e}_j)}} A(\st_v \Delta)_{*} \twoheadrightarrow A(\Delta)_{*+\bm{e}_j}.
\]
\end{lemma}
\begin{proof}
For each $v \in \Delta^{(\bm{e}_j)}$, the multiplication by $x_v$ induces a map $\varphi_v: \mathbbm{k}[\st_v \Delta]_* \overset{\cdot x_v}{\rightarrow} \mathbbm{k}[\Delta]_{*+\bm{e}_j}$. Consider their sum
\[
\varphi:\bigoplus_{v \in \Delta^{(\bm{e}_j)}} \mathbbm{k}[\st_v \Delta]_* \rightarrow \mathbbm{k}[\Delta]_{*+\bm{e}_j}
\]
over all $v \in \Delta^{(\bm{e}_j)}$.
Then $\varphi$ is surjective since every monomial of $\mathbbm{k}[\bm{x}]$ with degree at least $\bm{e}_j$ is divisible by some $x_v$ with $v \in \Delta^{(\bm{e}_j)}$.
So, $\varphi$ induces a surjection between the Artinian reductions.
\end{proof}
\begin{proof}[Proof of \Cref{thm:manifold almost HL}]
We have the following commuting diagram:
\[
\begin{tikzpicture}[auto]
\node (a) at (0,1.2) {$A(\Delta)_{\bm{b}+\bm{e}_j}$}; 
\node (b) at (6,1.2) {$A(\Delta)_{\bm{a}-\bm{b}}$};
\node (c) at (0,0) {$\bigoplus_{v \in \Delta^{(\bm{e}_j)}} A(\lk_v \Delta)_{\bm{b}}$}; 
\node (d) at (6,0) {$\bigoplus_{v \in \Delta^{(\bm{e}_j)}} A(\lk_v \Delta)_{\bm{a}-\bm{b}-\bm{e}_j}$};
\draw[->] (a) to node {$\times \bm{\ell}^{\bm{a}-2\bm{b}-\bm{e}_j}$} (b);
\draw[->] (c) to node[above] {$\times \bm{\ell}^{\bm{a}-2\bm{b}-\bm{e}_j}$} node[below] {$\cong$} (d);
\draw[->>] (c) to (a);
\draw[->>] (d) to (b);
\end{tikzpicture}
\]
Here, the vertical maps are the composite of an isomorphism in \Cref{lem:cone} and a surjection in \Cref{lem:partition}.
As, for each $v \in \Delta^{(\bm{e}_j)}$, $(\lk_v \Delta,\kappa_{|V(\lk_\tau \Delta)})$ is an $(\bm{a}-\bm{e}_j)$-balanaced homology sphere over $\mathbb{F}_2$, by \Cref{thm:main multiHL}, the map
\[
\times \bm{\ell}^{\bm{a}-2\bm{b}-\bm{e}_j} :A(\lk_v \Delta)_{\bm{b}} \rightarrow A(\lk_v \Delta)_{\bm{a}-\bm{b}-\bm{e}_j}
\]
is an isomorphism.
The horizontal map below in the diagram is a composite of these isomorphisms, so it is an isomorphism. Thus the horizontal map above is surjective.
\end{proof}

Let us derive a numerical consequence \Cref{thm:manifold flag_h} of \Cref{thm:manifold almost HL} in terms of flag $h''$-vector, a suitable modification of flag $h$-vector.
Given a field $\mathbbm{k}$, for an $\bm{a}$-balanced simplicial complex $(\Delta,\kappa)$, the \emph{flag $h'$-vector} $(h'_{\bm{b}})_{\bm{0} \leq \bm{b} \leq \bm{a}}$ and the \emph{flag $h''$-vector} $(h''_{\bm{b}})_{\bm{0} \leq \bm{b} \leq \bm{a}}$ is defined by 
\begin{align*}
h'_{\bm{b}}&=h_{\bm{b}} -\binom{\bm{a}}{\bm{b}} \left( \sum_{j=1}^{|\bm{b}|-1}(-1)^{|\bm{b}|-j}\widetilde{\beta}_{j-1} \right), \\
h''_{\bm{b}} &= 
    \begin{cases}
    h'_{\bm{b}} - \binom{\bm{a}}{\bm{b}} \widetilde{\beta}_{|\bm{b}|-1} & (\text{if } \bm{b} \neq \bm{a})\\
    h'_{\bm{b}} &(\text{if } \bm{b} = \bm{a})
    \end{cases}.
