text:
\section*{Introduction}

Let $H$ be a Hopf algebra with bijective antipode, $A$ be a left $H$-module algebra and $B$ be a left $H$-comodule algebra. There is a canonical algebra construction on the space $A\otimes B$ called the Takeuchi smash product of $A$ and $B$, denoted by $A\#B$ (see Definition \ref{defi-Take-product}).
Usual smash products and Ore extensions are examples of Takeuchi smash products (see Example \ref{ore-ext-Hopf-alg}).

If $A$ is a positively graded algebra and $G$ is a finite group acting gradedly on $A$,
Martinez-Villa \cite[Theorem 10]{Mar} proved that the Yoneda Ext-algebra of the skew  group algebra $A\#G$ is isomorphic to the skew group algebra $\Ext^\bullet_A(A_0, A_0) \# G$ under some mild conditions.
Bergh and Oppermann proved that the Yoneda Ext-algebra of a class of Takeuchi smash product (see Example \ref{a-class-twisted-tensor-product}) is isomorphic to the Takeuchi smash product of the Yoneda Ext-algebras \cite[Theorem 3.7]{BO}.
The main purpose of this paper is to give a precise description of the Yoneda Ext-algebras of the general Takeuchi smash products.

Let $M$ be a left $A\#H$-module and $X$ be a left $(H, B)$-Hopf module. Then there is a natural left $A\#B$-module structure on $M\otimes X$ denoted by $M\#X$, which is called  the Takeuchi smash product module of $M$ and $X$. If $P_\bullet \to M \to 0$ is a projective $A\#H$-module resolution of $M$, then $\Hom^\bullet_A(P_\bullet, P_\bullet)$ is a differential graded left $H$-module algebra and
$\Ext^\bullet_{A}(M, M) = \bigoplus_{n \geq 0} \Ext^n_{A}(M, M)$ is a graded left $H$-module algebra. If $X$ has a resolution $Q_\bullet \to X \to 0$ in the category of left $(H, B)$-Hopf modules such that each $Q_i$ is finitely generated as $B$-module, then $\Hom^\bullet_A(Q_\bullet, Q_\bullet)$ is a differential graded left $H$-comodule algebra and $\Ext^\bullet_{B}(X, X)$ is a graded left $H$-comodule algebra. Therefore, we can form the graded Takeuchi smash product algebra $\Ext^\bullet_{A}(M, M) \# \Ext^\bullet_{B}(X, X)$.

The following is one of the main results in this paper, which generalizes the result in the usual tensor product case,  and \cite[Theorem 10]{Mar}, \cite[Theorem 3.7]{BO}. Theorem \ref{Ext-alg of smash product} is the  right-sided version of it.

\begin{thm} \label{thm1}
	Suppose that $M$  is a left $A\#H$-module which is pseudo-coherent as $A$-module, and $X$ is a left $(H, B)$-Hopf module which is pseudo-coherent as $B$-module. Then, as graded algebras,
	$$\Ext^\bullet_{A\#B}(M\#X, M\#X) \cong \Ext^\bullet_{A}(M, M) \# \Ext^\bullet_{B}(X, X)$$
	where the last $\#$ means graded Takeuchi smash product.
\end{thm}
A module is called pseudo-coherent if it has a finitely generated projective resolution.

The other result in this paper is the graded Takeuchi smash product of AS-regular algebras is AS-regular (see Theorem \ref{AS-regular-TSP}). In the case that $H$ is trivial or the actions of $H$ is trivial, the result is well known.

\begin{thm} \label{thm2}  Let $A$ and $B$ be AS-regular algebras of dimension $d_1$ and $d_2$ respectively. If $H$ is a positively graded Hopf algebra, $A$ is an $H$-module graded algebra and $B$ is an $H$-comodule graded algebra,
then $A\#B$ is an AS-regular algebra of dimension $d_1+d_2$.
\end{thm}
The Nakayama automorphism of $A\#B$ is described as an application of Theorems \ref{thm1} and \ref{thm2}.

The paper is organized as follows.
In section $1$, we define Takeuchi smash products of algebras or modules, and give some basic facts.
In section $2$, we study homological properties of Takeuchi smash products, in particular, the Tor-groups and Ext-groups.
In section $3$, we introduce Takeuchi smash products of differential graded algebras, and prove Theorem \ref{Ext-alg of smash product}, that is, the Yoneda Ext-algebra of the Takeuchi smash product of modules is the graded Takeuchi smash product of the Yoneda Ext-algebras of the modules involved.
In section $4$, we prove that Takeuchi smash product preserves the AS-regularity, and describe the Nakayama automorphism of the product.

\section{Takeuchi smash products}

Throughout, let $k$ be a field. All algebras and modules are over $k$, and the unmarked tensor product $\otimes$ (resp. $\Hom$) means $\otimes_k$ (resp. $\Hom_k$).
For the basic concept and theory concerning Hopf algebras we refer to \cite{Sw} and \cite{Mo} as basic references.

\subsection{Takeuchi smash products of algebras}
Let $H$ be a Hopf algebra, $A$ be a left $H$-module algebra and $B$ be a left $H$-comodule algebra \cite[Definitions 4.1.1, 4.1.2]{Mo}. The following construction (see \cite[section 8]{Tak} and the references therein) is a generalization of smash products, which is now called Takeuchi smash product in literature.

\begin{defn} \label{defi-Take-product} The Takeuchi smash product $A\#B$ of $A$ and $B$ is the associative algebra with  underlying $k$-space $A\otimes B$, and multiplication defined by
$$(a\#b) (a'\#b') := \sum_{(b)}a(b_{-1} \rightharpoonup a') \# b_0b'$$
for any $a,a' \in A$ and $b,b' \in B$, where $\sum_{(b)} b_{-1} \otimes b_0 = \rho(b)$ is given by the comodule structure map of $B$.
\end{defn}

If $B=H$, the Takeuchi smash product $A\#B$ is the usual smash product $A\#H$ \cite[Definition 4.1.3]{Mo}.

\begin{ex}\label{ore-ext-Hopf-alg}
Any Ore extension $A[x, \sigma, \delta]$ can be viewed as a Takeuchi smash product, where $A$ is a $k$-algebra, $\sigma$ is an algebra automorphism of $A$ and $\delta$ is a $\sigma$-derivation of $A$.

In fact, let $H = k\langle g^{\pm 1}, X\rangle$ be the free algebra with the Hopf algebra structure given by
$$\Delta(g) = g\otimes g, \Delta(X) = g \otimes X + X \otimes 1, \varepsilon(g) = 1, \varepsilon(X) = 0, S(g) = g^{-1}, S(X) = -g^{-1}X.$$
Then  $A$ has a left $H$-module structure, given by $g \rightharpoonup a = \sigma (a) \, \textrm{ and } \,   X  \rightharpoonup a =  \delta(a).$
It is easy to see that
	$$g \rightharpoonup (ab) = (g \rightharpoonup a)(g \rightharpoonup b) \text{ and } X \rightharpoonup (ab) = (g \rightharpoonup a)(X \rightharpoonup b) + (X \rightharpoonup a)b$$
for all $a,b \in A$. So $m_A$ and $u_A$ are $H$-module morphisms, that is, with the action defined above $A$ is a left $H$-module algebra.

Let $B=k[X]$ be the subalgebra of $H$ generated by $X$. In fact, $B$ is a left coideal subalgebra of $H$. The Ore extension $A[x;\sigma, \delta]$ is isomorphic to the Takeuchi smash product $A\#B$, which is a subalgebra of $A \# H$.
\end{ex}

\begin{ex}\label{a-class-twisted-tensor-product}
	A tensor product of graded algebras twisted by a bicharacter can also be viewed as a Takeuchi smash product.
	Let $A$ and $B$ be two $k$-algebras that are graded by abelian groups $G$ and $L$ respectively:
	$A = \bigoplus_{g \in G} A_g$ and $B = \bigoplus_{l \in L} B_l.$

	Let $t: G \times L \to k^{\times}$ be a bicharacter, that is,
	$$t(g + g', l) = t(g, l)t(g', l)\, \text{ and } \, t(g, l + l') = t(g, l)t(g, l')$$
	for any $g, g' \in G$ and $l, l' \in L$.

	With this data, the twisted tensor product $A \otimes^t B$ (see \cite[Definition/Construction 2.2]{BO}) is by definition the vector space $A \otimes B$, with multiplication given by
	$$(a \otimes b) (a' \otimes b') = t(g, l)\, aa' \otimes bb',$$
	where $a \in A$, $a' \in A_g$, $b \in B_l$ and $b' \in B$.

	In fact, $B$ is a $kL$-comodule algebra, and $A$ is a $kL$-module algebra via $l \rightharpoonup a := t(g, l)a$ for any $g \in G$, $l \in L$ and $a \in A_g$.
	The twisted tensor product $A \otimes^t B$ is the Takeuchi smash product $A\#B$ over the Hopf algebra $kL$.
\end{ex}

In general, any Takeuchi smash product is a twisted tensor product \cite{CSV}.
Let $A$ and $B$ be two algebras, and $\tau: B \otimes A \longrightarrow A \otimes B$ be a linear map.
 Then there is a multiplication map on the vector space $A\otimes B$ given by
$$ m_{A\otimes^{\tau}B} := (m_A \otimes m_B) \circ (\id_A \otimes \tau \otimes \id_B)$$
where $m_A$ and $m_B$ are the multiplication maps of $A$ and $B$ respectively. If, with this multiplication, $A\otimes B$ is an associative algebra with unit $1_A \otimes 1_B$, then it is called a  {\it twisted tensor product} determined by $\tau$ and is denoted by  $A\otimes^{\tau}B$.
If $A\#B$ is a Takeuchi smash product, then it is the twisted tensor product $A\otimes^{\tau}B$ given by $\tau: B \otimes A \to A \otimes B, \, b \otimes a \mapsto \sum_{(b)} (b_{-1} \rightharpoonup a) \otimes b$.

\subsection{Takeuchi smash product of modules} Let $_A\mathcal{M}$ (resp. $\mathcal{M}_A$) denote the left (resp. right) $A$-module category for an algebra $A$.
Let $A$ be a left $H$-module algebra and $B$ be a left $H$-comodule algebra. Recall that a left $B$-module $X$ with a left $H$-coaction $\rho$ is called a left $(H,B)$-Hopf module \cite[Definition 8.5.1]{Mo} if $\rho(bx)=\rho(b)\rho(x)$ for all $b \in B$ and $x \in X$. Let $^H_B\mathcal{M}$ denote the left $(H,B)$-Hopf module category. The $(H,B)$-Hopf module category $^H\mathcal{M}_B$ is defined analogously.

Note that $A\#B \to (A\#H) \otimes B, a\#b \mapsto \sum_{(b)}(a\#b_{-1}) \otimes b_0$ is an algebra map.

For any $M \in {_{A\#H}\mathcal{M}}$ and $X \in {_{B}\mathcal{M}}$, there is a natural left $A\#B$-module structure on $M \otimes X$, given by
\begin{equation} \label{left AxB-mod-on-tensor}
a\#b \rightharpoonup m\otimes x = \sum_{(b)}a(b_{-1} \rightharpoonup m) \otimes b_0x = \sum_{(b)} (a \# b_{-1})m \otimes b_0x
\end{equation}
for any $a \in A$, $b \in B$, $m \in M$ and $x \in X$. This module, denoted  by $M\#X$, is called the {\it Takeuchi smash product} of $_{A\#H}M$ and $_BX$, where the element $m \otimes x$ in $M\#X$ is denoted by $m\#x$.

For any $M \in \mathcal{M}_{A\#H}$ and $X \in \mathcal{M}_B$, there is also a right $A\#B$-module structure on $M \otimes X$  given by
\begin{equation} \label{right AxB-mod-on-tensor-2}
(m \otimes x) (a \# b) = \sum_{(b)} m(a\#b_{-1}) \otimes xb_0.
\end{equation}
This module, denoted also by $M\#X$, is called the {\it Takeuchi smash product} of  $M_{A\#H}$ and $X_B$.

For any $N \in \mathcal{M}_A$ and $Y \in {^{H}\mathcal{M}_B}$, there is a natural right $A\#B$-module structure on $N \otimes Y$, given by
\begin{equation}\label{right AxB-mod-on-tensor}
n\otimes y \leftharpoonup a\#b = \sum_{(y)} n(y_{-1}\rightharpoonup a) \otimes y_0b
\end{equation}
for any $a \in A$, $b \in B$, $n \in N$ and $y \in Y$. This module, denoted  by $N\#Y$, is called the {\it Takeuchi smash product} of $N_A$ and $Y \in {^{H}\mathcal{M}_B}$, where the element $n \otimes y$ in $N\#Y$ is denoted by $n\#y$.

