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\title{A reduction of the Gorenstein symmetry conjecture\\to gendo-symmetric algebras}
\author{Qiyue Tang}
\date{}
\begin{document}
\maketitle
\begin{abstract}
Over any field, the Gorenstein symmetry conjecture for finite-dimensional algebras is equivalent to its restriction to gendo-symmetric algebras.
\end{abstract}
\begingroup
\renewcommand{\thefootnote}{}
\footnotetext{\textit{2020 Mathematics Subject Classification.} Primary 16E10; Secondary 16G10, 16D50.\newline
\textit{Keywords.} Gorenstein symmetry conjecture, gendo-symmetric algebra, trivial extension, injective dimension.}
\endgroup

\section{Introduction}
Let \(k\) be a field. All algebras in this paper are nonzero finite-dimensional associative \(k\)-algebras with identity, and all modules are finite-dimensional and unital. Modules are right modules unless a left action is specified. Write \(D=\Hom_k(-,k)\) for linear duality. For a right module \(M\) over an algebra \(R\), write \(\pd_R M\) and \(\id_R M\) for its projective and injective dimensions; these dimensions are allowed to be infinite. We display the side explicitly for regular modules. For right \(R\)-modules \(U,V\), the group \(\Ext_R^n(U,V)\), with integer \(n\ge0\), is the degree-\(n\) cohomology of \(\Hom_R(P_\bullet,V)\), where \(P_\bullet\to U\) is a projective resolution indexed in nonnegative degrees. In particular, \(\Ext_R^0(U,V)=\Hom_R(U,V)\).

For any algebra \(R\), the bimodule structure on \(DR\) is
\((r\varphi s)(t)=\varphi(str)\) for \(r,s,t\in R\) and \(\varphi\in DR\).
The Gorenstein symmetry conjecture asserts that every such algebra \(A\) satisfies
\begin{equation}\label{eq:gsc}
 \id({}_A A)<\infty
 \quad\Longleftrightarrow\quad
 \id(A_A)<\infty.
\end{equation}
A finite-dimensional algebra \(S\) is \emph{symmetric} if \(DS\cong S\) as \(S\)-bimodules. A right \(S\)-module \(M\) is a \emph{generator} if \(S\in\add M\), where \(\add M\) consists of direct summands of finite direct sums of copies of \(M\). An algebra is \emph{gendo-symmetric} if it is isomorphic to \(\End_S(M)\) for a generator \(M\) over a symmetric algebra \(S\). Products in endomorphism algebras are composition: \(\alpha\beta=\alpha\circ\beta\).

Gendo-symmetric algebras were studied by Fang and Koenig \cite{FK11,FK16}. Aihara, Chan and Honma \cite{ACH19} consider endomorphism algebras of generators over trivial extensions. For a self-injective algebra \(S\) and an indecomposable nonprojective \(S\)-module \(X\), Chen and Koenig \cite[Corollary~3.25]{CK15} prove Gorenstein symmetry for \(\End_S(S\oplus X)\). Han and Qin \cite[Corollary~8]{HQ14} give a reduction of the conjecture through recollements of derived categories. The reduction here uses the canonical quotient of a trivial extension and applies to every base algebra.

For an algebra \(A\), its trivial extension \(T(A)=A\ltimes DA\) is the vector space \(A\oplus DA\) with multiplication, for \(a,b\in A\) and \(\lambda,\mu\in DA\),
\[
 (a,\lambda)(b,\mu)=(ab,a\mu+\lambda b)
\]
and identity \((1,0)\). The projection \(\pi:T(A)\to A\), \((a,\lambda)\mapsto a\), makes \(A\) a right \(T(A)\)-module by \(x\cdot b=x\pi(b)\). Define
\begin{equation}\label{eq:construction}
 B=T(A),\qquad E(A)=\End_B(B\oplus A).
\end{equation}
When \(A\) is fixed, we write \(E=E(A)\).

\begin{theorem}\label{thm:main}
For every algebra \(A\), the algebra \(E=E(A)\) is gendo-symmetric and
\[
 \dim_k E=5\dim_k A.
\]
If \(d=\id({}_A A)<\infty\), then
\begin{equation}\label{eq:upper}
 \id({}_E E)\le 2d+2.
\end{equation}
Moreover,
\begin{equation}\label{eq:descent}
 \id(A_A)\le\id(E_E),
\end{equation}
where the inequality is interpreted in \(\{0,1,2,\ldots\}\cup\{\infty\}\).
\end{theorem}