\end{align*}
Note that for Cohen--Macaulay complex, both the flag $h'$-, $h''$-vector coincide with the flag $h$-vector.
For every $(d-1)$-dimensional simplicial complex, its $h'$-vector $(h'_i)_{0 \leq i \leq d}$ and $h''$-vector $(h''_i)_{0 \leq i \leq d}$ is defined as the flag $h'$-, $h''$-vector considered as a monochromatic simplicial complex.
Note that $h'_i=\sum_{|\bm{b}|=i} h'_{\bm{b}}$ and $h''_i=\sum_{|\bm{b}|=i} h''_{\bm{b}}$ hold.

For homology manifolds, algebraic interpretations of $h'$-, $h''$-vectors are given by Schenzel~\cite{Sch81} and Novik and Swartz~\cite{NS09}.
The following lemma is the balanced analogue of these interpretations about flag $h'$-, $h''$-vectors.
A homology $(d-1)$-manifold $\Delta$ over $\mathbbm{k}$ is \emph{orientable} if $\widetilde{\beta}_{d-1}$ is equal to the number of connected components of $\Delta$.
Recall that, for a graded $\mathbbm{k}[x_1,\ldots,x_n]$-module $M$, its socle is the submodule $\Soc(M)=\{a \in M: \mathfrak{m}a=0\}$, where $\mathfrak{m}=(x_1,\ldots,x_n)$ is the maximal graded ideal.
\begin{lemma} \label{lem:flag h', h''}
Let $(\Delta,\kappa)$ be an $\bm{a}$-balanced connected homology manifold over $\mathbbm{k}$ and let $\Theta$ be an $\mathbb{N}^m$-graded l.s.o.p.~for $\mathbbm{k}[\Delta]$.
Let $A=\mathbbm{k}[\Delta]/(\Theta)$ be an Artinian reduction of $\mathbbm{k}[\Delta]$ with respect to $\Theta$. Then,
\begin{enumerate}
\item[(i)] (Schenzel's formula~\cite[Theorem 3.1]{JMNS}) $h'_{\bm{b}}=\dim A_{\bm{b}}$ for each $\bm{b} \in \mathbb{N}^m$.
\item[(ii)] \cite[Corollary 3.3]{JMNS} $h''_{\bm{b}}=\dim A_{\bm{b}}/\Soc^\circ_{\bm{b}}$ for each $\bm{b} \in \mathbb{N}^m$, where $\Soc^\circ=\bigoplus_{\bm{0} \leq \bm{b} \lneq \bm{a}}\Soc(A)_{\bm{b}}$ denotes the internal socle of $A$.
\item[(iii)] (Dehn--Sommerville relation~\cite[Theorem 4.1]{JMNS}) Moreover if $\Delta$ is orientable, $h''_{\bm{b}}=h''_{\bm{a}-\bm{b}}$ for each $\bm{b} \in \mathbb{N}^m$.
\end{enumerate}
\end{lemma}
We remark that every homology manifold over a field of characteristic $2$ is orientable.

\begin{theorem} \label{thm:manifold flag_h}
For $\bm{a} \in \mathbb{N}^m_+$, let $(\Delta,\kappa)$ be an $\bm{a}$-balanced connected homology manifold over $\mathbb{F}_2$.
Let $\bm{b}, \bm{c} \in \mathbb{N}^m$ be integer vectors with $\bm{b} \lneq \bm{c} \lneq \bm{a}-\bm{b}$\footnote{$\bm{b} \lneq \bm{c}$ stands for $\bm{b} \leq \bm{c}$ and $\bm{b} \neq \bm{c}$.}.
Then we have $h''_{\bm{c}} \geq h''_{\bm{b}}+ \binom{\bm{a}}{\bm{b}}\widetilde{\beta}_{|\bm{b}|}$, where $\widetilde{\beta}_i$ is the $i$-th reduced betti number of $\Delta$ over $\mathbb{F}_2$.

In particular, 
\[
\frac{h_i''+\binom{d}{i}\widetilde{\beta}_{i}}{\binom{m+i-1}{i}} \leq \frac{h_{i+1}''}{\binom{m+i}{i+1}}
\]
holds for every nonnegative integer $i \in \mathbb{N}$ with $i\leq \min_{j=1}^m \frac{a_j-1}{2}$, where $d=|\bm{a}|$.