For any $N \in {_{A}\mathcal{M}}$ and $Y \in {_B^H\mathcal{M}}$, there is a left $A\#B$-module structure on $N \otimes Y$ which is given by
\begin{equation}
\label{left AxB-mod-on-tensor-2}(a \# b) (n \otimes y) = \sum_{(b)}( S^{-1}(b_{-1}y_{-1}) \rightharpoonup a)n \otimes b_0y_0.
 \end{equation}
This module, denoted also  by $N\#Y$, is called the {\it Takeuchi smash product} of $_AN$ and  $Y \in {_B^H\mathcal{M}}$.

\begin{rk} A right-sided version of Takeuchi smash product $B \# A$ is defined similarly for a right $H$-comodule algebra $B$ and a right $H$-module algebra $A$. There are also four types of Takeuchi smash product of modules similar to
the module structures defined by \eqref{left AxB-mod-on-tensor}, \eqref{right AxB-mod-on-tensor-2}, \eqref{right AxB-mod-on-tensor} and \eqref{left AxB-mod-on-tensor-2}. The module structures given by  \eqref{right AxB-mod-on-tensor} and \eqref{left AxB-mod-on-tensor-2} seem less natural than the two given by  \eqref{left AxB-mod-on-tensor} and \eqref{right AxB-mod-on-tensor-2}, but they are dual to the two natural ones in the right-sided version in some sense (in particular when $H$ is finite-dimensional).
\end{rk}

\subsection{Adjointness}
The following lemma is modified from the usual smash product case, the proof is routine.

\begin{lem} \label{Hom-Tensor-adj}
	Let $M \in {_{A\#H}\mathcal{M}}$, $X \in {}_B\mathcal{M}$ and $T \in {_{A\#B}\mathcal{M}}$.
	\begin{enumerate}
		\item There is a natural left $B$-module structure on $\Hom_A(M,T)$, given by
		\begin{equation} \label{B-mod-on-Hom}
			(b \rightharpoonup f)(m) = \sum\limits_{(b)}b_0f(S^{-1}b_{-1} \rightharpoonup m).
		\end{equation}
		\item The $\Hom$-Tensor adjoint isomorphism $\Hom(X, \Hom(M, T)) \stackrel{\cong}{\longrightarrow} \Hom(M \otimes X, T)$ restricts to a natural isomorphism
		$\Hom_B(X, \Hom_A(M,T)) \stackrel{\cong}{\longrightarrow} \Hom_{A\#B}(M\#X, T)$, that is, $\Hom_A(M, -): {_{A\#B}\mathcal{M}} \to {}_B\mathcal{M}$ and
$M \# -: {}_B\mathcal{M} \to  {_{A\#B}\mathcal{M}}$ is a pair of adjoint functors.
	\end{enumerate}
\end{lem}

Then, for any $M \in {_{A\#H}\mathcal{M}}$ and $T \in  {_{A\#B}\mathcal{M}}$, there exists a natural $B$-module structure on the Ext-groups $\Ext^n_A(M, T)$. In fact, the $H$-module structure on the Ext-groups can be induced by any resolution of $M$ in ${_{A\#H}\mathcal{M}}$ such that each term is projective $A$-module, or any resolution of $T$ in ${_{A\#B}\mathcal{M}}$ of which all terms are injective $A$-modules. And these two induced $B$-module structures on $\Ext^n_{A\#B}(M, T)$ coincide, as shown in the following lemma.

\begin{lem}\label{H-mod str on Ext-group}
	Let $M \in {_{A\#H}\mathcal{M}}$ and $T \in {_{A\#B}\mathcal{M}}$.
	Let $P_{\bullet} \to M \to 0$ and $P'_{\bullet} \to M \to 0$ be exact sequences in $_{A\#H}\mathcal{M}$, where all $P_n$ and $P'_n$ are projective $A$-modules,
	and let $0 \to T \to I^{\bullet}$ and $0 \to T \to I'^{\bullet}$ be exact sequences in $_{A\#B}\mathcal{M}$, where all $I^n$ and $I'^n$ are injective $A$-modules.
	Then, for any $n \in \mathbb{N}$, as $B$-modules,
	$$\mathrm{H}^n(\Hom_{A}(P_{\bullet}, T)) \cong \mathrm{H}^n(\Hom_{A}(M, I^{\bullet})) \cong \mathrm{H}^n(\Hom_{A}(P'_{\bullet}, T)) \cong \mathrm{H}^n(\Hom_{A}(M, I'^{\bullet})).$$
\end{lem}
\begin{proof}
	Consider the double complex $\Hom_A(P_{\bullet}, I^{\bullet})$ of $B$-modules. Then
	$$\text{H}^*(\Hom_{A}(P_{\bullet}, T)) \cong \text{H}^*(\Hom_{A}(M, I^{\bullet}))$$
	as $B$-modules.
\end{proof}

For any $M_{A\#H}$, $X_B$ and $T_{A\#B}$, there is  right module version of Lemma \ref{Hom-Tensor-adj} and Lemma \ref{H-mod str on Ext-group}.

There is also a version of Lemma \ref{Hom-Tensor-adj} for any $N \in {\mathcal{M}_A}$, $Y \in {^H\mathcal{M}_{B}}$ and $T \in {\mathcal{M}_{A\#B}}$, which can be viewed as a dual version.
\begin{lem} \label{right-Hom-Tensor-adj}
	Let $N \in {\mathcal{M}_A}$, $Y \in {^H\mathcal{M}_{B}}$ and $T \in {\mathcal{M}_{A\#B}}$.
	\begin{enumerate}
		\item There is a right $A$-module structure on $\Hom_{B^{op}}(Y,T)$ given by
		$$(f a)(y) = \sum\limits_{(y)}f(y_0) (S^{-1}y_{-1} \rightharpoonup a).$$
		\item The $\Hom$-Tensor adjoint isomorphism $\Hom(N, \Hom(Y, T)) \stackrel{\cong}{\longrightarrow} \Hom(N \otimes Y, T)$ restricts to an isomorphism
		$\Hom_{A^{op}}(N, \Hom_{B^{op}}(Y,T)) \stackrel{\cong}{\longrightarrow} \Hom_{(A\#B)^{op}}(N\#Y, T).$
	\end{enumerate}
\end{lem}

\begin{lem} \label{relative-resolution}
	Let $X \in {_B^H\mathcal{M}}$ (resp. $^H\mathcal{M}_B$).
	\begin{enumerate}
		\item \cite[Proposition 4.1]{CG} $X$ viewed as left (resp. right) $B$-module is finitely generated if and only if there is a finite-dimensional left $H$-comodule $V$ and a surjective morphism $\pi: B \otimes V  \twoheadrightarrow X$ in $_B^H \mathcal{M}$ (resp.  $\pi: V \otimes B \twoheadrightarrow X$ in $^H\mathcal{M}_B$).
		\item Suppose  $X$ viewed as left (resp. right) $B$-module is pseudo-coherent. Then there is an exact sequence in $_B^H\mathcal{M}$ (resp. $^H\mathcal{M}_B$), where all the $V_n's$ are finite-dimensional left $H$-comodules,
		$$ \cdots \to B \otimes V_n \to \cdots \to B \otimes V_0 \to X \to 0 \,\, (\textrm{resp.}  \cdots \to V_n \otimes B \to \cdots \to V_0 \otimes B \to X \to 0).$$
	\end{enumerate}
\end{lem}

\begin{proof}
	(1) Let $\{x_1,\dots,x_m\}$ be a set generators of $_BX$.  By the Finiteness Theorem \cite[5.1.1]{Mo}, there exists a finite-dimensional  left $H$-comodule $V \subseteq X$ such that $\{x_1,\dots,x_m\} \subseteq V$. Hence a surjective morphism $\pi: B \otimes V \twoheadrightarrow X$ in $_B^H\mathcal{M}$ exists.

	(2) By (1), there exists a finite-dimensional  left $H$-comodule $V_0$ with a surjective morphism $\pi : B \otimes V_0 \twoheadrightarrow X$ in $_B^H\mathcal{M}$. By Schanuel's lemma, the kernel of $\pi$ is also a pseudo-coherent $B$-module. Then the conclusion follows by induction.

\end{proof}

\subsection{Comodule structure on the B-morphism space of (H, B)-Hopf modules} The material in this subsection is essentially modified from \cite{Ul}, especially \cite[Lemma 2.2]{Ul}.

For any $X, Y \in {_B^{H}\mathcal{M}}$, we try to endow $\Hom_B(X,Y)$ with a left $H$-comodule structure, say, when $_BX$ is finitely generated. So, firstly we have to define a map $\Hom_B(X, Y) \to H \otimes \Hom_B(X, Y).$
We use the natural embedding $$H \otimes \Hom_B(X,Y) \subseteq \Hom_B(X, H \otimes Y), \, h \otimes f \mapsto \big(x \mapsto  h \otimes f(x)\big).$$

Let $\rho=\rho_Y: \Hom_B(X, Y) \to \Hom_B(X, H \otimes .Y)$ be the  morphism given by
\begin{equation} \label{H-comod-on-HOM}
	\rho(f)(x) = \sum_{(x),(f(x_0))} (Sx_{-1}) f(x_0)_{-1} \otimes f(x_0)_0
\end{equation}
where $f \in \Hom_B(X, Y)$ and $x \in X$.
Let
$$\HOM_B(X,Y) := \{ f \in \Hom_B(X,Y) \mid \rho(f) \in H \otimes \Hom_B(X,Y) \subseteq \Hom_B(X, H \otimes Y) \}.$$
Then, $\HOM_B(X,Y)$ is a left $H$-comodule. A proof of this is added in the following for the convenience.

\begin{lem} \label{H-comod-str-on-Hom}
Let $X, Y \in {}_B^H\mathcal{M}$.
	\begin{enumerate}
		\item $\HOM_B(X, Y)$ has a left $H$-comodule structure as given by \eqref{H-comod-on-HOM}.
		\item If $_BX$ is finitely generated, then $\Hom_B(X, Y) = \HOM_B(X, Y)$, and so it is a left $H$-comodule (see also \cite[Proposition 4.2]{CG}).
	\end{enumerate}
\end{lem}
\begin{proof} (1)
	For any $f \in \HOM_B(X,Y)$, suppose $\rho(f) = \sum_i h^i \otimes f^i \in H \otimes \Hom_B(X,Y)$, where $\{h^i\} \subseteq H$ and $\{f^i\} \subseteq \Hom_B(X,Y)$  are $k$-linear independent respectively.
	Then $$\rho(f)(x)=\sum_i h^i \otimes f^i(x) = \sum_{(x),(f(x_0))} (Sx_{-1}) f(x_0)_{-1} \otimes f(x_0)_0, \, \textrm{ and }$$

	$$\sum_{(x)} Sx_{-1} \otimes \rho(f)(x_0) = \sum_{i, (x)} Sx_{-1} \otimes h^i \otimes f^i(x_0)= \sum_{ (x), (f(x_0)} Sx_{-2} \otimes (Sx_{-1})f(x_0)_{-1} \otimes f(x_0)_0.$$
	By acting by $\rho$ on the last component of both sides in the last equality, and switching the first two components obtained, then
	\begin{equation}\label{equality-for-next}
		\sum_i \sum_{(x), (f^i(x_0))} h^i \otimes (Sx_{-1})f^i(x_0)_{-1} \otimes f^i(x_0)_0
		=\sum_{(x), (f(x_0))} (Sx_{-1})f(x_0)_{-2} \otimes (Sx_{-2})f(x_0)_{-1} \otimes f(x_0)_0.
	\end{equation}
	Consider the images of $\sum_{i, (h^i)} h^i_1 \otimes h^i_2 \otimes f^i \in H \otimes H \otimes \Hom_B(X,Y)$ and $\sum_i h^i \otimes \rho(f^i) \in H \otimes \Hom_B(X, H \otimes Y)$ in $\Hom_B(X, H \otimes H \otimes Y)$. By acting on $x \in X$, then, on one hand,
$$(\Delta \otimes 1)\rho(f)(x) = (\Delta \otimes 1)\sum_i(h^i \otimes f^i(x))
= \sum_i\sum_{(h^i)} h^i_1 \otimes h^i_2 \otimes f^i(x).$$
On the other hand,
\begin{align*}
  &(\Delta \otimes 1)\rho(f)(x)=\sum_{(x), (f(x_0))} (Sx_{-1})f(x_0)_{-2} \otimes (Sx_{-2})f(x_0)_{-1} \otimes f(x_0)_0 \\
		\stackrel{\eqref{equality-for-next}}{=} &\sum_i \sum_{(x), (f^i(x_0))} h^i \otimes (Sx_{-1})f^i(x_0)_{-1} \otimes f^i(x_0)_0 = \sum_i h^i \otimes \rho(f^i)(x) \\
=& \sum_i (1_H \otimes \rho) (h^i \otimes f^i)(x)\\
		= &(1_H \otimes \rho) \rho(f)(x).
	\end{align*}
	Now it is easy to see that $f^i \in \HOM_B(X,Y)$ for all $i$, and $\HOM_B(X,Y)$ is a left $H$-comodule.