The proof is completed in Section~4.

\begin{corollary}\label{cor:reduction}
For each field \(k\), the following statements are equivalent:
\begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item Every finite-dimensional \(k\)-algebra satisfies \eqref{eq:gsc}.
 \item Every finite-dimensional gendo-symmetric \(k\)-algebra satisfies \eqref{eq:gsc}.
 \item The algebra \(E(A)\) satisfies \eqref{eq:gsc} for every finite-dimensional \(k\)-algebra \(A\).
\end{enumerate}
\end{corollary}
\begin{proof}
Theorem~\ref{thm:main} gives the implications (i)\(\Rightarrow\)(ii)\(\Rightarrow\)(iii).
Assume (iii), and let \(\id({}_A A)=d<\infty\).
Then \eqref{eq:upper} and (iii) give \(\id(E(A)_{E(A)})<\infty\), and \eqref{eq:descent} gives \(\id(A_A)<\infty\).
Applying this implication to \(A^{\mathrm{op}}\), the algebra with reversed multiplication, proves the converse direction of \eqref{eq:gsc}.
\end{proof}

\paragraph{AI contributions.}
The mathematical construction and proofs were generated and reviewed with AI assistance in ChatGPT. Codex was used to recheck the module calculations and prepare this manuscript.

\section{The endomorphism algebra}
Let \(e,f\in E\) be the endomorphisms defined by \(e(z,t)=(z,0)\) and \(f(z,t)=(0,t)\) for \(z\in B\), \(t\in A\). Thus \(e+f=1\), \(ef=fe=0\), and \(e^2=e\), \(f^2=f\). We identify \(DA\) with the square-zero ideal \(\{(0,\lambda):\lambda\in DA\}\) of \(B\).

The elements \(z\in B\) satisfying \(zDA=0\) are precisely those in \(DA\). Indeed, if \((a,\lambda)(0,\mu)=0\) for every \(\mu\in DA\), then \(a\mu=0\) for all \(\mu\). Evaluating at \(1\) gives \(\mu(a)=0\) for every linear functional \(\mu\), and hence \(a=0\). A \(B\)-linear map \(A\to B\) is determined by the image of \(1\), which must annihilate \(DA\) on the right. Conversely, each \(u\in DA\) defines a \(B\)-linear map \(t\mapsto ut\): for \(t\in A\) and \(b\in B\), one has \(u(t\pi(b))=(ut)b\). Evaluation at \(1\) therefore gives
\[
 \Hom_B(A,B)=DA,\qquad
 \Hom_B(B,A)=A,\qquad
 \End_B(A)=A.
\]
Together with \(\End_B(B)=B\), these identifications give
\begin{equation}\label{eq:blocks}
 E=\begin{pmatrix}B&DA\\ A&A\end{pmatrix}.
\end{equation}
Here \(b\in B\) represents left multiplication on \(B\), \(u\in DA\) represents the map \(A\to B\), \(t\mapsto ut\), \(v\in A\) represents \(B\to A\), \(z\mapsto v\pi(z)\), and the lower right entry \(a\in A\) represents left multiplication by \(a\) on \(A\).

In particular, if \(b,b'\in B\), \(u,u'\in DA\), and \(v,v',a,a'\in A\), multiplication is
\begin{equation}\label{eq:product}
 \begin{pmatrix}b&u\\v&a\end{pmatrix}
 \begin{pmatrix}b'&u'\\v'&a'\end{pmatrix}
 =
 \begin{pmatrix}
 bb'+uv'&bu'+ua'\\
 v\pi(b')+av'&aa'
 \end{pmatrix}.
\end{equation}
The \(B\)-actions on \(DA\) in this formula factor through \(\pi\).
The product \(uv'\) in the upper left block is included in the ideal \(DA\subset B\).
Every composite \(A\to B\to A\) is zero, since the first map lands in \(DA\) and the second kills \(DA\).