\end{theorem}
\begin{proof}[Proof of \Cref{thm:manifold flag_h}]
Let $\bm{b}, \bm{c} \in \mathbb{N}^m$ be integer vectors with $\bm{b} \lneq \bm{c} \lneq \bm{a}-\bm{b}$.
Then there is some $j\in[m]$ with $\bm{b}+\bm{e}_j \leq \bm{c}$.
By \Cref{thm:manifold almost HL} the compositie $A(\Delta)_{\bm{b}+\bm{e}_j} \overset{\times \bm{\ell}^{\bm{c}-\bm{b}-\bm{e}_j}}{\longrightarrow} A(\Delta)_{\bm{c}} \overset{\times \bm{\ell}^{\bm{a}-\bm{b}-\bm{c}}}{\longrightarrow} A(\Delta)_{\bm{a}-\bm{b}}$ is surjective, so the latter map $\times \bm{\ell}^{\bm{a}-\bm{b}-\bm{c}}:A(\Delta)_{\bm{c}} \rightarrow A(\Delta)_{\bm{a}-\bm{b}}$ is surjective.
Since $\bm{c} \lneq \bm{a}-\bm{b}$, the degree $\bm{c}$ component of the socle is contained in the kernel of this map. 
Thus by \Cref{lem:flag h', h''}, we have the inequality
\begin{equation}
h''_{\bm{c}}=\dim A(\Delta)_{\bm{c}} - \dim \Soc(A(\Delta))_{\bm{c}} \geq \dim A(\Delta)_{\bm{a}-\bm{b}} = h'_{\bm{a}-\bm{b}}. \label{eq:mfd_h}
\end{equation}
All that remains is to convert (\ref{eq:mfd_h}) to the inequality between flag $h''$-vectors.
For this, if $\bm{b}\neq\bm{0}$, by \Cref{lem:flag h', h''} (iii), we have 
\begin{align}
h'_{\bm{a}-\bm{b}} &=h''_{\bm{a}-\bm{b}}+\binom{\bm{a}}{\bm{b}}\widetilde{\beta}_{|\bm{a}-\bm{b}|-1} & \notag \\
&=h''_{\bm{b}} + \binom{\bm{a}}{\bm{b}}\widetilde{\beta}_{|\bm{b}|} & (\widetilde{\beta}_{k}=\widetilde{\beta}_{d-k-1} \text{ for } k \geq 1 \text{ by Poincar\'{e} duality}), \label{eq:mfd_betti}
\end{align}
while the same equality (\ref{eq:mfd_betti}) also holds for $\bm{b}=\bm{0}$ since $h'_{\bm{a}}=1$ and $h''_{\bm{0}}+\binom{\bm{a}}{\bm{0}}\beta_{0}=1+0=1$. By (\ref{eq:mfd_h}) and (\ref{eq:mfd_betti}), the desired inequality $h''_{\bm{c}} \geq h''_{\bm{b}}+\binom{\bm{a}}{\bm{b}}\widetilde{\beta}_{|\bm{b}|}$ follows.

The inequality between $h''$-vectors is obtained by taking a weighted sum of the inequalies $h''_{\bm{b}} + \binom{\bm{a}}{\bm{b}}\widetilde{\beta}_{|\bm{b}|} \leq h''_{\bm{b}+\bm{e}_j}$ over all $\bm{b} \in \mathbb{N}^m$ with $|\bm{b}|=i$ and $j \in [m]$ as follows:
\[
(m+i) (h''_i+\binom{d}{i}\widetilde{\beta}_i)
=\sum_{|\bm{b}|=i}\sum_{j=1}^m (b_j+1)\left(h''_{\bm{b}}+\binom{\bm{a}}{\bm{b}}\widetilde{\beta}_i \right) \leq \sum_{|\bm{b}|=i}\sum_{j=1}^m (b_j+1)h''_{\bm{b}+\bm{e}_j}
=(i+1) h''_{i+1}.
\]
\end{proof}

\subsection{Simplicial Cycle}
For a $(d-1)$-dimensional simplicial complex $\Delta$ with $\widetilde{H}_{d-1}(\Delta;\mathbbm{k}) \neq 0$, we call a nonzero element $\mu \in \widetilde{H}_{d-1}(\Delta;\mathbbm{k})$ a \emph{simplicial cycle}.