	(2)

	By Lemma \ref{relative-resolution}, there exists a finite-dimensional left $H$-subcomodule $V$ of $X$, and a surjective morphism $\pi: B \otimes V \to X$ in ${_B^{H}\mathcal{M}}$. So, $\Hom_B(X,Y) \stackrel{\pi^*}{\longrightarrow} \Hom_B(B \otimes V, Y)$ is injective,

	and the following diagram is commutative.
	$$\xymatrix{
		\Hom_B(X, Y) \ar[d]^{\rho_X} \ar@{>->}[r]^{\pi^*} & \Hom_B(B \otimes V, Y) \ar[d]^{\rho_{B \otimes V}} \\
		\Hom_B(X, H \otimes Y) \ar@{>->}[r]^{\pi^*} & \Hom_B(B \otimes V, H \otimes Y)\\
		H \otimes \Hom_B(X,Y) \ar[r]^{1_H \otimes \pi^*} \ar@{>->}[u] & H \otimes \Hom_B(B \otimes V, Y)  \ar[u]^{=}
	}
	$$

	Now for any $f \in \Hom_B(X,Y)$, suppose $\rho_{B \otimes V}(f \pi) = \sum h^i \otimes g^i$, where $\{h^i\} \in H$ and $\{g^i\} \in \Hom_B(B \otimes V,Y)$
	are $k$-linear independent respectively.
	Then, for any $x' \in \ker \pi$, $$\sum h^i \otimes g^i(x') = \rho_X(f)(\pi(x')) = 0.$$ It follows that $g^i(x') = 0$ for any $i$, and so $g^i = f^i\pi$ for some $f^i \in \Hom_B(X,Y)$.
	Since $\Hom_B(X,Y) \stackrel{\pi^*}{\longrightarrow} \Hom_B(B \otimes V, Y)$ is injective, $\rho_X(f)=\sum h^i \otimes f^i \in H \otimes\Hom_B(X,Y)$ and $\Hom_B(X,Y) = \HOM_B(X,Y)$.
\end{proof}

If fact, $\HOM_B(-,-):  ({_B^{H}\mathcal{M}})^{op} \times {_B^{H}\mathcal{M}}  \to  {_{H}\mathcal{M}}$ is  a functor.
For any $X, X' \in {_B^{H}\mathcal{M}}$ with $_BX$ pseudo-coherent, by Lemma \ref{H-comod-str-on-Hom}, there is a natural $H$-comodule structure on $\Ext^*_B(X,X')$. The $H$-comodule structure on the Ext-groups is independent of the choices of the resolutions of $X$ in ${_B^{H}\mathcal{M}}$ such that each term is finitely generated projective as $B$-module and the resolutions of $X'$ in ${_B^{H}\mathcal{M}}$ of which all terms are injective as $B$-modules.

\begin{lem}\label{H-comod str on Ext-group}
	Let $X, X' \in {_B^{H}\mathcal{M}}$. Suppose that $P_{\bullet} \to X \to 0$, $P'_{\bullet} \to X \to 0$, $0 \to X' \to I^{\bullet}$ and $0 \to X' \to I'^{\bullet}$ are exact sequences
	in $_B^H\mathcal{M}$, where all $P_n$ and $P'_n$ are finitely generated projective as $B$-module, and all $I^n$ and $I'^n$ are injective as $B$-module. Then, for any $n \in \mathbb{N}$, as $H$-comodules,
	$$\mathrm{H}^n(\Hom_{B}(P_{\bullet}, X')) \cong \mathrm{H}^n(\Hom_{B}(X, I^{\bullet})) \cong \mathrm{H}^n(\Hom_{B}(P'_{\bullet}, X')) \cong \mathrm{H}^n(\Hom_{B}(X, I'^{\bullet})).$$
\end{lem}
\begin{proof}
	Similar to the proof of Lemma \ref{H-mod str on Ext-group}.
\end{proof}

\section{Some homological properties of Takeuchi smash products}

In this section, we consider some (homological) properties of Takeuchi smash products of modules, especially, the decompositions of the Tor-groups and Ext-groups of Takeuchi smash products of modules.

Let $A$ be a left $H$-module algebra and $B$ be a left $H$-comodule algebra.
With the natural module structure, $_A(A\#B)$ and $(A\#B)_B$ are obviously free modules.

\begin{lem}\label{proj-Takeuchi-smash}
 Both $(A\#B)_A$ and ${}_B(A\#B)$ are free modules.
\end{lem}
\begin{proof}
	The conclusion follows from the $B$-$A$-bimodule isomorphism $\varphi: {}_B(A\#B)_A \to {}_B B \otimes A_A$ defined by $\varphi(a \# b) = \sum_{(b)} b_0 \otimes (S^{-1}b_{-1} \rightharpoonup a)$.
\end{proof}

\begin{prop} \label{Ext-spec-seq-of-smash-prod-mod}
	For any $M \in {_{A\#H}\mathcal{M}}$, $X \in {_{B}\mathcal{M}}$ and $T \in {_{A\#B}\mathcal{M}}$,
	there is a convergent spectral sequence
	$$\Ext_B^p(X, \Ext_A^q(M,T)) \Rightarrow \Ext^{p+q}_{A\#B}(M\#X, T).$$
\end{prop}
\begin{proof}
	Let $P_{\bullet}$ be a projective resolution of $_BX$, and let $I^{\bullet}$ be an injective resolution of $_{A\#B}T$. Since $A\#B$ is a flat right $A$-module by Lemma \ref{proj-Takeuchi-smash}, each $I^i$ is injective as an $A$-module.
	Since $\Hom_A(M, I^i)$ is an injective $B$-module by Lemma \ref{Hom-Tensor-adj}, then the double complex $$\Hom_{A\#B}(M\#P_{\bullet}, I^{\bullet}) \cong \Hom_{B}(P_{\bullet}, \Hom_A(M, I^{\bullet}))$$
	yields the spectral sequence.
\end{proof}

\begin{cor}\label{proj-mod of smash product}
	For any $X \in {_{B}\mathcal{M}}$ and $M \in {_{A\#H}\mathcal{M}}$, $\pd_{A\#B}(M\#X) \leq \pd_A M + \pd_B X$. In particular, if both $_AM$ and $_BX$ are projective modules, then $M\#X$ is a projective $A\#B$-module.
\end{cor}

There is also right module versions of
Proposition \ref{Ext-spec-seq-of-smash-prod-mod} and Corollary \ref{proj-mod of smash product}.

\begin{prop} \label{right-case-spec-seq}
	For any $N \in {\mathcal{M}_A}$, $Y \in {^H\mathcal{M}_B}$ and $T \in {\mathcal{M}_{A\#B}}$,
	there is a convergent spectral sequence
	$$\Ext_{A^{op}}^p(N, \Ext_{B^{op}}^q(Y,T)) \Rightarrow \Ext^{p+q}_{(A\#B)^{op}}(N\#Y, T).$$
\end{prop}

\begin{cor}\label{proj-mod of smash product'}
	For any $N \in {\mathcal{M}_A}$, $Y \in {^H\mathcal{M}_{B}}$, $\pd_{(A\#B)^{op}}(N\#Y) \leq \pd_{A^{op}} N + \pd_{B^{op}} Y$. In particular, if both $N_A$ and $Y_B$ are projective modules, then $N\#Y$ is a projective right $A\#B$-module.
\end{cor}

\begin{prop}\label{decom-Tor-group}
	Suppose $M \in {_{A\#H}\mathcal{M}}$, $X \in {_{B}\mathcal{M}}$, $N \in \mathcal{M}_A$ and $Y \in {^{H}\mathcal{M}_B}$. Then,
	\begin{enumerate}
		\item $(N\#Y) \otimes_{A\#B} (M\#X) \cong (N \otimes_A M) \otimes (Y \otimes_B X)$.
		\item $\Tor^{A\#B}_n(N\#Y, M\#X) \cong \bigoplus\limits_{p+q=n} \Tor^{A}_p(N, M) \otimes \Tor^{B}_q(Y, X)$.
	\end{enumerate}
\end{prop}
\begin{proof}
	(1) It is easy to check that the map $\phi : (N\#Y) \otimes_{A\#B} (M\#X) \to (N \otimes_A M) \otimes (Y \otimes_B X)$
	$$(n\#y) \otimes_{A\#B} (m\#x) \mapsto \sum_{(y)} (n \otimes_A y_{-1} \rightharpoonup m) \otimes (y_0 \otimes_B x)$$
	is well-defined. The map $\phi$ is an isomorphism with the inverse $\phi^{-1}$  given by
	$$ (n \otimes_A m) \otimes (y \otimes_B x) \mapsto \sum_{(y)} (n\# y_0) \otimes_{A\#B} (S^{-1}y_{-1} \rightharpoonup m \, \# x).$$

	(2) Let $P_{\bullet}$ be a projective resolution of $_{A\#H}M$, and $Q_{\bullet}$ be a projective resolution of $_BX$. By Corollary \ref{proj-mod of smash product},
	the total complex $\mathrm{Tot}(P_{\bullet} \# Q_{\bullet})$ is a projective resolution of $_{A\#B}(M\#X)$. It follows from  (1) that we have an isomorphism of complexes
	$$(N\#Y) \otimes_{A\#B} \mathrm{Tot}(P_{\bullet} \# Q_{\bullet}) \cong \mathrm{Tot}((N \otimes_A P_{\bullet}) \otimes (Y \otimes_B Q_{\bullet})).$$
	The conclusion follows by taking the homology.
\end{proof}

\subsection{Ext-groups of Takeuchi smash product modules}

 As a special case in Lemma \ref{Hom-Tensor-adj} (1) where $B=H$, $\Hom_A(M,M')$ is a left $H$-module for $M, M' \in {_{A\#H}\mathcal{M}}$,  with the $H$-action given by
		$$(h \rightharpoonup f)(m) = \sum h_2 \rightharpoonup f(S^{-1}h_1 \rightharpoonup m).$$

\begin{lem}\label{an-iso-for-Ext}
	Let $M, M' \in {_{A\#H}\mathcal{M}}$, $X \in {_B^{H}\mathcal{M}}$, and $X' \in {_{B}\mathcal{M}}$.
There is a morphism
      \begin{align*}
\Phi: \Hom_A(M, M') \otimes \Hom_B(X, X') & \longrightarrow  \Hom_{B}(X, \Hom_A(M,M') \otimes X') \\
 f \otimes g & \mapsto [f, g]: x \mapsto (x_{-1} \rightharpoonup f) \otimes g(x_0)
      \end{align*}
  where $\Hom_A(M,M')\otimes X'$ is viewed as a left $B$-module via the algebra map
		$\rho: B \to H \otimes B$.

 If  ${}_BX$ is finitely presented, then $\Phi$ is an isomorphism.
\end{lem}
\begin{proof} For any $k$-vector space $U$, the natural map $U \otimes \Hom_B(X, X') \to \Hom_B(X, U \otimes .X'), u \otimes g\mapsto \big([u, g], x \mapsto u \otimes g(x)\big)$ is an isomorphism provided ${}_BX$ is finitely presented.

If $U$ is a left $H$-module, then
$\Hom_B(X, U \otimes .X') \cong \Hom_B(X, U \otimes X'), \, F \mapsto \big({\bar F}: x \mapsto \sum x_{-1} \rightharpoonup F(x_0)\big)$, where $U \otimes X'$ in the second $\Hom$ is viewed as left $B$-module via the algebra map $\rho: B \to H \otimes B$. The inverse map is $G \mapsto  \big({\tilde G}: x \mapsto  \sum Sx_{-1} \rightharpoonup G(x_0)\big)$.