Consequently the map
\begin{equation}\label{eq:q}
 q:E\longrightarrow A,\qquad
 \begin{pmatrix}b&u\\v&a\end{pmatrix}\longmapsto a
\end{equation}
is an algebra epimorphism.
For a right \(A\)-module \(X\), denote by \(X_q\) the same vector space with right \(E\)-action \(x r=xq(r)\).

\begin{lemma}\label{lem:symmetric}
The algebra \(B\) is symmetric. The algebra \(E\) is gendo-symmetric and has dimension \(5\dim_k A\).
\end{lemma}
\begin{proof}
Define the linear functional \(\tau:B\to k\) by
\(\tau(a,\lambda)=\lambda(1)\). Its associated bilinear form is
\[
 \tau\bigl((a,\lambda)(b,\mu)\bigr)=\mu(a)+\lambda(b).
\]
This form is symmetric. If it vanishes for all \((b,\mu)\), setting \(b=0\) gives \(a=0\), and setting \(\mu=0\) gives \(\lambda=0\); hence it is nondegenerate. Thus \(b\mapsto[c\mapsto\tau(cb)]\) is a bijection \(B\to DB\). For \(r,s,b,c\in B\), associativity and symmetry give
\[
 \tau((scr)b)=\tau(s(crb))=\tau((crb)s)=\tau(c(rbs)).
\]
By the dual action defined in Section~1, this is the bimodule identity for the map.
Since \(B\oplus A\) contains \(B\), it is a generator, proving that \(E\) is gendo-symmetric.
The upper left block in \eqref{eq:blocks} has dimension \(2\dim_k A\); each of the other three blocks has dimension \(\dim_k A\). Their sum is \(5\dim_k A\).
\end{proof}

For a right \(B\)-module \(X\), set
\[
 H(X)=\Hom_B(B\oplus A,X).
\]
It is a right \(E\)-module by precomposition, \(h r=h\circ r\).
If \(X\) is a right \(A\)-module, its inflation along \(\pi\) has action \(xb=x\pi(b)\) for \(x\in X\), \(b\in B\). A map \(B\oplus A\to X\) then has the form \((z,t)\mapsto x\pi(z)+yt\) for unique \(x,y\in X\). Precomposing with the block matrix in \eqref{eq:product} gives
\begin{equation}\label{eq:Haction}
 H(X)=X\oplus X,\qquad
 (x,y)\begin{pmatrix}b&u\\v&a\end{pmatrix}
 =(x\pi(b)+yv,ya).
\end{equation}
For a homomorphism \(g:X\to Y\) of inflated modules, \(H(g)\) is \((x,y)\mapsto(g(x),g(y))\); hence this restriction of \(H\) is exact.
The identifications \(H(B)=eE\) and \(H(A)=fE\) send a map into a summand of \(B\oplus A\) to the corresponding row of an endomorphism. In particular, \(H\) takes projective \(A\)-modules to projective \(E\)-modules: it preserves finite direct sums and direct summands and sends \(A\) to \(fE\).

\section{The finite injective dimension}
Linear duality is exact and exchanges finite projective modules with finite injective modules on the opposite side. Dualizing resolutions therefore gives
\[
 \id({}_A A)=\pd_A DA,\qquad
 \id({}_E E)=\pd_E DE.
\]
Here both projective dimensions are those of right modules, and \(DA\), regarded as an ideal of \(B\), has the action inflated along \(\pi\).

\begin{lemma}\label{lem:quotient}
Suppose \(d=\pd_A DA<\infty\). Then
\[
 \pd_E H(DA)\le d,\qquad
 \pd_E A_q\le d+2,\qquad
 \pd_E(DA)_q\le 2d+2.
\]
If \(K\) is the kernel of \(H(A)\to A_q\), \((x,y)\mapsto y\), then
\(\pd_E K\le d+1\).
\end{lemma}
\begin{proof}
Choose an exact sequence
\[
 0\longrightarrow P_d\longrightarrow P_{d-1}\longrightarrow\cdots
 \longrightarrow P_0\longrightarrow DA\longrightarrow0
\]
of right \(A\)-modules with each \(P_i\) projective. For \(d=0\), this means \(P_0\cong DA\). Applying \(H\) to the modules inflated along \(\pi\) gives an exact sequence with \(E\)-projective terms \(H(P_i)\), proving \(\pd_E H(DA)\le d\).

Left exactness of \(H\), applied to \(0\to DA\to B\xrightarrow{\pi}A\to0\), together with the following cokernel calculation gives
\begin{equation}\label{eq:firstseq}
 0\longrightarrow H(DA)\longrightarrow eE
 \xrightarrow{H(\pi)}fE\longrightarrow A_q\longrightarrow0.
\end{equation}
We use row coordinates \(eE=B\oplus DA\) and \(fE=A\oplus A\) from \eqref{eq:blocks}.
The map \(H(\pi)\) sends \((b,u)\) to \((\pi(b),0)\). Its image is \(K=\{(x,0):x\in A\}\), and projection to the second coordinate identifies its cokernel with \(A_q\). Splicing the resolution of \(H(DA)\) into \eqref{eq:firstseq} gives \(\pd_E K\le d+1\) and \(\pd_E A_q\le d+2\).

Inflate the same length-\(d\) resolution of \(DA\) along \(q\). Every term is a direct summand of a finite sum of copies of \(A_q\), and thus has \(E\)-projective dimension at most \(d+2\).
For an exact sequence \(0\to L\to M\to N\to0\) of right \(E\)-modules, the Ext long exact sequence gives
\(\pd_E N\le\max\{\pd_E M,\pd_E L+1\}\).
To apply this bound explicitly, put \(Z_0=(DA)_q\) and, for \(0\le i<d\), let \(Z_{i+1}\) be the kernel of the induced epimorphism \((P_i)_q\to Z_i\). Then \(Z_d\cong(P_d)_q\), and the short exact sequences
\[
 0\longrightarrow Z_{i+1}\longrightarrow(P_i)_q
 \longrightarrow Z_i\longrightarrow0\qquad(0\le i<d)
\]
give, by descending induction, \(\pd_E Z_i\le d+2+d-i\). Taking \(i=0\) proves the last bound. If \(d=0\), it follows directly from \((DA)_q\cong(P_0)_q\).
\end{proof}

\begin{proposition}\label{prop:finite}
If \(d=\id({}_A A)<\infty\), then \(\id({}_E E)\le2d+2\).
\end{proposition}
\begin{proof}
The decomposition \({}_E E=Ee\oplus Ef\) gives
\(DE=D(Ee)\oplus D(Ef)\) as right \(E\)-modules.
For either left module \(Ee\) or \(Ef\), its dual has right action \((\varphi r)(s)=\varphi(rs)\), where \(r\in E\), \(s\) belongs to that left module, and \(\varphi\) is a linear functional on it.

First, \(D(Ee)\cong eE\).
Use row coordinates \(eE=B\oplus DA\) and column coordinates \(Ee=B\oplus A\). For \(c,z\in B\), \(\lambda\in DA\), and \(t\in A\), define
\begin{equation}\label{eq:pairing}
 \langle(c,\lambda),(z,t)\rangle=\tau(cz)+\lambda(t).
\end{equation}
It is nondegenerate because the trace pairing on \(B\) and the evaluation pairing between \(DA\) and \(A\) are nondegenerate. If \(r=\left(\begin{smallmatrix}b&u\\v&a\end{smallmatrix}\right)\), then
\[
 (c,\lambda)r=(cb+\lambda v,cu+\lambda a),\qquad
 r(z,t)=(bz+ut,v\pi(z)+at).
\]
The identities
\[
 \tau(\lambda vz)=\lambda(v\pi(z)),\qquad
 \tau(cut)=(cu)(t),\qquad
 (\lambda a)(t)=\lambda(at)
\]
give
\begin{align*}
 \langle(c,\lambda)r,(z,t)\rangle
 &=\tau(cbz)+\lambda(v\pi(z))+(cu)(t)+\lambda(at)\\
 &=\langle(c,\lambda),r(z,t)\rangle.
\end{align*} Thus \eqref{eq:pairing} identifies \(eE\) with \(D(Ee)\) as a right \(E\)-module, and this summand is projective.

Next, in column coordinates \(Ef=DA\oplus A\), an element \((w,t)\), with \(w\in DA\) and \(t\in A\), has action
\[
 \begin{pmatrix}b&u\\v&a\end{pmatrix}(w,t)=(bw+ut,at).
\]
For \(x\in A\) and \(\lambda\in DA\), the pairing
\[
 ((x,\lambda),(w,t))\longmapsto w(x)+\lambda(t)
\]
identifies \(D(Ef)\) with \(A\oplus DA\). Its dual action is determined by
\[
 (bw+ut)(x)+\lambda(at)
 =w(x\pi(b))+u(tx)+\lambda(at)
 =w(x\pi(b))+(xu+\lambda a)(t).
\]
Thus its right action is
\begin{equation}\label{eq:Jaction}
 (x,\lambda)\begin{pmatrix}b&u\\v&a\end{pmatrix}
 =(x\pi(b),xu+\lambda a).
\end{equation}
Comparison with \eqref{eq:Haction} shows that
\[
 D(Ef)\longrightarrow H(A),\qquad (x,\lambda)\longmapsto(x,0)
\]
is \(E\)-linear, with kernel \((DA)_q\) and image \(K\) from Lemma~\ref{lem:quotient}.
We obtain a short exact sequence
\[
 0\longrightarrow(DA)_q\longrightarrow D(Ef)
 \longrightarrow K\longrightarrow0.
\]
For a short exact sequence \(0\to L\to M\to N\to0\), the terms \(\Ext_E^n(N,V)\) and \(\Ext_E^n(L,V)\) adjacent to \(\Ext_E^n(M,V)\) vanish when \(n>\max\{\pd_E L,\pd_E N\}\), for every \(V\). Hence \(\pd_E M\le\max\{\pd_E L,\pd_E N\}\). Lemma~\ref{lem:quotient} therefore gives
\[
 \pd_E D(Ef)\le\max\{2d+2,d+1\}=2d+2.
\]
Together with the projectivity of \(D(Ee)\), this proves
\[
 \id({}_E E)=\pd_E DE\le2d+2.\qedhere
\]
\end{proof}

\section{Descent along the quotient summand}
The lower right block identifies the corner algebra \(fEf\) with \(A\). Left multiplication by this corner makes \(fE\) an \(A\)-\(E\)-bimodule. In row coordinates \(fE=A\oplus A\), its left action is \(a(x,y)=(ax,ay)\), so \({}_A fE\) is free of rank two. Write \(\mathrm{mod}\text{-}R\) for the category of finite-dimensional right modules over an algebra \(R\), and define
\[
 F:\mathrm{mod}\text{-}A\longrightarrow\mathrm{mod}\text{-}E,
 \qquad F(U)=U\otimes_A fE.
\]
This functor is exact because \({}_A fE\) is free. It preserves projectives because \(F(A)\cong fE\) is a direct summand of \(E_E\).
For a right \(E\)-module \(N\), the subspace \(Nf\) is a right \(A=fEf\)-module, and tensor--Hom adjunction gives
\begin{equation}\label{eq:adj}
 \Hom_E(F(U),N)\cong\Hom_A(U,Nf).
\end{equation}
Explicitly, an \(E\)-linear map \(\varphi:F(U)\to N\) is sent to \(u\mapsto\varphi(u\otimes f)\), whose image lies in \(Nf\). Conversely, an \(A\)-linear map \(\psi:U\to Nf\) gives
\[
 u\otimes fr\longmapsto\psi(u)r\qquad(u\in U,\ r\in E).
\]
This is well-defined: \(\psi(u)=\psi(u)f\) makes the value depend only on \(fr\), and \(\psi(ua)r=\psi(u)ar\) gives the tensor balancing relation for \(a\in A=fEf\). These constructions are inverse and natural. Right multiplication by \(f\) is an idempotent projection on \(fE\), with image \(fEf=A\). It follows that
\[
 (F(U))f\cong U\otimes_A fEf\cong U,
 \qquad u\otimes a\longmapsto ua.
\]
The inverse of the last map is \(u\mapsto u\otimes f\); also \((fE)f=A\).

\begin{proposition}\label{prop:descent}
For every finite-dimensional right \(A\)-module \(U\),
\[
 \pd_E F(U)=\pd_A U,
\]
including the value \(\infty\). Moreover, \(\id(A_A)\le\id(E_E)\).
\end{proposition}
\begin{proof}
Applying \(F\) to a projective resolution of \(U\) gives an \(E\)-projective resolution of \(F(U)\).
Applying \eqref{eq:adj} term by term consequently gives, for every right \(E\)-module \(N\) and every integer \(n\ge0\),
\begin{equation}\label{eq:extadj}
 \Ext_E^n(F(U),N)\cong\Ext_A^n(U,Nf).
\end{equation}
The resolution already gives \(\pd_E F(U)\le\pd_A U\).
If \(\pd_E F(U)\le r\) for a nonnegative integer \(r\), take \(N=F(V)\) for an arbitrary finite right \(A\)-module \(V\).
Then \eqref{eq:extadj} implies \(\Ext_A^n(U,V)=0\) for all \(n>r\), so \(\pd_A U\le r\).
Indeed, for \(r>0\) let \(\Omega^r U\) be defined by an exact sequence \(0\to\Omega^r U\to P_{r-1}\to\cdots\to P_0\to U\to0\) with finite projective terms, and set \(\Omega^0U=U\). Choose an epimorphism \(P\to\Omega^r U\) with \(P\) finite projective and kernel \(L\). Dimension shifting gives
\[
 \Ext_A^1(\Omega^r U,L)\cong\Ext_A^{r+1}(U,L)=0.
\]
Thus \(0\to L\to P\to\Omega^r U\to0\) splits, so \(\Omega^r U\) is projective.
This proves the equality, also when either dimension is infinite.

For the second assertion we may assume \(m=\id(E_E)<\infty\).
Set \(N=fE\) in \eqref{eq:extadj}. Since \(fE\) is a direct summand of \(E_E\), we obtain
\[
 \Ext_A^n(U,A)\cong\Ext_E^n(F(U),fE)=0
 \qquad(n>m)
\]
for every finite right \(A\)-module \(U\).
Take an injective resolution of \(A_A\) with finite terms, and let \(C\) be its \(m\)-th cosyzygy: for \(m>0\), it occurs in the exact sequence
\[
 0\longrightarrow A\longrightarrow I^0\longrightarrow\cdots
 \longrightarrow I^{m-1}\longrightarrow C\longrightarrow0;
\]
for \(m=0\), set \(C=A\). For every right ideal \(I\subseteq A\), dimension shifting gives
\[
 \Ext_A^1(A/I,C)\cong\Ext_A^{m+1}(A/I,A)=0.
\]
The module \(A/I\) is finite, so the preceding vanishing applies. The exact sequence obtained by applying \(\Hom_A(-,C)\) to \(0\to I\to A\to A/I\to0\) shows that every map \(I\to C\) extends to \(A\to C\). By Baer's criterion, \(C\) is injective, and hence \(\id(A_A)\le m\).
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Lemma~\ref{lem:symmetric} proves the structural and dimension assertions.
Proposition~\ref{prop:finite} proves \eqref{eq:upper}, and Proposition~\ref{prop:descent} proves \eqref{eq:descent}.
\end{proof}

\begin{thebibliography}{9}
\bibitem{ACH19}
T.~Aihara, A.~Chan and T.~Honma,
\emph{On the gendo-symmetric algebra of a trivial extension algebra},
conference report, 2019, 6 pp.
\href{https://ring-theory-japan.com/ring/oldmeeting/2019/report2019/AIHARA-AARON-HONMA2.pdf}{Author report}.

\bibitem{CK15}
H.~Chen and S.~Koenig,
\emph{Ortho-symmetric modules, Gorenstein algebras and derived equivalences},
preprint (2015),
\href{https://arxiv.org/abs/1506.03337v1}{arXiv:1506.03337v1}.

\bibitem{FK11}
M.~Fang and S.~Koenig,
\emph{Endomorphism algebras of generators over symmetric algebras},
J. Algebra \textbf{332} (2011), 428--433.
\href{https://doi.org/10.1016/j.jalgebra.2011.02.031}{doi:10.1016/j.jalgebra.2011.02.031}.

\bibitem{FK16}
M.~Fang and S.~Koenig,
\emph{Gendo-symmetric algebras, canonical comultiplication, bar cocomplex and dominant dimension},
Trans. Amer. Math. Soc. \textbf{368} (2016), no.~7, 5037--5055.
\href{https://doi.org/10.1090/tran/6504}{doi:10.1090/tran/6504}.
\bibitem{HQ14}
Y.~Han and Y.~Qin,
\emph{Reducing homological conjectures by \(n\)-recollements},
preprint (2014),
\href{https://arxiv.org/abs/1410.3223v2}{arXiv:1410.3223v2}.
\end{thebibliography}

\bigskip
\begingroup\small
\noindent\textsc{Zhili College, Tsinghua University, Beijing 100084, China}\par
\noindent\textit{Email address:}\enspace
\mbox{\href{mailto:tangqy24@mails.tsinghua.edu.cn}{\texttt{tangqy24@mails.tsinghua.edu.cn}}}\par
\endgroup
\end{document}