Let $\Delta$ be a $(d-1)$-dimensional simplicial complex, and let $A=\mathbbm{k}[\Delta]/(\Theta)$ be an Artinian reduction of $\mathbbm{k}[\Delta]$.
A simplicial cycle $\mu$ of $\Delta$ defines a nonzero linear function $\Psi_\mu:A_{d} \rightarrow \mathbbm{k}$ by 
\begin{equation}
\Psi_\mu(x_\sigma) = \frac{\mu(\sigma)}{[\sigma]} \label{eq:cycle}
\end{equation}
for each (oriented) facet $\sigma \in \Delta$, where $[\sigma]$ is computed as in \Cref{lem:Lee formula} (note that the linear function on $A_d$ is determined by the values of squarefree monomials~\cite[Theorem 9]{Lee}).
In general, a nonzero linear function $\varphi:\mathbbm{k}[\bm{x}]_d \rightarrow \mathbbm{k}$ determines a standard Artinian Gorenstein graded algebra $\mathbbm{k}[\bm{x}]/I$ by the condition that $f \in \mathbbm{k}[\bm{x}]_k$ is in $I$ if and only if either $k \geq d+1$ or $\varphi(fg)=0$ for all $g \in \mathbbm{k}[\bm{x}]_{d-k}$.
Thus the map $\Psi_\mu$ (by concatenating a projection $\mathbbm{k}[\Delta]_{d} \twoheadrightarrow A_{d}$ to it) determines an Artinian Gorenstein $\mathbbm{k}$-algebra denoted as $B(\mu)$, which is called the \emph{Gorensteinification} of $A$.
Note that if $\Delta$ is a homology sphere, $A$ itself is Gorenstein, so we have $A=B(\mu)$ for nonzero $\mu \in \widetilde{H}_{d-1}(\Delta;\mathbbm{k})$.
If $A$ is $\mathbb{N}^m$-graded, $B(\mu)$ is also $\mathbb{N}^m$-graded and satisfies multigraded Poincar\'{e} duality.

\Cref{thm:main multiHL} is generalized to a simplicial cycle over a field of characteristic $2$ as follows.
\begin{theorem} \label{thm:simplcial cycle}
For an $\bm{a}$-balanced simplicial complex $(\Delta,\kappa)$, let $\mu$ be a simplicial cycle of $\Delta$ over a field $\mathbbm{k}$ of characteristic $2$.
Let $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ be the generic $\mathbb{N}^m$-graded Artinian reduction of $\mathbbm{k}[\Delta]$ and let $B(\mu)$ be the Gorensteinification of $A$ with respect to $\mu$.
Then for generic elements $\ell_j \in B(\mu)_{\bm{e}_j}$ for $j=1,\ldots,m$, the multiplication map
\[
\times \bm{\ell}^{\bm{a}-2\bm{b}}: B(\mu)_{\bm{b}} \rightarrow B(\mu)_{\bm{a}-\bm{b}}
\]
is an isomorphism for every $\bm{b}\in\mathbb{N}^m$ with $\bm{b} \leq \frac{\bm{a}}{2}$.
\end{theorem}
\begin{proof}[Proof Sketch]
We only give a proof sketch as the discussion is almost the same as in Section 3-5.
By the similar discussion as in~\cite[Section 2.6]{KX}, by introducing the cone vertex, the simplicial cycle $\mu$ can be written as a weighted connected sum of simplex boundaries with the weights in $\mathbbm{k}$.
Accordingly, the map $\Psi_\mu$ is written as the weighted sum of the evaluation map of simplex boundaries. 
Hence, $\Psi=\Psi_\mu$ also satisfies the differential formula in \Cref{lem:diff}, and thus \Cref{cor:diff general}.
In the argument in Section 5, we have only used the fact that $A$ has multigraded Poincar\'{e} duality, which is now passed to $B(\mu)$. Hence we can reproduce the argument in Section 5 by replacing $A$ with $B(\mu)$.
\end{proof}
\begin{remark}
To deduce a (generalized) lower bound type inequality from \Cref{thm:simplcial cycle}, it is a key to understand the values $\dim A_*$, $\dim A_* - \dim B(\mu)_*$ and $\dim \Soc B(\mu)_*$.
Unfortunately, we have little knowledge of these numbers other than the case of homology manifolds, except for partial cases e.g. normal pseudomanifolds~\cite{Mu}, minimal cycle complexes~\cite{Fog,Oba}.
The understanding of these numbers would also be important in the perspective of skeletal rigidity~\cite{TW}.
\end{remark}
From \Cref{thm:simplcial cycle}, we can obtain the following multigraded version of top-heavy Lefschetz property for doubly Cohen--Macaulay complexes.
A simplicial complex $\Delta$ is called a \emph{doubly Cohen--Macaulay} complex (or \emph{$2$-CM} complex) over a field $\mathbbm{k}$ if $\Delta$ is Cohen--Macaulay over $\mathbbm{k}$ and for each vertex $v$, $\Delta-v$ is Cohen--Macaulay over $\mathbbm{k}$ and has the same dimension as $\Delta$.
\begin{theorem} \label{thm:2CM lefschetz}
For $\bm{a} \in \mathbb{N}^m_+$, let $(\Delta,\kappa)$ be an $\bm{a}$-balanced doubly Cohen--Macaulay complex over a field $\mathbbm{k}$ of characteristic $2$.
Let $A=\widetilde{\mathbbm{k}}[\Delta]/(\Theta)$ be the generic $\mathbb{N}^m$-graded Artinian reduction of $\mathbbm{k}[\Delta]$.
Then, for generic elements $\ell_j \in A_{\bm{e}_j}$ for $j=1,\ldots,m$, the multiplication map
\[
\times \bm{\ell}^{\bm{a}-2\bm{b}}: A_{\bm{b}} \rightarrow A_{\bm{a}-\bm{b}}
\]
is injective for any $\bm{b} \in \mathbb{N}^m$ with $\bm{b} \leq \frac{\bm{a}}{2}$.
\end{theorem}
\begin{proof}
Let $d=|\bm{a}|$ and let $\mu_1,\ldots,\mu_k$ be a basis of $\widetilde{H}_{d-1}(\Delta;\mathbbm{k})$.
It is known that an Artinian reduction of the Stanley--Reisner ring of a doubly Cohen--Macaulay complex is a level ring~\cite[Section III.3]{Sta-book}, that is, $\Soc (A)=A_{\bm{a}}$.
This implies that for any nonzero $x \in A_{\bm{b}}$, there is $y\in A_{\bm{a}-\bm{b}}$ such that $(\Psi_{\mu_1}(xy),\ldots,\Psi_{\mu_k}(xy))\neq0$.
Thus the map $A(\Delta)_* \rightarrow \bigoplus_{i=1}^k B(\mu_i)_*$ is injective.
Consider the following commuting diagram:
\[
\begin{tikzpicture}[auto]
\node (a) at (0,1.2) {$A(\Delta)_{\bm{b}}$}; 
\node (b) at (4,1.2) {$A(\Delta)_{\bm{a}-\bm{b}}$};
\node (c) at (0,0) {$\bigoplus_{l=1}^k B(\mu_l)_{\bm{b}}$}; 
\node (d) at (4,0) {$\bigoplus_{l=1}^k B(\mu_l)_{\bm{a}-\bm{b}}$};
\draw[->] (a) to node {$\times \bm{\ell}^{\bm{a}-2\bm{b}}$} (b);
\draw[->] (c) to node[above] {$\times \bm{\ell}^{\bm{a}-2\bm{b}}$} node[below] {$\cong$} (d);
\draw[right hook->] (a) to (c);
\draw[right hook->] (b) to (d);
\end{tikzpicture}
\]
As the bottom map is an isomorphism by \Cref{thm:simplcial cycle}, the top map is injective.
\end{proof}
Now \Cref{thm:2CM_flag h} is readily derived.
\begin{proof}[Proof of \Cref{thm:2CM_flag h}]
For $\bm{b} \leq \bm{c} \leq \bm{a}-\bm{b}$, \Cref{thm:2CM lefschetz} implies that the linear map $\times \bm{\ell}^{\bm{b}-\bm{c}}:A_{\bm{b}} \rightarrow A_{\bm{c}}$ is injective. Thus $h_{\bm{b}}=\dim A_{\bm{b}} \leq \dim A_{\bm{c}} = h_{\bm{c}}$ follows.
The inequality between the $h$-vectors follows by the same argument as in the proof of \Cref{cor:h-vec}.
\end{proof}