The composition of above two maps with $U=\Hom_A(M, M')$ is the given morphism $\Phi$.
\end{proof}

\begin{prop}\label{decom-Ext-group}
	Let $M, M' \in {_{A\#H}\mathcal{M}}$, $X \in {_B^{H}\mathcal{M}}$, and $X' \in {_{B}\mathcal{M}}$.
	\begin{enumerate}
\item There is a natural morphism
		$$\psi: \Hom_A(M, M') \otimes \Hom_B(X, X') \longrightarrow \Hom_{A\#B}(M\#X, M'\# X')$$
		$$f \otimes g \mapsto \big(m\#x \mapsto \sum_{(x)} (x_{-1}\rightharpoonup f)(m) \# g(x_0)\big).$$

If $M$ viewed as $A$-module and $X$ viewed as $B$-module are finitely presented, then $\psi$ is an isomorphism.
		\item Suppose that $_AM$ and $_BX$ are pseudo-coherent modules. Then for any $n \in \mathbb{N}$
		$$ \Ext^n_{A\#B}(M\#X, M'\# X') \cong \bigoplus_{p+q=n} \Ext^p_{A}(M, M') \otimes \Ext^q_{B}(X, X').$$
	\end{enumerate}
\end{prop}

\begin{proof}
(1) As noted before, $\Hom_A(M,M')\otimes X'$ is viewed as a left $B$-module  by the algebra morphism
		$\rho: B \to H \otimes B$, that is,  $b \rightharpoonup (f \otimes y) := \sum (b_{-1} \rightharpoonup f) \otimes b_0y.$
 By \eqref{B-mod-on-Hom}, $\Hom_A(M, M'\#X')$ is a left $B$-module.
 Then the natural  morphism $\sigma: \Hom_A(M,M') \otimes X' \to \Hom_A(M,M'\#X'), f \otimes y \mapsto \big(m \mapsto f(m) \# y \big)$ is a $B$-module morphism. If $_AM$ is finitely presented as an $A$-module, then $\sigma$ is  an  isomorphism.

The composition $\psi$ in the following commutative diagram is the morphism as claimed.
$$
\xymatrix{
  \Hom_A(M,M') \otimes \Hom_B(X,X')  \ar@{.>}[d]^{\psi}\quad  \ar[r]^{\text{Lemma \ref{an-iso-for-Ext}}} & \quad \Hom_{B}(X, \Hom_A(M,M')\#X') \ar[d]^{\sigma_*}  \\
  \Hom_{A\#B}(M\#X, M'\#X')
                & \Hom_{B}(X, \Hom_A(M,M'\#X')) \ar[l]^{\cong}_{\text{Lemma \ref{Hom-Tensor-adj}}}
                 }
$$
If both $_AM$ and $_BX$ are finitely presented, then $\psi$ is an isomorphism.

   (2) By Lemma \ref{relative-resolution}(2), there is an exact sequence
		$ \cdots \to Q_n \to \cdots \to Q_0 \to X \to 0$ in $_B^H\mathcal{M}$ with $Q_n=B \otimes V_n$ where $V_n$ is some finite-dimensional $H$-comodule.
		Let $P_{\bullet}$ be a projective resolution of $_{A\#H}M$. It follows from Corollary \ref{proj-mod of smash product} that $\mathrm{Tot}(P_{\bullet} \# Q_{\bullet})$ is a projective resolution of $_{A\#B}(M\#X)$.
Since ${_AM}$ is pseudo-coherent, there is a finitely generated $A$-projective resolution ${\widetilde P}_{\bullet} \to M \to 0$ of $M$. Then, $ P_{\bullet}$ and ${\widetilde P}_{\bullet}$ are homotopic equivalent, and the following is a commutative diagram of complexes
$$
\xymatrix{
  \Hom_A(P_{\bullet},M')\otimes X' \ar[r]^{\sigma_\bullet}  \ar[d]^{\text{homo. equ.}} & \Hom_A(P_{\bullet}, M'\# X') \ar[d]^{\text{homo. equ.}}   \\
 \Hom_A({\widetilde P}_{\bullet},M')\otimes X'  \ar[r]^{\sigma_\bullet}_{\cong}
                & \Hom_A({\widetilde P}_{\bullet},M'\# X')
                 }
$$
It follows that $ \Hom_A(P_{\bullet},M')\otimes X' \stackrel{\sigma_\bullet}{\longrightarrow} \Hom_A(P_{\bullet}, M'\# X')$ is a quasi-isomorphism of $B$-module complexes. Hence
$$\mathrm{Tot}(\Hom_{B}(Q_{\bullet}, \Hom_A(P_{\bullet},M')\otimes X'))\xrightarrow{\sim} \mathrm{Tot}(\Hom_{B}(Q_{\bullet}, \Hom_A(P_{\bullet},M'\#X')))$$
where $\xrightarrow{\sim}$ means quasi-isomorphism.
		\begin{align*}
		 & \Hom_{A\#B}(\mathrm{Tot}(P_{\bullet} \# Q_{\bullet}), M'\#X') \\
		\cong & \mathrm{Tot}(\Hom_{B}(Q_{\bullet}, \Hom_A(P_{\bullet},M'\#X')))  \quad \text{ (by Lemma \ref{Hom-Tensor-adj})} \\
		\xleftarrow{\sim} & \mathrm{Tot}(\Hom_{B}(Q_{\bullet}, \Hom_A(P_{\bullet},M')\otimes X'))  \\
		\cong & \mathrm{Tot}(\Hom_A(P_{\bullet},M') \otimes \Hom_{B}(Q_{\bullet}, X')) \quad \text{ (as $Q_n=B \otimes V_n$ is finitely generated projective)}.
		\end{align*}
	 The conclusion follows by taking cohomologies.
\end{proof}

\begin{cor
	Suppose further both $A$ and $B$ are semi-primary rings over a perfect field $k$. If the Jacobson radical $J_A$ of $A$ is an $H$-submodule, and the Jacobson radical $J_B$ of $B$ is an $H$-subcomodule, then $\gld A\#B = \gld A + \gld B$.
\end{cor}
\begin{proof}
	By assumption, $J := A \# J_B + J_A \# B$ is a nilpotent ideal of $A\#B$. Since $k$ is a perfect field, $(A\#B)/ J \cong A/J_A \# B/J_B$ is a semisimple ring. Hence $A\#B$ is also a semi-primary ring. By \cite[Corollary 12]{Aus}, $\gld A\#B = \sup\{ n \mid \Ext^n_{A\#B}((A\#B)/J, (A\#B)/J) \}$. The conclusion follows from Proposition \ref{decom-Ext-group}.
\end{proof}

Also we have the right version of Proposition \ref{decom-Ext-group}.

For any $M \in {\mathcal{M}}_{A\#H}$, $M$ is sometimes viewed as a left $H$-module via $h \rightharpoonup m := m(1 \# Sh)$. Thus $h \rightharpoonup (ma) = \sum\limits_{(h)} (h_1 \rightharpoonup m)(h_2 \rightharpoonup a)$. For any $M, N \in \mathcal{M}_{A\#H}$, $\Hom_{A^{op}}(M,N) \in {_H\mathcal{M}}$, with the $H$-action given by
\begin{equation}\label{H-mod on right Hom set}
(h \rightharpoonup f)(m) = \sum_{(h)} h_1 \rightharpoonup f(Sh_2 \rightharpoonup m)
\end{equation}
where $m \in M, h \in H, f \in \Hom_{A^{op}}(M,N)$.

\begin{prop}\label{decom-Ext-group'}
	Let $M \in {\mathcal{M}}_{A\#H}$, $N \in {\mathcal{M}}_{A}$, and $X, Y \in {^{H}\mathcal{M}_B}$. Suppose $X_B$ is finitely generated. Then,
	\begin{enumerate}
		\item $\Hom_{B^{op}}(X,Y) \in {^H\mathcal{M}}$, with the $H$-coaction given by
		\begin{eqnarray}\label{H-comod on right Hom set}
			\rho(f)(x) = \sum_{(x)} f(x_0)_{-1}S^{-1}x_{-1} \otimes f(x_0)_0
		\end{eqnarray}
		for any $x \in X$ and $f \in \Hom_A(M,N)$.
		\item There exists a natural morphism
		$$\phi: \Hom_{A^{op}}(M,N) \otimes \Hom_{B^{op}}(X,Y) \longrightarrow \Hom_{(A\#B)^{op}}(M\#X, N\#Y)$$
		$$f \otimes g \mapsto \big(m\#x \mapsto \sum_{(g)} f(m (1 \# Sg_{-1})) \# g_0(x)\big)$$
		for any $m \in M, x \in X$. If $M_A$ and $X_B$ are finitely presented modules, then $\phi$ is an isomorphism.
		\item Suppose that $M_A$ and $X_B$ are pseudo-coherent modules. Then for any $n \in \mathbb{N}$
		$$ \Ext^n_{(A\#B)^{op}}(M\#X, N\#Y) \cong \bigoplus_{p+q=n} \Ext^p_{A^{op}}(M, N) \otimes \Ext^q_{B^{op}}(X, Y).$$
	\end{enumerate}
\end{prop}

\section{Yoneda Ext-algebras of Takeuchi smash products}

\subsection{Smash products of differential graded algebras}

A graded algebra $\Lambda = \bigoplus _{n \in \Z} \Lambda_n$ is called a graded $H$-module algebra, if $\Lambda$ is an $H$-module algebra, and $h \rightharpoonup \Lambda_n \subseteq \Lambda_n$ for all $h \in H$ and $n \in \Z$. For any non-zero homogeneous element $\lambda \in \Lambda_n$, $|\lambda| := n$ is called the degree of $\lambda$.

Let $\Lambda$ be a differential graded algebra (or dga, for short), that is, $\Lambda = \bigoplus \Lambda_n$ is a graded algebra with a graded map $d: \Lambda \to \Lambda$ of degree $1$, such that $d^2 = 0$ and the Leibnitz rule holds:
$$d(\lambda \cdot \lambda') = d(\lambda) \cdot \lambda' + (-1)^{|\lambda|} \lambda \cdot d(\lambda'),  \, \text{ for any  homogeneous element } \lambda \text{ and } \lambda' \in \Lambda.$$
$\Lambda$ is called a {\it differential graded left $H$-module algebra}, if $\Lambda = \bigoplus \Lambda_n$ is a graded $H$-module algebra and the differential $d$ is an $H$-morphism, that is, $d(h \rightharpoonup \lambda) = h \rightharpoonup d(\lambda)$ for all $h \in H$ and $\lambda \in \Lambda$.

{\it Differential graded left $H$-comodule algebras} are defined analogously.

Suppose that $\Lambda$ is a differential graded left $H$-module algebra, and $\Gamma$ is a differential graded left $H$-comodule algebra. Then
$\Lambda\#\Gamma$ is also a (differential) graded algebra with the  multiplication defined by
$$(\lambda\#\gamma)(\lambda'\#\gamma') = \sum_{(\gamma)} (-1)^{|\gamma||\lambda'|} \lambda(\gamma_{-1} \rightharpoonup \lambda') \# \gamma_0 \gamma'$$
and the differential $d_{\Lambda \# \Gamma}$ defined by
$$d_{\Lambda \# \Gamma}(\lambda\#\gamma) = d_{\Lambda}(\lambda) \# \gamma + (-1)^{|\lambda|} \lambda \# d_{\Gamma}(\gamma)$$
for any homogeneous elements $\lambda, \lambda' \in \Lambda$, and $\gamma, \gamma' \in \Gamma$.

Let $R$ be a ring,  $P_{\bullet}$ and $Q_{\bullet}$ be two $R$-module complexes. Consider the graded space $\Hom^{\bullet}_R(P_{\bullet}, Q_{\bullet})$ with
$$ \Hom^n_R(P_{\bullet}, Q_{\bullet}) = \prod_i \Hom_R(P_i, Q_{i-n}). $$
The differentials on $P_{\bullet}$ and $Q_{\bullet}$ which are of degree $-1$ give rise to a differential $d_{\Hom}$ of degree $1$ on $\Hom^{\bullet}_R(P_{\bullet}, Q_{\bullet})$, where
$d_{\Hom}(f) = d^Q \circ f - (-1)^{|f|}f \circ d^P = (d^Q_{i-|f|} \circ f_i - (-1)^{|f|}f_{i-1} \circ d^P_i)$
for $f =(f_i)_i$, and $f_i \in \Hom_R(P_i, Q_{i-|f|})$.

In fact, $\End^{\bullet}_R(P_{\bullet}) = \Hom^{\bullet}_R(P_{\bullet},P_{\bullet})$ is a dga, as  $d_{\Hom}(fg) = d_{\Hom}(f)g + (-1)^{|f|} fd_{\Hom}(g)$  for any $f,g \in \End^{\bullet}_R(P_{\bullet})$. So, the cohomology group $\mathrm{H}^*(\End^{\bullet}_R(P_{\bullet}))$ has a graded algebra structure.

\begin{prop}\label{Hom-mod-comod-dga}
	Let $M_{\bullet}$ be a complex in $\mathcal{M}_{A\#H}$, and $X_{\bullet}$ be a complex in $^H\mathcal{M}_B$. Suppose that each term $X_i$ viewed as a $B^{op}$-module is finitely generated. Then
	\begin{enumerate}
		\item $\End_{A^{op}}^{\bullet}(M_{\bullet})$ is a differential graded left $H$-module algebra, with the module structure given in \eqref{H-mod on right Hom set}.
		\item $\End_{B^{op}}^{\bullet}(X_{\bullet})$ is a differential graded left $H$-comodule algebra, with the comodule structure given in \eqref{H-comod on right Hom set}.
		\item There is a differential graded algebra morphism
		$$\varphi: \End_{A^{op}}^{\bullet}(M_{\bullet}) \# \End_{B^{op}}^{\bullet}(X_{\bullet}) \to \End_{(A\#B)^{op}}^{\bullet}(\mathrm{Tot}(M_{\bullet} \# X_{\bullet}))$$
		$$f \# g \mapsto \big( m\#x \mapsto (-1)^{|g| |m|} \sum\limits_{(g)} f(g_{-1} \rightharpoonup m) \# g_0(x)\big),$$
		which is induced by Proposition \ref{decom-Ext-group'} (2).
	\end{enumerate}
\end{prop}
\begin{proof}
	(1) For any $h \in H, f, f' \in \End_{A^{op}}^{\bullet}(M_{\bullet})$ and $m \in M_i$,
	\begin{align*}
		&((h \rightharpoonup (ff'))(m)  = \sum\limits_{(h)}h_1 \rightharpoonup\big(ff'(Sh_2 \rightharpoonup m)\big) \\
 = & \sum\limits_{(h)} h_1 \rightharpoonup f\big((Sh_2) h_3\rightharpoonup f'(Sh_4 \rightharpoonup m)\big) = \sum\limits_{(h)} h_1 \rightharpoonup f\big(Sh_2 \rightharpoonup \big((h_3\rightharpoonup f')(m)\big)\big) \\
		 = & \sum\limits_{(h)} (h_1 \rightharpoonup f)(h_2 \rightharpoonup f')(m).
	\end{align*}
It follows that $h \rightharpoonup (ff')=\sum\limits_{(h)} (h_1 \rightharpoonup f)(h_2 \rightharpoonup f')$.

Since $(h \rightharpoonup \id_{M_{\bullet}})(m) = \sum\limits_{(h)}h_1 \rightharpoonup \id_{M_{\bullet}}(Sh_2 \rightharpoonup m)
= \varepsilon(h) (m)$, $(h \rightharpoonup \id_{M_{\bullet}})=\varepsilon(h) \id_{M_{\bullet}}$.

Hence $\End_{A^{op}}^{\bullet}(M_{\bullet})$ is a graded $H$-module algebra.
In fact, the differential $d_{\Hom}$ is an $H$-morphism.
	\begin{align*}
		& (d_{\Hom}(h \rightharpoonup f))(m)  = d^M \circ (h \rightharpoonup f)(m) - (-1)^{|f|}(h \rightharpoonup f) \circ d^M(m) \\
		 = &\sum_{(h)} d^M(h_1 \rightharpoonup f(Sh_2 \rightharpoonup m)) - (-1)^{|f|} h_1 \rightharpoonup f(Sh_2 \rightharpoonup d^M(m)) \\
		= &\sum_{(h)} h_1 \rightharpoonup (d^M  f (Sh_2 \rightharpoonup m)) - (-1)^{|f|}h_1 \rightharpoonup (f  d^M(Sh_2 \rightharpoonup m)) \\
		= & ( h \rightharpoonup d_{\Hom}(f))(m).
	\end{align*}
	It follows that $\End_{A^{op}}^{\bullet}(M_{\bullet})$ is a differential graded left $H$-module algebra.

	(2) For any $g,g' \in \End_{B^{op}}^{\bullet}(X_{\bullet})$ and  $x \in X_{\bullet}$, by \eqref{H-comod on right Hom set}, $$\rho(g')(x) = \sum g'_{-1} \otimes g'_0(x) = \sum\limits_{(x)}  g'(x_0)_{-1}S^{-1}x_{-1} \otimes g'(x_0)_0,$$
	and $$\rho(g)\big(g'(x_0)_0\big)= \sum g_{-1}\otimes g_0\big(g'(x_0)_0\big)=\sum\limits_{(g'(x_0)_0)}  g(g'(x_0)_0)_{-1}(S^{-1}g'(x_0)_{-1}) \otimes g(g'(x_0)_0)_0.$$
Then
	\begin{align*}
		& \sum\limits_{(g),(g')} g_{-1}g'_{-1} \otimes g_0g'_0(x) = \sum\limits_{(g),(x),g'(x_0)} g_{-1} g'(x_0)_{-1}S^{-1}x_{-1} \otimes g_0(g'(x_0)_0)\\
		 = & \sum\limits_{(x),(g'(x_0)),(g(g'(x_0)_0))} g(g'(x_0)_0)_{-1}(S^{-1}g'(x_0)_{-1}) g'(x_0)_{-2}S^{-1}x_{-1} \otimes g(g'(x_0)_0)_0 \\
		 = & \sum\limits_{(x),(gg'(x_0))} g(g'(x_0))_{-1}S^{-1}x_{-1} \otimes g(g'(x_0))_0 \\
		 = & \sum\limits_{(gg')} (gg')_{-1} \otimes (gg')_0(x).
	\end{align*}
	It follows that  $\rho(gg') = \rho(g) \rho(g')$.

Obviously, $(\varepsilon \otimes 1)\rho(g) = g$. Hence $\End_{B^{op}}^{\bullet}(X_{\bullet})$ is a graded left $H$-comodule algebra. The differential $d_{\Hom}$ is also an $H$-comodule morphism as shown next. Since
	\begin{align*}
		&\rho(d^X \circ g)(x)  = \sum_{(x)} (d^X \circ g)(x_0)_{-1}S^{-1}x_{-1} \otimes (d^X \circ g)(x_0)_0 \\
		 = & \sum_{(x)} g(x_0)_{-1}S^{-1}x_{-1} \otimes d^X (g(x_0)_0) \\
		 = & \sum_{(x)} g_{-1} \otimes d^X  (g_0(x)),

	\end{align*}
it follows that
\begin{align*}
		&\rho(d_{\Hom} (g))(x)  = \sum_{(x)} d_{\Hom} (g)(x_0)_{-1}S^{-1}x_{-1} \otimes d_{\Hom} ( g)(x_0)_0 \\
		 = & \sum_{(x)} g(x_0)_{-1}S^{-1}x_{-1} \otimes d_{\Hom} (g(x_0)_0) \\
		 = & \sum_{(x)} g_{-1} \otimes d_{\Hom} (g_0)(x).
	\end{align*}
Hence $\rho(d_{\Hom} (g))= \sum_{(g)} g_{-1} \otimes d_{\Hom} (g_0)$ and $\End_{B^{op}}^{\bullet}(X_{\bullet})$ is a differential graded left $H$-comodule algebra.

	(3) For any $f,f' \in \End_{A^{op}}^{\bullet}(M_{\bullet}), g,g' \in \End_{B^{op}}^{\bullet}(X_{\bullet}), m \in M_i, x \in X_l$,
	\begin{align*}
	& \varphi\big((f\#g) (f'\#g')\big)(m\#x)  = (-1)^{|g||f'|}\varphi\big(\sum_{(g)} f(g_{-1} \rightharpoonup f') \# g_0g'\big)(m\#x) \\
		 = &(-1)^{|g||f'|} (-1)^{|m|(|g| + |g'|)} \sum_{(g),(g')} f(g_{-2} \rightharpoonup f')(g_{-1}g'_{-1} \rightharpoonup m) \# g_0g'_0(x) \\
		 = &(-1)^{|g||f'|}(-1)^{|m|(|g| + |g'|)} \sum_{(g),(g')} f\big(g_{-1} \rightharpoonup f'(g'_{-1} \rightharpoonup m)\big) \# g_0g'_0(x) \\
		 = &(-1)^{|m||g'|} \sum_{(g')} \phi(f\#g)\big(f'(g'_{-1} \rightharpoonup m) \# g'_0(x)\big) \\
		 = & \varphi(f\#g)\varphi(f'\#g')(m\#x).
	\end{align*}
Hence  $\varphi\big((f\#g) (f'\#g')\big) = \varphi(f\#g)\varphi(f'\#g').$

	Clearly $\varphi(\id_{M_{\bullet}} \# \id_{X_{\bullet}}) = \id_{\mathrm{Tot}(M_{\bullet}\#X_{\bullet})}$. So $\varphi$ is an algebra morphism.

It is left to show that $\varphi$ is a cochain map.
	\begin{align*}
		& (d_{\Hom} \circ \varphi (f \# g))(m \# x) \\
		= & d^{M\#X}(\varphi (f \# g)(m \# x)) - (-1)^{|f| + |g|} \varphi (f \# g)(d^{M\#X} (m \# x)) \\
		= & (-1)^{|g| |m|} d^{M\#X}\big(\sum_{(g)} f(g_{-1} \rightharpoonup m) \# g_0(x)\big)
		    - (-1)^{|f| + |g|} \varphi (f \# g)\big(d^{M}(m) \# x + (-1)^{|m|} m \# d^X(x)\big) \\
		= & (-1)^{|g| |m|} \big(\sum_{(g)} d^M(f(g_{-1} \rightharpoonup m)) \# g_0(x) + (-1)^{|f|+|m|} f(g_{-1} \rightharpoonup m) \# d^X(g_0(x))\big) \\
		& - (-1)^{|f| + |g|} \sum_{(g)} \big( (-1)^{|g|(|m| + 1)}f(g_{-1} \rightharpoonup d^{M}(m)) \# g_0(x) + (-1)^{|m| + |g||m|} f(g_{-1} \rightharpoonup m) \# g_0(d^X(x)) \big) \\
		= &\sum_{(g)} (-1)^{|g||m|} d_{\Hom}(f)(g_{-1} \rightharpoonup m) \# g_0(x) + (-1)^{|f| + (|g|+1)|m|} f(g_{-1} \rightharpoonup m) \# d_{\Hom}(g_0)(x) \\
		= & \varphi (d_{\Hom}(f) \# g + (-1)^{|f|} f \# d_{\Hom}(g))(m \# x) \\
		= & \varphi (d_{\Hom}(f \# g))(m \# x).
	\end{align*}
	Therefore, $d_{\Hom} \circ \varphi  = \varphi \circ d_{\Hom}$, and the conclusion follows.
\end{proof}

By taking the homology, then $\Ext^{\bullet}_{A^{op}}(M, M) = \bigoplus_{n \geq 0} \Ext^n_{A^{op}}(M, M)$ is a graded left $H$-module algebra, and $\Ext^{\bullet}_{B^{op}}(X, X)$ is a graded left $H$-comodule algebra. Hence we can form the graded Takeuchi smash product algebra $\Ext^{\bullet}_{A^{op}}(M, M) \# \Ext^{\bullet}_{B^{op}}(X, X)$.

\begin{thm}\label{Ext-alg of smash product}
	Suppose that $M \in {\mathcal{M}_{A\#H}}$ is pseudo-coherent as $A$-module, and $X \in {^H\mathcal{M}_B}$ is pseudo-coherent as $B$-module. Then, as graded algebras,
	$$\Ext^{\bullet}_{(A\#B)^{op}}(M\#X, M\#X) \cong \Ext^{\bullet}_{A^{op}}(M, M) \# \Ext^{\bullet}_{B^{op}}(X, X).$$
\end{thm}
\begin{proof}
	Let $P_{\bullet}$ be a projective resolution of $M_{A\#H}$. By Lemma \ref{relative-resolution}, there exists an exact sequence
	$$\cdots \longrightarrow Q_n \longrightarrow \cdots \longrightarrow Q_0 \longrightarrow X \longrightarrow 0 $$
	in ${^H\mathcal{M}_B}$, where all the $Q_n$'s are finitely generated free as $B$-module.

	By Proposition \ref{Hom-mod-comod-dga}, $\End^{\bullet}_{A^{op}}(P_{\bullet})$ is a differential graded left $H$-module algebra, and $\End^{\bullet}_{B^{op}}(Q_{\bullet})$ is a differential graded left $H$-comodule algebra, and
	$$\varphi: \End^{\bullet}_{A^{op}}(P_{\bullet}) \# \End^{\bullet}_{B^{op}}(Q_{\bullet}) \to \End^{\bullet}_{(A\#B)^{op}}(\mathrm{Tot}(P_{\bullet}\#Q_{\bullet}))$$
	$$f \# g \mapsto \big(p\#q \mapsto (-1)^{|g||p|}\sum_{(g)} f(g_{-1} \rightharpoonup p) \# g_0(q)\big)$$
	is a differential graded algebra morphism.
	By Proposition \ref{decom-Ext-group'} (3), it is a quasi-isomorphism.
	Then the conclusion follows by taking the homology.
\end{proof}

\section{Application to Takeuchi smash products of AS-regular algebras}
We recall some definitions first.
\subsection{Artin-Schelter regular algebras}
A graded algebra $A = \bigoplus_{n \in \Z} A_n$ is called {\it connected}, if $A_{<0} = 0$ and $A_0$ is the base field $k$. For any connected graded algebra $A$,  the left global dimension and right global dimension of $A$ are equal, which are equal to the projective dimension of the trivial module $k \cong A/A_{\geq 1}$ or $k_A$.

\begin{defn}
	A connected graded algebra $A$ is called {\it Artin-Schelter regular} (for short, AS-regular) of dimension $d$ for some integer $d \geq 0$, if
	\begin{enumerate}
		\item $A$ has global dimension $d$;
		\item $\Ext^i_{A}(k,A) \cong
		\begin{cases}
			0 & i \neq d \\
			k & i = d.
		\end{cases}$
	\end{enumerate}

	If $A$ is a left AS-regular algebra of dimension $d$, then it follows from the Ischebeck's spectral sequence that the right version of the condition (2) holds.
\end{defn}

Let $A$ be a connected graded algebra. By \cite[Lemma 1.2]{RRZ1}, $A$ is AS-regular of dimension $d$ if and only if it is a graded skew Calabi-Yau algebra of dimension $d$, which is defined in the following.

\begin{defn} A $k$-algebra $A$ is called {\it skew Calabi-Yau algebra} of dimension $d$, if
	\begin{itemize}
		\item [(i)] $A$ is {\it homologically smooth}, that is, as $A^e$-module, $A$ has a finitely generated projective resolution of finite length;
		\item [(ii)] there exists an automorphism $\mu$ of $A$, such that
		$$\Ext^i_{A^e}(A,A^e) \cong
		\begin{cases}{\tiny {\large }}
			0, & i \neq d\\
			A^{\mu}, & i = d
		\end{cases}$$
		as $A$-$A$-bimodules.
	\end{itemize}
The automorphism $\mu$ is unique up to an inner automorphism, which is called a {\it Nakayama automorphism} of $A$, denoted sometimes by $\mu_A$.

Graded skew Calabi-Yau algebras are defined similarly in the graded module category for graded algebras.
\end{defn}
\subsection{Takeuchi smash products of AS-regular algebras}
A Hopf algebra $H = \bigoplus_{n \in \Z} H_n$ is said to be a {\it graded Hopf algebra} if both of its algebra and coalgebra structure are graded with respect to the grading $H = \bigoplus_{n \in \Z} H_n$, and its antipode is also graded.
Let $H$ be a graded Hopf algebra, and $A$ be a graded algebra which is also an $H$-module algebra. If $H_m \rightharpoonup A_n \subseteq A_{m+n}$ for all $n$ and $m$, then $A$ is called an {\it $H$-module graded algebra}.
Let $B$ be a graded algebra with an $H$-comodule algebra structure $\rho$. If $\rho(B_n) \subseteq \bigoplus_{i} H_i \otimes B_{n-i}$ for all $n$, then $B$ is called {\it an $H$-comodule graded algebra}. In this case,
it is clear that the Takeuchi smash product $A \# B$ is a graded algebra with $(A\#B)_n = \bigoplus_i A_i \otimes B_{n-i}$.

The Hopf algebra $H = k\langle g^{\pm 1}, X\rangle$ given in Example \ref{ore-ext-Hopf-alg} is a graded Hopf algebra by setting $\deg(g) = 0$ and $\deg(X) = 1$. Suppose $A$ is a graded algebra, with a graded automorphism $\sigma$ of degree $0$ and a graded $\sigma$-derivation $\delta$ of degree $1$. Then $A$ is an $H$-module graded algebra, $B=k[X]$ is an $H$-comodule graded algebra, and the Ore extension $A[x, \sigma, \delta]$ is the graded Takeuchi smash product $A \# B$.

\begin{thm}\label{AS-regular-TSP} Let $A$ and $B$ be AS-regular algebras of dimension $d_1$ and $d_2$ respectively. If $H$ is a graded Hopf algebra, $A$ is an $H$-module graded algebra and $B$ is an $H$-comodule graded algebra, such that $H \rightharpoonup A_{\geq 1} \subseteq A_{\geq 1}$ or $\rho(B_{\geq 1}) \subseteq H \otimes B_{\geq 1}$, then $A\#B$ is an AS-regular algebra of dimension $d_1+d_2$.
\end{thm}
\begin{proof}
	Suppose $H \rightharpoonup A_{\geq 1} \subseteq A_{\geq 1}$ (which is automatic if $H$ is positively graded). Then the trivial module $k \cong A/A_{\geq 1}$ is an $A\#H$-module.
	By Corollary \ref{proj-mod of smash product},  $\pd_{A\#B} \, k \leq \pd_A \, k + \pd_B \, k = d_1 + d_2$.
	It follows from the spectral sequence in Proposition \ref{Ext-spec-seq-of-smash-prod-mod} that
	$$\Ext^i_{A\#B}(k, A\#B) \cong \begin{cases}
		0, & i \neq d_1+d_2 \\
		\Ext^{d_2}_B(k, \Ext^{d_1}_A(k, A\#B)), & i = d_1+d_2.
	\end{cases}$$

	We claim that $\Ext^{d_1}_A(k, A\#B) \cong B$ as left $B$-modules. Let $P_{\bullet}$ be a projective resolution of $_{A\#H}k$. Then there is a natural $B$-module complex morphism
	$$\Hom_{A}(P_{\bullet}, A) \otimes B \To \Hom_A(P_{\bullet}, A\#B)$$
	where the $B$-module structure on $\Hom_{A}(P_i, A) \otimes B$ is given by $b \rightharpoonup (f \otimes b') = \sum_{(b)} (b_{-1} \rightharpoonup f) \otimes b_0 b'$, and  the $B$-module structure on $\Hom_{A}(P_i, A \# B)$ is given by \eqref{B-mod-on-Hom}. Since $_Ak$ is pseudo-coherent by \cite[Proposition 3.1]{SZ}, it follows that  $\Hom_{A}(P_{\bullet}, A) \otimes B \To \Hom_A(P_{\bullet}, A\#B)$ is a quasi-isomorphism of $B$-module complex. Thus, with the induced $B$-module structures, $\Ext^{d_1}_A(k, A\#B) \cong \Ext^{d_1}_A(k, A) \otimes B$ as $B$-modules.
	Since $\Ext_A^{d_1}(k, A)$ is an one-dimensional $k$-vector space, then there is algebra morphism $\alpha: H \to k$, such that $h \rightharpoonup e = \alpha(h) e$ for all $h \in H$ and $e \in \Ext_A^{d_1}(k, A)$.
	This implies that $\Ext_A^{d_1}(k, A\#B) \cong {^{\Xi^l_{\alpha}} B}$, where $\Xi^l_{\alpha} : B \to B$ is the algebra automorphism defined by $\Xi^l_{\alpha}(b) = \sum_{(b)} \alpha(b_{-1}) b_0$.
	Thus $\Ext_A^{d_1}(k, A\#B) \cong B$ as left $B$-modules, the claim is proved.

	Since $B$ is AS-regular, then $\Ext^{d_1+d_2}_{A\#B}(k, A\#B) \cong \Ext^{d_2}_B(k, B) \cong k$. It follows that $A\#B$ is a $(d_1+d_2)$-dimensional AS-regular algebra.

	Suppose $\rho(B_{\geq 1}) \subseteq H \otimes B_{\geq 1}$. Then $k \cong B/{B_{\geqslant 1}} \in ^H\!\mathcal{M}_B$. Then
	it follows Corollary \ref{proj-mod of smash product'} and Proposition \ref{right-case-spec-seq}  that  $\pd_{(A\#B)^{op}} \, k \leq \pd_{A^{op}} \, k + \pd_{B^{op}} \, k = d_1 + d_2$ and
	$$\Ext^i_{(A\#B)^{op}}(k, A\#B) \cong \begin{cases}
		0, & i \neq d_1+d_2 \\
		\Ext^{d_1}_{A^{op}}(k, \Ext^{d_2}_{B^{op}}(k, A\#B)), & i = d_1+d_2.
	\end{cases}$$

    Since $A$ is AS-regular, by \cite[Proposition 3.1]{SZ}, $k_B$ is pseudo-coherent. It follows from Lemma \ref{relative-resolution} that $k \cong B/{B_{\geqslant 1}}$ has a  resolution $Q_\bullet \to k \to 0$ in $^H\!\mathcal{M}_B$ such that $Q_n = V_n \otimes B$ for some finite-dimensional left $H$-comodule $V_n$. Then, $\Hom_{B^{op}}(Q_n, A\#B)$ has an  $A^{op}$-module structure given by Lemma \ref{right-Hom-Tensor-adj}, that is,
    $(fa)(y)=f(y_0)(S^{-1}y_{-1} \rightharpoonup a)$, and  $\Hom_{B^{op}}(Q_n, B)$ has a left $H$-comodule structure given by   $\rho(g)(y)=\sum_{(y),(g(y_0))} (g(y_0)_{-1} S^{-1}y_{-1}) \otimes g(y_0)_0$ as defined in \eqref{H-comod on right Hom set} (see Proposition \ref{decom-Ext-group'}).

    If we endow $A \otimes \Hom_{B^{op}}(Q_n, B)$ with a right $A$-module structure by
    $$(a \otimes g) a'= a( g_{-1}\rightharpoonup a') \otimes g_0.$$
    Then,
   $A \otimes \Hom_{B^{op}}(Q_n, B) \to \Hom_{B^{op}}(Q_n, A\#B), a \otimes g \mapsto \big([a, g]: Q \to A\#B, y \mapsto a\# g(y)\big)$
	is an $A^{op}$-module isomorphism. In fact.
    $$[a( g_{-1}\rightharpoonup a'), g_0](y)= a( g_{-1}\rightharpoonup a')  \# g_0(y)= a( g(y_0)_{-1} S^{-1}y_{-1}\rightharpoonup a') \# g(y_0)_0,\, \textrm{ and }$$
	$$([a, g] a')(y) =\big([a, g](y_0)\big) (S^{-1} y_{-1} \rightharpoonup a')=a\big(g(y_0)_{-1}S^{-1} y_{-1} \rightharpoonup a'\big) \#  g(y_0)_0.$$
It follows that $\Ext^{d_2}_{B^{op}}(k, A\#B) \cong A \otimes \Ext^{d_2}_{B^{op}}(k, B) \cong (A \otimes k)_A \cong A_A.$ Therefore,
	$$\Ext^{d_1+d_2}_{(A\#B)^{op}}(k, A\#B) \cong \Ext^{d_1}_{A^{op}}(k, \Ext^{d_2}_{B^{op}}(k, A\#B)) \cong \Ext^{d_1}_{A^{op}}(k, A) \cong k.$$
Hence  $A\#B$ is an AS-regular algebra of dimension $d_1+d_2$.
\end{proof}

So graded Ore extensions preserve the AS-regularity, which  was proved in \cite[Proposition 2]{AST}. The ungraded case was proved in \cite{LWW} in terms of skew Calabi-Yau algebras.
The Yoneda Ext-algebras recover many properties of AS-regular algebras as it is well known (see, say \cite{LPWZ1, LPWZ2}), for instance,  the Nakayama automorphisms.

\begin{thm}\cite[Theorem 4.2]{RRZ2}
	Let $A$ be a noetherian AS-regular algebra generated in degree $1$. Let $\mu_A$ and $\mu_E$ be the Nakayama automorphism of $A$ and $\Ext^\bullet_A(k, k)$ respectively.
	Then $\mu_A|_{A_1} = (\mu_E|_{\Ext^{1}_A(k, k)})^*$ where $A_1$ is identified with $\Ext^{1}_A(k, k)^*$.
\end{thm}

As an application of Theorems \ref{Ext-alg of smash product} and \ref{AS-regular-TSP}, we can describe the Nakayama automorphism of the Takeuchi smash product of AS-regular algebras with a similar proof to Theorem \cite[Theorem 4.2]{SZL}.
For the definitions of homological determinants and codeterminant we refer \cite{KKZ} as the reference.

\begin{prop}
Suppose $A$ and $B$ are noetherian AS-regular algebras generated in degree $1$, and  $A$ is a graded $H$-module algebra, $B$ is a graded $H$-comodule for some Hopf algebra $H$. If $A\#B$ is noetherian, then the Nakayama automorphism $\mu_{A\#B}$ of $A\#B$ is given by
	$$\mu_{A\#B}(a\#b) = \sum_{(b)} \mu_A(g \rightharpoonup a) \# \hdet(b_{-1})\mu_B(b_0),$$
	where $\mu_A$ and $\mu_B$ are the Nakayama automorphisms of $A$ and $B$ respectively, $\hdet \in H^*$ and $g \in H$ are the homological determinant of the $H$-action on $A$ and the homological codeterminant of the $H$-coaction on $B$ respectively.
\end{prop}

\section*{Acknowledgements} This research is partially supported by the National Science Foundation of China (Grant No. 11771085) and the National Key Research and Development Program of China (Grant No. 2020YFA0713200).

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\end{document}


text:
\section{Introduction}
Throughout this paper we work in the piecewise linear category.
We shall study knots and links in the three-dimensional Euclidean space $\Bbb{R}^{3}$.
For the standard definitions and results of knots and links, we refer to \cite{A}, \cite{BZ}, \cite{C}, \cite{Kaw}, \cite{Lic2}, \cite{L}, \cite{M} and \cite{R}.

For the purpose of defining a numerical invariant of knots and links, we need to prepare following terms.
A link diagram is {\em ordered} if an order is given to its components.
A link diagram is {\em based} if a basepoint (different from the crossing points) is specified on each component.
A link diagram is {\em oriented} if an orientation is specified on each component.

Let $L$ be a link, and let $\tilde{L}$ be a based ordered oriented link diagram of $L$.
The {\it descending diagram} of $\tilde{L}$ is obtained as follows, and will be denoted by $d(\tilde{L})$.
Beginning at the basepoint of the first component of $\tilde{L}$ and proceeding in the direction specified by the orientation, change the crossings as necessary so that each crossing is first encountered as an over-crossing.
Continue this procedure with the remaining components in the sequence determined by the ordering, proceeding from the basepoint in the direction determined by the orientation, changing crossings so that ultimately every crossing is first encountered as an over-crossing.
The result is the descending diagram $d(\tilde{L})$ obtained from $\tilde{L}$. An example is shown in Figure \ref{example1}.
Note that $d(\tilde{L})$ is a diagram of a trivial link.

\begin{figure}[htbp]
	\begin{center}
	\begin{tabular}{cc}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.4\linewidth]{example1.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.4\linewidth]{example2.eps}\\
	$\tilde{L}$ & $d(\tilde{L})$
	\end{tabular}
	\end{center}
	\caption{}
	\label{example1}
\end{figure}

Prof. Shin'ichi Suzuki mentioned in \cite[2.16 (ii)]{S} that the smallest number of crossing changes to obtain the trivial link may not always be provided with the number of difference crossing between a link diagram and the descending diagram.
That was why we defined the next numerical invariant in \cite{MO}.

\begin{definition}
Let $L$ be a link and let $\tilde{L}$ be a based ordered oriented link diagram of $L$. The {\em ascending number} of $\tilde{L}$ is defined as the number of different crossings between $\tilde{L}$ and $d(\tilde{L})$, and denoted by $a(\tilde{L})$.
The {\em ascending number} of $L$ is defined as the minimum number of $a(\tilde{L})$ over all based ordered oriented link diagram $\tilde{L}$ of $L$, and denoted by $a(L)$.
\end{definition}

The definition and the above note imply that $a(L) \geq u(L)$,
where $u(L)$ denotes the unknotting number of $L$.
Accordingly $L$ is a trivial link if and only if $a(L) = 0$.
A difference of $a(L)$ and $u(L)$ can become no matter how big.
Actually, the mention of S. Suzuki will be demonstrated in Corollary \ref{difference}.

Now this paper is organized as follows. In section 2 we
state all results. In section 3 we give all proofs of the
results. In section 4 we show a table of the ascending
number, the unknotting number, and the bridge number of prime knots with eight crossings or less.

\section{Results}

\begin{proposition}\label{crossing}
For a nontrivial knot $K$, we have
\[ a(K) \leq \left[\frac{c(K)-1}{2}\right].\]
For a link $L$, we have
\[ a(L) \leq \left[\frac{c(L)}{2}\right],\]
where $c(L)$ denotes the minimum crossing number of $L$,
and $[x]$ denotes the greatest integer which does not
exceed $x$.
\end{proposition}

The following theorem is the fundamental inequality between the ascending
number and the bridge number.

\begin{theorem}\label{bridge}
For an $n$-component link $L$, we have
\[a(L) \geq b(L)-n,\]
where $b(L)$ is the bridge number of $L$.
\end{theorem}

The following corollary asserts the difference between
$a(K)$ and $u(K)$.

\begin{corollary}\label{difference}
For any nonnegative integer $n$, there is a knot $K$
such that $a(K)-u(K) \geq n$.
\end{corollary}

For the connected sum, we have the following proposition.

\begin{proposition}\label{subadditive}
For a knot, the ascending number is subadditive with
respect to connected sum \#, i.e. $a(K_{1}\#K_{2}) \leq
a(K_{1})+a(K_{2}).$
\end{proposition}

It is natural to ask whether the ascending number is
additive with respect to the connected sum.

\begin{conjecture}
For any knots $K_{1}, K_{2}$,
\[a(K_{1}\# K_{2}) = a(K_{1}) + a(K_{2}).\]
\end{conjecture}

The following corollary solves the above question
partially.

\begin{corollary}\label{subanswer}
Let $K_{i}\ (i = 1,2)$ be a knot. If $K_{i}$ satisfies
$a(K_{i}) = b(K_{i}) - 1\ (i = 1,2)$, then
\[a(K_{1}\# K_{2}) = a(K_{1}) + a(K_{2}).\]
\end{corollary}

In consequence of this, we have the following corollary.

\begin{corollary}\label{exist}
For any nonnegative integer $n$, there is a knot $K$ such that
$a(K) = n$.
\end{corollary}

The following theorem characterizes the ascending number one link.
We note that Tat Sang Fung (\cite{Fun}) also obtained the same result for knot case.
In fact, he showed that almost descending knots, i.e. knots with diagrams descending except at one crossing, are twist knots.

\begin{theorem}\label{one}
Let $L$ be an $n$-component link. Then the ascending
number of $L$ is one if and only if $L$ is
\begin{align*}
& (\text{twist knot})\circ O^{n-1},
 \text{\quad if L is completely splittable}\\
\intertext{or}
& (\text{Hopf link})\circ O^{n-2},
 \text{\quad otherwise,}
\end{align*}
where $\circ$ denotes the split union, and $O^{n}$
denotes an $n$-component trivial link.

In paticular, the ascending number of a knot $K$ is one if
and only if $K$ is a twist knot.
\end{theorem}

The following theorem determines the ascending number for
torus knots.

\begin{theorem}\label{torus}

Let $p$ and $q$ be coprime positive integers, and let
$T_{p,q}$ be a $(p,q)$-torus knot. Then we have
\[ a(T_{p,q})=\frac{(p-1)(q-1)}{2}. \]
\end{theorem}

\section{Proofs}

\begin{proof}(of Proposition \ref{crossing})
At the beginning, we prove the proposition for a
link. Let $L$ be a link, and let $\tilde{L}$ be a based
ordered oriented link diagram of $L$ with minimum crossings.
Consider a based ordered oriented link diagram which is
obtained by reversing the order and the orientation of
$\tilde{L}$, and let it be denoted by $-\tilde{L}$. Then
it holds that $c(L) = a(\tilde{L}) + a(-\tilde{L})$.
Indeed $d(-\tilde{L})$ is obtained by reflecting
$d(\tilde{L})$ in the plane. Hence
\[\frac{c(L)}{2} \geq \min\{a(\tilde{L}),a(-\tilde{L})\} \geq
a(L).\]

Next we prove the proposition for a knot. Let $K$ be a
knot, and let $\tilde{K}$ be a knot diagram of $K$ with
minimum crossings. Choose a crossing of $\tilde{K}$ and
specify a basepoint and an orientation on $\tilde{K}$ in
the following fashion.
Whenever we begin the basepoint of $\tilde{K}$ and proceed
in the direction specified by the orientation, we
first encounter the crossing as an over-crossing.

On the other hand let $-\tilde{K'}$ be a based oriented
knot diagram which is obtained by reversing the
orientation of $\tilde{K}$ and sliding the base point of
$\tilde{K}$, as follows. Whenever we begin the basepoint of
$-\tilde{K'}$ and proceed in the direction specified by the
orientation, we first encounter the crossing as an
over-crossing. See Figure \ref{reverse}.

\begin{figure}[htbp]
	\begin{center}
	\begin{tabular}{cc}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.25\linewidth]{reverse1.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.25\linewidth]{reverse2.eps}\\
	$\tilde{K}$ & $-\tilde{K}'$
	\end{tabular}
	\end{center}
	\caption{}
	\label{reverse}
\end{figure}

Then it holds that $c(K) - 1 = a(\tilde{K}) +
a(-\tilde{K'})$. Indeed a knot consists of a single circle.
Hence

\[
\frac{c(k)-1}{2} \geq \min\{a(\tilde{K}),a(-\tilde{K'})\}
\geq a(K).
\]

This completes the proof.
\end{proof}

\begin{proof}(of Theorem \ref{bridge})
Let $\tilde{L}$ be a based ordered oriented link diagram of $L=K_1\cup \cdots \cup K_n$ satisfying $a(\tilde{L})=a(L)$.

First, we make an $n$-bridge presentation of a trivial link $L'=K_1'\cup \cdots \cup K_n'$ from the descending diagram $d(\tilde{L})$ by isotoping on $z$-coordinate so that $L'$ satisfy the next conditions.
We note that a similar presentation can be seen in \cite[Section 3.1]{A}.

\begin{enumerate}
\item For each $i\in \{1,\ldots,n\}$, $K_i'$ is parametrized by an embedding $f_i:[0,1]\to \Bbb{R}^3$ so that the orientation and basepoint of $p(f_i([0,1]))$ are in consistency with that of $d(\tilde{K_i})$, where $p$ denotes the projection $\Bbb{R}^3\to \Bbb{R}^2\times \{0\},\ (x,y,z)\mapsto (x,y,0)$.
\item For any $x,\ y\in [0,1]$ such that $x<y$, $h(f_i(x))>h(f_i(y))$, where $h$ denotes the hight function $\Bbb{R}^3\to \{0\}^2\times \Bbb{R},\ (x,y,z)\mapsto (0,0,z)$.
\item For each $i\in \{1,\ldots,n-1\}$, $h(f_i(\{1\}))>h(f_{i+1}(\{0\}))$.
\item For each $i\in \{1,\ldots,n\}$, $f_i(\{0\})$ and $f_i(\{1\})$ are connected by a vertical arc $P_i$ contained in $p^{-1}($base point of $d(\tilde{K_i}))$. Thus, $K_i'=f_i([0,1])\cup P_i$
\end{enumerate}

Next, we deform $L'$ to obtain an $(n+a(L))$-bridge presentation of $L$ as follows.
Let $\mathcal{C}$ be the set of crossings of $\tilde{L}$ which are different from that of $d(\tilde{L})$.
For each crossing $c\in \mathcal{C}$, we isotope the corresponding under-crossing of $L'$ to exceed the corresponding over-crossing.
Here, we may assume that such an operation accrues exactly one maximal point.
As a result, we obtain a presentation of $L$ whose bridge number is equal to $n+a(L)$.
Figure \ref{example2} shows a realization of $\tilde{L}$ and $d(\tilde{L})$ in Figure \ref{example1}.

\begin{figure}[htbp]
	\begin{center}
	\begin{tabular}{cc}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.3\linewidth]{L.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.3\linewidth]{L2.eps}\\
	$L$ & $L'$
	\end{tabular}
	\end{center}
	\caption{}
	\label{example2}
\end{figure}

Hence we have
\[
b(L)\le n+a(L).
\]

This arrives at the inequality of Theorem \ref{bridge}.
\end{proof}

\begin{proof}(of Corollary \ref{difference})
We will show that a composite knot $K$ which is a connected sum of $n$ $8_{17}$'s satisfies the inequality of Corollary \ref{difference}.
It is known that $u(8_{17})=1$ and $b(8_{17})=3$.
By Schubert's formula $b(K_1\#K_2)=b(K_1)+b(K_2)-1$ (\cite{Sch}),
\[
b(K)=nb(8_{17})-(n-1)=2n+1.
\]

Since an inequality $u(K_1\#K_2)\le u(K_1)+u(K_2)$ generally holds,
\[
u(K)\le n u(8_{17})=n.
\]

Hence, by Theorem \ref{bridge},
\[
a(K)-u(K)\ge (b(K)-1)-u(K)\ge 2n-n=n.
\]
\end{proof}

\begin{proof}(of Proposition \ref{subadditive})
Let $\tilde{K_1}$ and $\tilde{K_2}$ be based oriented diagram of knots $K_1$ and $K_2$ such that $a(\tilde{K_i})=a(K_i)$ for $i=1,2$.
By considering diagrams on the 2-sphere, we may assume that basepoints of $K_1$ and $K_2$ are in regions containing a point at infinity.
Then we perform a connected sum $\tilde{K_1}$ and $\tilde{K_2}$ to obtain a based oriented diagram $\tilde{K_1\# K_2}$ as Figure \ref{connected sum}.

\begin{figure}[htbp]
	\begin{center}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.7\linewidth]{connected.eps}	\end{center}
	\caption{}
	\label{connected sum}
\end{figure}

Therefore
\[
a(K_1\# K_2)\le a(\tilde{K_1}\#\tilde{K_2})=a(\tilde{K_1})+a(\tilde{K_2})=a(K_1)+a(K_2).
\]
\end{proof}

\begin{proof}(of Corollary \ref{subanswer})
By Schubert's formula and Theorem \ref{bridge},
\[
a(K_1\# K_2)\ge b(K_1\# K_2)-1 =(b(K_1)-1)+(b(K_2)-1)=a(K_1)+a(K_2).
\]
Then, Proposition \ref{subadditive} gives the equality.
\end{proof}

\begin{proof}(of Corollary \ref{exist})
Let $K$ be a connected sum of $n$ $3_1$'s.
Since $a(3_1)=1$ and $b(3_1)=2$, $3_1$ satisfies the supposition of Corollary \ref{subanswer}.
Hence, we have $a(3_1)=n$.
\end{proof}

\begin{proof}(of Theorem \ref{one})
The ascending number of the Hopf link is obviously equal to 1.
The ascending number of a twist knot is also equal to 1 by the transformation in Figure \ref{twist}.

\begin{figure}[htbp]
	\begin{center}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.8\linewidth]{twist.eps}	\end{center}
	\caption{}
	\label{twist}
\end{figure}

Conversely, suppose that $a(L)=1$ and let $\tilde{L}$ be a based ordered oriented diagram of $L$ with $a(\tilde{L})=1$.
Then there is just one crossing $c$ of $\tilde{L}$ that is a difference between $\tilde{L}$ and $d(\tilde{L})$.
We make an $n$-bridge presentation of a trivial link $L'$ from the descending diagram $d(\tilde{L})$ and deform $L'$ to obtain an $(n+1)$-bridge presentation of $L$ in the same way as the proof of Theorem \ref{bridge}.

There are two cases.

\begin{enumerate}
\item The crossing $c$ consists of an over-crossing and an under-crossing of $\tilde{K_i}$.
\item The crossing $c$ consists of an over-crossing of $\tilde{K_j}$ and an under-crossing of $\tilde{K_i}$ $(i<j)$.
\end{enumerate}

In Case 1, $L$ is completely splittable and by a deformation in Figure \ref{twist2}, $K_i$ is a twist knot.
Here, we may assume that a basepoint of $\tilde{K_i}$ is in the left most by considering a diagram on the 2-sphere.

\begin{figure}[htbp]
	\begin{center}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.9\linewidth]{twist3.eps}	\end{center}
	\caption{}
	\label{twist2}
\end{figure}

In Case 2, $L$ is a split union of a Hopf link $K_i\cup K_j$ and $(n-2)$-component trivial link by a deformation in Figure \ref{Hopf2}.

\begin{figure}[htbp]
	\begin{center}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.9\linewidth]{Hopf3.eps}	\end{center}
	\caption{}
	\label{Hopf2}
\end{figure}

This completes the proof of Theorem \ref{one}.
\end{proof}

\begin{proof}(of Theorem \ref{torus})
Let $p,\ q$ be coprime positive integers $(p<q)$, $K$ a $(p,q)$-torus knot and $\tilde{K}$ a standard diagram with $(p-1)q$-crossings.

We assign a basepoint and an orientation to $\tilde{K}$ so that whenever we begin the basepoint and proceed by the orientation, we first encounter $(p-1)$-crossings as over-crossings.
We denote a resultant based oriented diagram by $\tilde{K_a}$ (Figure \ref{torus2}).

On the other hand, let $\tilde{K_b}$ be a based oriented diagram which is obtained by reversing the orientation of $\tilde{K_a}$ and sliding the base point of $\tilde{K_a}$ so that whenever we begin the basepoint of $\tilde{K_b}$ and proceed by the orientation, we first encounter the same $(p-1)$-crossings as over-crossings (Figure \ref{torus2}).

\begin{figure}[htbp]
	\begin{center}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.5\linewidth]{torus.eps}	\end{center}
	\caption{$(4,7)$-torus knot}
	\label{torus2}
\end{figure}

Then, it holds that $a(\tilde{K_a})+a(\tilde{K_b})=(p-1)(q-1)$.
Hence
\[
a(\tilde{K})\le \min \{ a(\tilde{K_a}), a(\tilde{K_b}) \} \le \frac{(p-1)(q-1)}{2}
\]

On the other hand, by a solution of Milnor's conjecture $u(K)=(p-1)(q-1)/2$ (\cite{KM}),
\[
\frac{(p-1)(q-1)}{2} = u(K) \le a(K) \le a(\tilde{K})
\]
Hence, we obtain Theorem \ref{torus}.
\end{proof}

\section{Table of the ascending number of knots}

Here we present a table of the ascending number, unknotting number and bridge number of knots with $8$-crossings or less.
The unknotting numbers are cited from \cite{S} except for $u(8_{10})=2$ from \cite[Corollary 1.2]{OS} and $u(8_{16})=2$ from an unpublished result by J. R. Rickard (\cite{Lic}, \cite{K}).
The bridge numbers are cited from \cite{M}.
For the ascending numbers, $a(8_4)=2$ was determined by Miki Okuda (\cite{O}), $a(8_{13})=2$ was determined by Sachie Fujimura (\cite{F}) and she also pointed out $a(8_{20})=2$, where they belong to Laboratory of Prof. Takao Matumoto, Department of Mathematics, School of Science, Hiroshima University.
Miki Okuda also determined $a(9_3)=3$, $a(9_4)=2$, $a(9_6)=3$, $a(9_7)=2$, $a(9_{47})=2$ and $a(9_{48})=2$.

\bigskip
\begin{minipage}{4cm}
\begin{tabular}{|c||c|c|c|}
\hline
$K$ & $a(K)$ & $u(K)$ & $b(K)$\\
\hline
$3_1$ & 1 & 1 & 2\\
\hline
$4_1$ & 1 & 1 & 2\\
\hline
$5_1$ & 2 & 2 & 2\\
\hline
$5_2$ & 1 & 1 & 2\\
\hline
$6_1$ & 1 & 1 & 2\\
\hline
$6_2$ & 2 & 1 & 2\\
\hline
$6_3$ & 2 & 1 & 2\\
\hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}{6cm}
\begin{tabular}{|c||c|c|c|}
\hline
$K$ & $a(K)$ & $u(K)$ & $b(K)$\\
\hline
$7_1$ & 3 & 3 & 2\\
\hline
$7_2$ & 1 & 1 & 2\\
\hline
$7_3$ & 2 & 2 & 2\\
\hline
$7_4$ & 2 & 2 & 2\\
\hline
$7_5$ & 2 & 2 & 2\\
\hline
$7_6$ & 2 & 1 & 2\\
\hline
$7_7$ & 2 & 1 & 2\\
\hline
\end{tabular}
\end{minipage}
\bigskip

\begin{minipage}{4cm}
\begin{tabular}{|c||c|c|c|}
\hline
$K$ & $a(K)$ & $u(K)$ & $b(K)$\\
\hline
$8_1$ & 1 & 1 & 2\\
\hline
$8_2$ & 2 or 3 & 2 & 2\\
\hline
$8_3$ & 2 & 2 & 2\\
\hline
$8_4$ & 2 & 2 & 2\\
\hline
$8_5$ & 2 or 3 & 2 & 3\\
\hline
$8_6$ & 2 & 2 & 2\\
\hline
$8_7$ & 2 or 3 & 1 & 2\\
\hline
$8_8$ & 2 & 2 & 2\\
\hline
$8_9$ & 2 or 3 & 1 & 2\\
\hline
$8_{10}$ & 2 or 3 & 2 & 3\\
\hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}{6cm}
\begin{tabular}{|c||c|c|c|}
\hline
$K$ & $a(K)$ & $u(K)$ & $b(K)$\\
\hline
$8_{11}$ & 2 & 1 & 2\\
\hline
$8_{12}$ & 2 & 2 & 2\\
\hline
$8_{13}$ & 2 & 1 & 2\\
\hline
$8_{14}$ & 2 & 1 & 2\\
\hline
$8_{15}$ & 2 & 2 & 3\\
\hline
$8_{16}$ & 2 or 3 & 2 & 3\\
\hline
$8_{17}$ & 2 or 3 & 1 & 3\\
\hline
$8_{18}$ & 2 & 2 & 2\\
\hline
$8_{19}$ & 3 & 3 & 3\\
\hline
$8_{20}$ & 2 & 1 & 3\\
\hline
$8_{21}$ & 2 & 1 & 3\\
\hline
\end{tabular}
\end{minipage}
\bigskip

We present a based oriented knot diagram which gives the minimal ascending number.
We omit knot diagrams which are same as minimal crossing diagram.
As S. Suzuki remarked in \cite[2.16 (i)]{S}, the minimal ascending diagram is not unique.

\begin{figure}[htbp]
	\begin{center}
	\begin{tabular}{ccc}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.28\linewidth]{5_2.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.3\linewidth]{6_1.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.29\linewidth]{7_2.eps}\\
	$5_2$ & $6_1$ & $7_2$\\
	\end{tabular}
	\end{center}
	\label{minimal}
\end{figure}

\begin{figure}[htbp]
	\begin{center}
	\begin{tabular}{ccc}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.32\linewidth]{7_3.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.28\linewidth]{7_4.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.28\linewidth]{7_5.eps}\\
	$7_3$ & $7_4$ & $7_5$\\
	\end{tabular}
	\end{center}
	\label{minimal}
\end{figure}

\begin{figure}[htbp]
	\begin{center}
	\begin{tabular}{ccc}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.3\linewidth]{7_6.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.3\linewidth]{8_1.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.28\linewidth]{8_3.eps}\\
	$7_6$ & $8_1$ & $8_3$\\
	\end{tabular}
	\end{center}
	\label{minimal}
\end{figure}

\begin{figure}[htbp]
	\begin{center}
	\begin{tabular}{ccc}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.33\linewidth]{8_4.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.26\linewidth]{8_6.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.28\linewidth]{8_8.eps}\\
	$8_4$ & $8_6$ & $8_8$\\
	\end{tabular}
	\end{center}
	\label{minimal}
\end{figure}

\begin{figure}[htbp]
	\begin{center}
	\begin{tabular}{ccc}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.33\linewidth]{8_11.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.25\linewidth]{8_12.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.25\linewidth]{8_13.eps}\\
	$8_{11}$ & $8_{12}$ & $8_{13}$\\
	\end{tabular}
	\end{center}
	\label{minimal}
\end{figure}

\begin{figure}[htbp]
	\begin{center}
	\begin{tabular}{ccc}
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.3\linewidth]{8_14.eps}&
	\includegraphics[trim=0mm 0mm 0mm 0mm, width=.3\linewidth]{8_15.eps}&
	\\
	$8_{14}$ & $8_{15}$ & \\
	\end{tabular}
	\end{center}
	\label{minimal}
\end{figure}

\bigskip

\noindent{\bf Acknowledgement.}
I would like to thank Ryo Nikkuni for informimg me a result of Tat Sang Fung about Theorem \ref{one}.

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv