\section{Introduction}

  Let $R$ be a Noetherian ring, $M$ an $R$-module, $\frak{p}\in\textmd{Spec}R$.
H. Bass defined so called Bass numbers $\mu_{i}(\frak{p},M)$ by
using the minimal injective resolution of $M$ for all integers
$i\geq 0$, and proved that $$\mu_{i}(\frak{p},M)=\textmd{dim}_{k(\frak{p})}\textmd{Ext}_{R_{\frak{p}}}^{i}(k(\frak{p}),M_{\frak{p}}).$$ If M is a finitely generated
$R$-module, the Betti numbers $\beta_{i}(\frak{p},M)$ is defined by
using the minimal free resolution of $M_{\frak{p}}$ for all $i\geq
0$, and we have $$\beta_{i}(\frak{p},M)=\textmd{dim}_{k(\frak{p})}\textmd{Tor}^{R_{\frak{p}}}_{i}(k(\frak{p}),M_{\frak{p}}).$$ In addition, E. Enochs and J. Z. Xu defined the Dual Bass numbers $\pi_{i}(\frak{p},M)$ by using the minimal flat resolution of $M$ for all $i\geq 0$ and showed that $$\pi_{i}(\frak{p},M)=\textmd{dim}_{k(\frak{p})}\textmd{Tor}^{R_{\frak{p}}}_{i}(k(\frak{p}),\textmd{Hom}_{R}(R_{\frak{p}},M))$$ for any Cotorsion $R$-module $M$ in \cite{EX}. J. Z. Xu characterized Gorenstein rings and strongly Cotorsion modules by vanishing properties of $\pi_{i}(\frak{p},M)$. The author have studied the vanishing properties of Dual Bass numbers in \cite{L}.

  Cohen Macaulay modules over Noetherian rings are important
objects in commutative algebra and algebraic geometry. In duality, Z. M. Tang and H. Zakeri introduced the concept of Co-Cohen Macaulay modules and studied the properties
of this in \cite{TZ} and \cite{T}. An elementary and important property of Cohen Macaulay modules is
that the localization preserves the Cohen Macaulayness. The dual question for Artinian modules is to ask wether the co-localization of Co-Cohen Macaulay modules preserves the Co-Cohen Macaulayness. We
show that this statement is true under a certain conditions (Proposition 3.5).

In addition, we give a characterization of Co-Cohen Macaulay modules by vanishing properties of Dual Bass numbers which have relation to the maximal length of co-regular sequence (Theorem 3.8). This is dual to the theory of Cohen Macaulay modules over Noetherian rings.


\newpage

\section{Preliminaries}

Now, we first recall some definitions and results which we need in this paper.

   The concept of Krull dimension (Kdim) for Artinian modules was introduced by R. N. Roberts in
\cite{Roberts}. Later, Kirby \cite{Kirby} changed the terminology of Roberts and referred to Noetherian dimension (Ndim) to avoid any confusion with well-known Krull dimension defined for finitely generated modules. Let $R$ be a ring, $M$ a $R$-module. The Noetherian dimension of $M$, denoted by $\textmd{Ndim}_R~M$, is defined inductively as follows: when $M=0$, put $\textmd{Ndim}_R~M=-1$. Then by induction, for an integer $d\geq 0$, we put $\textmd{Ndim}_R~M=d$, if $\textmd{Ndim}_R~M=d$ is false and for every ascending chain $M_0\subseteq M_1\subseteq\cdots$ of submodules of $M$, there exists a positive integer $n_0$ such that $\textmd{Ndim}_R(M_{n+1}/M_n)<d$ for all $n>n_0$. Therefore $\textmd{Ndim}_R~M=0$ if and only if $M$ is a non-zero Noetherian module.

   L. Melkersson and P. Schenzel introduced the co-localization of modules in \cite{MS}. Let $R$ be a
ring, $S\subseteq R$ a multiplicative set, and $M$ an $R$-module.
The $R_S$-module $\textmd{Hom}_{R}(R_S,M)$ is called the $\mathbf{co}$-$\mathbf{localization}$ of $M$ with respect to $S$, and defined the $\mathbf{co}$-$\mathbf{support}$ of $M$ by
$$\textmd{Cos}_{R}~M=\{~\frak{p}\in\textmd{Spec}~R~|~\textmd{Hom}_{R}(R_{\frak{p}},M)\neq 0~\}.$$
In \cite{L}, we define the $\mathbf{Co}$-$\mathbf{dimension}$ of $M$ as
$$\textmd{Co-dim}_{R}~M=\textmd{sup}\{~\textmd{dim}~R/\frak{p}~|~\frak{p}\in
\textmd{Cos}_{R}~M~\}.$$

If $R$ is a Noetherian ring and $M$ an Artinian $R$-module, then $\textmd{Att}_{R}~M$, $\textmd{Cos}_{R}~M$, $\textmd{Ann}_{R}~M$ have the same minimal elements by Corollary 4.3 in \cite{CN}.

Let $R$ be a ring (not necessarily Noetherian),
$S\subseteq R$ a multiplicative set, $M$ an Artinian $R$-module. Then we have
$$\textmd{Co-dim}_{S^{-1}R}\textmd{Hom}_{R}(S^{-1}R,M)
=\textmd{sup}\{\textmd{dim}S^{-1}R/S^{-1}\frak{p}|~\frak{p}\in\textmd{Cos}_{R}~M,\frak{p}\cap S=\emptyset~\}$$
\hspace{148pt}$=\textmd{sup}\{~\textmd{dim}~S^{-1}R/S^{-1}\frak{p}~|~\frak{p}\in\textmd{Att}_{R}~M,~\frak{p}\cap S=\emptyset~\}$\\
Let $\frak{p}\in\textmd{Cos}_{R}~M$, we denote
$$\textmd{ht}_{M}\frak{p} =\textmd{sup}\{~n~|~\frak{p}_{0}\subsetneq \frak{p}_{1}\subsetneq\cdots\subsetneq \frak{p}_{n},~\frak{p}_{i}\in\textmd{Cos}_{R}~M~\text{for}~i=0,1,\cdots,n~\}.$$
It is obvious that $\textmd{ht}_{M}\frak{p}=\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)$.

We except that $\textmd{Ndim}_R~M=\textmd{Co-dim}_{R}~M$ for any Artinian module $M$. Unfortunately, this equality may not be holds in general. In fact, there exists an Artinian module $M$ over a Noetherian local ring $(R,\frak{m})$ such that $\textmd{Ndim}_R~M<\textmd{Co-dim}_{R}~M$, See example 4.1 in \cite{CN3}.
However, we have the following proposition.

\begin{Proposition} Let $R$ be a Noetherian ring, $M$ an Artinian $R$-module such that
$\textmd{Ann}_{R}~(0:_{M}\frak{p})=\frak{p}$ for any $\frak{p}\in\textmd{V}(\textmd{Ann}_{R}~M)$.
Then $\textmd{Ndim}_R~M=\textmd{Co-dim}_{R}~M$.
\end{Proposition}
\begin{pf} We first note that $\textmd{Co-dim}_{R}~M=\textmd{dim}~R/(\textmd{Ann}_R~M)$. By Proposition in \cite{CN3}, we have the inequality $\textmd{Ndim}_R~M\leq\textmd{dim}~R/(\textmd{Ann}_R~M)$.

  On the other hand, for any ideal $\frak{a}\subsetneq R$, we have $(\frak{a}+\textmd{Ann}_R(M))\subseteq
\textmd{Ann}_R(0:_M\frak{a})$, and $\textmd{Ann}_R(0:_M\frak{a})\subseteq\textmd{Ann}_{R}~(0:_{M}\frak{p})=\frak{p}$ for any $(\frak{a}+\textmd{Ann}_R~M)\subseteq\frak{p}\in\textmd{Spec}~R$ by hypothesis. Thus
$$\sqrt{\frak{a}+\textmd{Ann}_R~M}=\sqrt{\textmd{Ann}_R(0:_M\frak{a})}.$$
Let $\textmd{Ndim}_R~M=d$. Then there exist elements $x_1,\cdots,x_d\in J(M)$ such that $$(0:_M(x_1,\cdots,x_d))\neq 0$$ have finite length by Theorem 4.1 in \cite{T}. Let $\frak{a}=(x_1,\cdots,x_d)R$. Then
$$0=\textmd{dim}_R~(0:_M(x_1,\cdots,x_d))=\textmd{dim}_R~R/((x_1,\cdots,x_d)+\textmd{Ann}_R~M)\geq \textmd{dim} R/(\textmd{Ann}_R~M)-d.$$
Thus $\textmd{dim}_R/(\textmd{Ann}_R~M)\leq d=\textmd{Ndim}_R~M$. Hence, $$\textmd{Co-dim}_{R}~M=\textmd{dim}~R/(\textmd{Ann}_R~M)=\textmd{Ndim}_R~M.$$
\end{pf}

N. T. Cuong and N. T. Dung showed that the above
condition is not holds for any Artinian modules, and they also given some sufficient conditions for that in \cite{CDN}.

(\cite{CDN}, Proposition 2.1) Let $(R,\frak{m})$ be a Noetherian local
ring, $M$ an Artinian $R$-module, if one of the following cases
happens:

$(i)$. $R$ is complete with respect to $m$-adic topology.

$(ii)$. $M$ contain a submodule which is isomorphic to the injective
hull of $R/\frak{m}$.\\
Then $\textmd{Ann}_{R}~(0:_{M}\frak{p})=\frak{p}$ for any $\frak{p}\in\textmd{V}(\textmd{Ann}_{R}~M)$.

\begin{Definition} A Noetherian ring $R$ is called a \textbf{U ring}, if for any Artinian $R$-module $M$,
such that $\textmd{Ann}_{R}~(0:_{M}\frak{p})=\frak{p}$ for any $\frak{p}\in\textmd{V}(\textmd{Ann}_{R}M)$.
\end{Definition}

Hence, any complete Noetherian local rings are $U$ rings.\\

Let $R$ be a ring, $M$ an Artinian ring. R. Y. Shorp in \cite{S3} showed that $\textmd{Supp}_{R}~M$ is a finite subset of $\textmd{Max}~R$, and if $\textmd{Supp}_{R}~M=\{~\frak{m}_{i}\in\textmd{Max}~R~|~i=1,2,\cdots,s\}$, then $M=M_{1}\oplus\cdots\oplus M_{s}$, where
$$M_{i}=\Gamma_{\frak{m}_{i}}M=\{~x\in M~|~\text{there exists some integer}~n\geq 0,~\text{such that}~\frak{m}_{i}^{n}x=0\}.$$

Now, We collect some results of Artinian modules and the vanishing properties of Dual Bass numbers, which we need in this paper. The proofs of Lemma 2.3 can be found in \cite{T}, and the proofs of Lemma 2.4 and Lemma 2.5 can be founded in \cite{L}.

\begin{Lemma} Let $R$ be a ring, $M$ an Artinian $R$-module. Denote $\textmd{J}(M)=\bigcap\limits_{\frak{m}\in Supp_RM}\frak{m}$. Then

$(1)$. Let $M=N_1+N_2+\cdots+N_n$ be a minimal secondary presentation of $M$. Then
$\textmd{Cograde}_{R}(\textmd{J}(M),M)\leq\textmd{Ndim}_{R}~N_i\leq\textmd{Ndim}_{R}~M$,
for $i=1,\cdots,n$.

$(2)$. $\textmd{Cograde}_{R}(J(M),M)=\textmd{min}\{\textmd{Cograde}_{R}(\frak{m_i},M_i)~|~i=1,\cdots,s~\}$.

$(3)$. Let $I\subseteq J(M)$ be an ideal of $R$. Then $(0:_MI)\neq 0$.

$(4)$. Let $x_1,x_2,\cdots, x_r\in J(M)$ be a $M$-co-regular sequence. Then $M$ is a Co-Cohen Macaulay $R$-module if and only if $(0:_M(x_1,x_2,\cdots, x_r))$ is a Co-Cohen Macaulay $R$-module.

$(5)$. $\textmd{Ndim}_{R}~M$ equal to the least number of generators of an ideal $I\subseteq\textmd{J}(M)$ such that $(0:_MI)$ has finite length.
\end{Lemma}

\begin{Lemma} Let $R$ be a Noetherian ring, $M$ an Artinian $R$-module. Then

$(1)$. The following conditions are equivalent:

\hspace*{20pt}$(a)$. $\textmd{Ann}_{R}~(0:_{M}\frak{p})=\frak{p}$ for any $\frak{p}\in \textmd{V}(\textmd{Ann}_{R}~M)$.

\hspace*{20pt}$(b)$. $\textmd{Cos}_{R}~(0:_{M}I)=\textmd{Cos}_{R}~M\cap\textmd{V}(I)$
for any ideal $I$ of $R$.

In this case, $\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)<\infty$ for any $\frak{p}\in\textmd{Cos}_{R}~M$.

$(2)$. Let $\frak{p}\in Cos_R~M$. If $\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)<\infty$, then
$$\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)\leq\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M).$$

If $\pi_{i}(\frak{p},M)>0$, then
$$\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)\leq
i\leq\textmd{fd}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M),$$
and $$\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)=\textmd{inf}~\{~i~|~\pi_{i}(\frak{p},M)>0\}.$$
where $\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)=\textmd{inf}~\{~i~|~\textmd{Tor}_{i}^{R_{\frak{p}}}(k(\frak{p}),\textmd{Hom}_{R}(R_{\frak{p}},M))\neq 0~\}$ is equal to the length of any maximal $\textmd{Hom}_{R}(R_{\frak{p}},M)$-quasi co-regular sequence contained in the maximal ideal $\frak{p}R_{\frak{p}}$ of $R_{\frak{p}}$, and $\textmd{fd}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)$ is the flat dimension of $R_{\frak{p}}$-module $\textmd{Hom}_{R}(R_{\frak{p}},M)$.

$(3)$. Let $I\subsetneq R$ a proper ideal of $R$. Then
$$\textmd{Cograde}_{R}(I,M)=\textmd{inf}~\{\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}
(R_{\frak{p}},M)~|~{\frak{p}}\in\textmd{Cos}_{R}~M\cap\textmd{V}(I)\}$$
\hspace*{82pt}$=\textmd{inf~}\{\textmd{Cograde}_{R_{\frak{p}}}
(IR_{\frak{p}},\textmd{Hom}_{R}(R_{\frak{p}},M))~|~\frak{p}\in\textmd{Cos}_{R}~M\cap\textmd{V}(I)\},$\\
where $\textmd{Cograde}_{R}(I,M)$ is equal to the common length of any maximal $M$-co-regular sequence contained in $I$.

$(4)$. $\textmd{fd}_{R}~M=\textmd{sup}\{~\textmd{fd}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)~|~\frak{p}\in\textmd{Spec}~R~\}$\\
\hspace*{65pt}$=\textmd{sup}\{~\textmd{fd}_{R_{\frak{m}}}\textmd{Hom}_{R}(R_{\frak{m}},M)~|~\frak{m}\in\textmd{Max}~R~\}$.
\end{Lemma}

\begin{Lemma} Let $R$ be a $U$ ring, $M$ an Artinian $R$-module. Let $\frak{p},\frak{q}\in\textmd{Spec}~R$ such that $\frak{q}\subsetneq\frak{p},~\textmd{ht}(\frak{p}/\frak{q})=s$. Then $\pi_{i}(\frak{q},M)\neq 0\Longrightarrow\pi_{i+s}(\frak{p},M)\neq 0$.
\end{Lemma}

\vspace{0.5cm}




\section{Co-localization of Co-Cohen Macaulay Modules}

Cohen-Macaulay modules over Noetherian rings are important objects in commutative algebra. In duality, Z. Tang and H. Zakeri introduced the Co-Cohen Macaulay modules over local rings in \cite{TZ}. Z. Tang in \cite{T} establish the theory of Co-Cohen Macaulay modules over general rings.

\begin{Definition}$($\cite{T}, Definition 5.3$)$ Let $R$ be a ring. An Artinian
$R$-module $M$ is called a \textbf{Co-Cohen-Macaulay $R$-module}, if $\textmd{Cograde}_{R}(\textmd{J}(M),M)=\textmd{Ndim}_{R}~M$, where $\textmd{Cograde}_{R}(\textmd{J}(M),M)$ is equal to the common length of any maximal $M$-co-regular sequence contained in $I$.
\end{Definition}

\begin{Proposition}Let $R$ be a ring, $M$ an Artinian $R$-module. If $M$ has a decomposition $M=M_{1}\oplus\cdots\oplus M_{s}$ and $\textmd{Supp}_{R}~M=\{~\frak{m}_{i}\in\textmd{Max}~R~|~i=1,2,\cdots,s\}$, where $M_{i}=\Gamma_{\frak{m}_{i}}M=\{~x\in M~|~\text{there exists some integer}~n\geq 0,~\text{such that}~\frak{m}_{i}^{n}x=0\}$. Then $M$ is a Co-Cohen-Macaulay $R$-module if and
only if $M_{i}$ is a Co-Cohen-Macaulay $R$-module and
$\textmd{Ndim}_{R}~M=\textmd{Ndim}_{R}~M_{i}$ for $i=1,2,\cdots,s$.
\end{Proposition}

\begin{pf} Notice that $\textmd{Ndim}_{R}~M=\textmd{max}\{~\textmd{Ndim}_{R}~M_{i}~|~i=1,2,\cdots,s\}$ by Proposition 1 in \cite{Roberts}.
Since $M$ is a Co-Cohen Macaulay $R$-module, and $\textmd{J}(M_{i})=\frak{m}_{i}$ for any $1\leq i\leq s$, by Lemma 2.3(1) and Lemma2.2(2), we get
$$\textmd{Ndim}_{R}~M=\textmd{Ndim}_{R}~M_{i}=\textmd{Cograde}_{R}(\frak{m}_{i},M_{i})=\textmd{Cograde}_{R}(\textmd{J}(M),M)$$
for $i=1,2,\cdots,s$. Hence, $M_{i}$ is a Co-Cohen Macaulay $R$-module,
and $\textmd{Ndim}_{R}~M=\textmd{Ndim}_{R}~M_{i},~i=1,2,\cdots,s$. It is similar to prove the other side.
\end{pf}

\begin{Proposition}Let $R$ be a ring, $M$ a Co-Cohen Macaulay $R$-module, and $x\in
R$ an $M$-co-regular element. Then $(0:_{M}x)$ is a Co-Cohen Macaulay $R$-module, and
$\textmd{Ndim}_{R}(0:_{M}x)=\textmd{Ndim}_{R}~M-1$.
\end{Proposition}

\begin{pf} It is obvious that $(0:_{M}x)=(0:_{M_{1}}x)\oplus\cdots\oplus
(0:_{M_{s}}x)\neq 0$. We claim that
$$x\in \frak{m}_{i}~\text{if and only if}~(0:_{M_{i}}x)\neq 0~\text{for}~i=1,2,\cdots,s.$$
If $x\in \frak{m}_{i}$, then by Lemma 2.3(3) we get $(0:_{M_{i}}x)\neq 0$. On the other hand, let $x\in R-\frak{m}_{i}$. Assume that $(0:_{M_{i}}x)\neq 0$. Let $u$ be a non-zero element in $(0:_{M_{i}}x)$, then there exists $n\geq 0$ such that $\frak{m}_{i}^{n}u=0$. Thus $(\frak{m}_{i}^{n},x)u=0$. Since $\frak{m}_{i}$ is a maximal ideal, we get $(\frak{m}_{i}^{n},x)=R$. It follows that $u=0$. Hence $(0:_{M_{i}}x)=0$.

After renumbering, we can assume that there exists an integer $r$, $1\leq r\leq s$, such that
$x\in\frak{m}_i$ for $i=1,2,\cdots,r$, and $x\not\in\frak{m}_i$ for $i=r+1,r+2,\cdots,s$.
Then $(0:_{M}x)=(0:_{M_{1}}x)\oplus\cdots\oplus(0:_{M_{r}}x)$. As $x$ is an $M$ co-regular element, so $x$ is also an $M_{i}$ co-regular element, $i=1,2,\cdots,r$. By Lemma 2.3(4), we have
$(0:_{M_{i}}x)$ is a Co-Cohen Macaulay Artinian $R$-module for $i=1,2,\cdots,r$. Thus
$$\textmd{Ndim}_{R}(0:_{M_{i}}x)=\textmd{Cograde}_{R}(\frak{m}_{i},(0:_{M_i}x))=\textmd{Cograde}_{R}(\frak{m}_i,M_i)-1=\textmd{Ndim}_{R}~M_{i}-1,$$
for $i=1,2,\cdots,r$. Hence, by Proposition 3.2, $(0:_{M}x)$ is a
Co-Cohen Macaulay $R$-module, and
$\textmd{Ndim}_{R}(0:_{M}x)=\textmd{Ndim}_{R}~M-1$.
\end{pf}\\

It is well-known that operation of localization preserves the Cohen Macaulayness
of Cohen Macaulay modules over Noetherian rings. Since co-localization
is dual to localization, hence there is a natural question as following: Whether co-localization preserves the Co-Cohen Macaulayness of Co-Cohen Macaulay modules?

The main content of this section is to study the properties of co-localization of Co-Cohen Macaulay modules, and give a affrmative answer to above question to a certain extent.

\begin{Proposition} Let $R$ be a Noetherian ring, $M$ an Artinian $R$-module such that
$\textmd{Ann}_{R}~(0:_{M}\frak{p})=\frak{p}$ for any $\frak{p}\in\textmd{V}(\textmd{Ann}_{R}~M)$.
If $\frak{p}\in\textmd{Cos}_{R}~M$,and $x/s\in \frak{p}R_{\frak{p}}$ is a $\textmd{Hom}_{R}(R_{\frak{p}},M)$ co-regular element. Then
$$\textmd{Co-dim}_{R_{\frak{p}}}(0:_{\tiny{\textmd{Hom}_{R}(R_{\frak{p}},M)}}x/s)
=\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)-1.$$
\end{Proposition}

\begin{pf} Since $(0:_{\tiny{\textmd{Hom}_{R}(R_{\frak{p}},M)}}x/s)\cong\textmd{Hom}_{R}(R_{\frak{p}},(0:_{M}x))$, we have
$$\textmd{Co-dim}_{R_{\frak{p}}}(0:_{\tiny{\textmd{Hom}_{R}(R_{\frak{p}},M)}}x/s)=\textmd{ht}(\frak{p}/\textmd{Ann}_{R}~(0:_{M}x)).$$
Moreover, by Lemma 2.4(1) we have
$$\textmd{V}(\textmd{Ann}_{R}~(0:_{M}x))=\textmd{V}(\textmd{Ann}_{R}~M)\cap\textmd{V}((x))=\textmd{V}(\textmd{Ann}_{R}~M,x)$$
Hence
$\sqrt{\textmd{Ann}_{R}~(0:_{M}x)}=\sqrt{(\textmd{Ann}_{R}~M,x)}$. On
the other hand, since
$$\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)=\textmd{ht}(\frak{p}/\textmd{Ann}_{R}~M)).$$
we only need to show that
$\textmd{ht}(\frak{p}/(\textmd{Ann}_{R}~M,x))=\textmd{ht}(\frak{p}/\textmd{Ann}_{R}~M)-1$.

Since $x/s\in \frak{p}R_{p}$ is a $\textmd{Hom}_{R}(R_{\frak{p}},M)$ co-regular element, we get $$x/s\in \frak{p}R_{\frak{p}}-\bigcup\limits_{\tiny{\frak{q}\subseteq \frak{p}\atop \frak{q}\in\textmd{Att}_{R}~M}}\frak{q}R_{\frak{p}}.$$ It follows that
$x\in \frak{p}-\bigcup\limits_{\tiny{\frak{q}\subseteq \frak{p}\atop \frak{q}\in\textmd{Att}_{R}~M}}\frak{q}$. Hence
$\textmd{ht}(\frak{p}/(\textmd{Ann}_{R}~M,x))\leq\textmd{ht}(\frak{p}/\textmd{Ann}_{R}~M)-1$. On the other hand, by theory of system of parameters we get
$$\textmd{ht}(\frak{p}/(\textmd{Ann}_{R}~M,x))\geq\textmd{ht}(\frak{p}/\textmd{Ann}_{R}~M)-1.$$
\end{pf}

The following Proposition show that co-localization preserves Co-Cohen Macaulayness under a certain conditions.

\begin{Proposition}Let $R$ be a Noetherian ring, $M$ a Co-Cohen Macaulay $R$-module, such that $\textmd{Ann}_{R}~(0:_{M}\frak{p})=\frak{p}$
for any $\frak{p}\in\textmd{V}(\textmd{Ann}_{R}~M)$. Then for any
$\frak{p}\in\textmd{Cos}_{R}~M$, $$\textmd{Cograde}_{R}(\frak{p},M)=\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M) =\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M).$$
\end{Proposition}

\begin{pf} Let $\frak{p}\in\textmd{Cos}_{R}~M$, then we have $\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)<\infty$, and
$$\textmd{Cograde}_{R}(\frak{p},M)\leq\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M) \leq\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M).$$
We only need to show that $\textmd{Cograde}_{R}(\frak{p},M)=\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)$.

Let $n=\textmd{Cograde}_{R}(\frak{p},M)$, we use induction
on $n$. For the case $n=0$, there exists $\frak{Q}\in\textmd{Att}_{R}~M$ such that $\frak{p}\subseteq {\frak Q}$. Since $\frak{p}\in\textmd{Cos}_{R}~M$, then for any $\frak{q}\in\textmd{Att}_{R}~M$ with $\frak{q}\subseteq \frak{p}$, we have $\frak{q}=\frak{p}={\frak Q}$ by Lemma 2.3(1). Thus
$\textmd{Att}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)=\{\frak{p}R_{\frak{p}}\}$. Hence, $\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)=0$.

Suppose that $n>0$, and the Proposition holds for $n-1$. Let $x\in\frak{p}$ be a $M$-co-regular element, then $\textmd{Cograde}_{R}(\frak{p},(0:_{M}x))=n-1$. Hence, by induction hypothesis, we get $$\textmd{Cograde}_{R}(\frak{p},(0:_{M}x))=\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},(0:_{M}x)).$$ On the other hand, by Proposition 3.4 in \cite{L}, $x/1\in \frak{p}R_{\frak{p}}$ is a $\textmd{Hom}_{R}(R_{\frak{p}},M)$-quasi co-regular element. Since $(0:_{\tiny{\textmd{Hom}_{R}(R_{\frak{p}},M)}}\frak{p}R_{\frak{p}})\neq 0$, $x/1$ is a $\textmd{Hom}_{R}(R_{\frak{p}},M)$ co-regular element. Then we have $\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},(0:_{M}x))
=\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)-1$
by Proposition 3.4. Hence $\textmd{Cograde}_{R}(\frak{p},M)=\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)$.
\end{pf}\\

To obtain the main result of this section, we first prove the following Lemma.

\begin{Lemma}Let $R$ be a Noetherian ring, $M$ an Artinian $R$-module. Then
$\textmd{Supp}_{R}~M\subseteq\textmd{Cos}_{R}~M$. In particular, $\textmd{Cos}_{R}~M\cap\textmd{V}(\textmd{J}(M))=\textmd{Supp}_{R}~M$.
\end{Lemma}

\begin{pf}
Let $M=N_1+N_2+\cdots+N_n$ be a minimal secondary presentation of $M$ such that $\textmd{Att}_{R}~M=\{\frak{p}_i~|~i=1,2,\cdots,n.\}$, where $\frak{p}_i=\sqrt{\textmd{Ann}_{R}~N_i}$, for $i=1,2,\cdots,n$. On the other hand, we have decomposition $M=M_{1}\oplus\cdots\oplus M_{s}$, $\textmd{Supp}_{R}~M=\{~\frak{m}_{i}\in\textmd{Max}~R~|~i=1,2,\cdots,s\}$, where $$M_{i}=\Gamma_{\frak{m}_{i}}M=\{~x\in M~|~\text{there exists some integer}~n\geq 0,~\frak{m}_{i}^{n}x=0\}.$$
Assume that there exists $\frak{m}_j\in\textmd{Supp}_{R}~M$ such that $\frak{m}_j\not\in\textmd{Cos}_{R}~M$, then $\frak{p}_i\nsubseteq\frak{m}_j$ for any $1\leq i\leq n$. It follows that $\bigcap_{i=1}^n\frak{p}_i\nsubseteq\frak{m}_j$. Let $x\in\bigcap_{i=1}^n\frak{p}_i-\frak{m}_j$, then $x^l\cdot M=0$ for some integer $l$. This contradict to $(0:_{M_j}x)=0$ by proof of Proposition 3.3. Hence, $\textmd{Supp}_{R}~M\subseteq\textmd{Cos}_{R}~M$.
\end{pf}

\begin{Proposition} Let $R$ be a Noetherian ring, $M$ an Artinian $R$-module such that $\textmd{Ann}_{R}~(0:_{M}\frak{p})=\frak{p}$ for any $\frak{p}\in\textmd{V}(\textmd{Ann}_{R}~M)$.
Then $$\textmd{max}\{\textmd{Cograde}_{R}(\frak{p},M)~|~\frak{p}\in\textmd{Cos}_{R}~M\}
=\textmd{max}\{\textmd{Cograde}_{R}(\frak{m},M)~|~\frak{m}\in\textmd{Supp}_{R}~M\}$$
\end{Proposition}

\begin{pf}
Since $\textmd{Ann}_{R}~(0:_{M}\frak{p})=\frak{p}$ for any $\frak{p}\in\textmd{V}(\textmd{Ann}_{R}~M)$, we have $$\textmd{Cograde}_{R}(\frak{p},M)<\infty~\text{for any}~\frak{p}\in\textmd{Cos}_{R}~M$$.
By Lemma 3.6, we only need to show that for any $\frak{p}\in\textmd{Cos}_{R}~M$, there exists $\frak{m}\in\textmd{Supp}_{R}~M$ such that $\textmd{Cograde}_{R}(\frak{p},M)\leq\textmd{Cograde}_{R}(\frak{m},M)$.

If $x$ is a $M$ co-regular element, then we have $x\in\bigcup\limits_{\tiny{\frak{m}_{i}\in\textmd{Supp}_{R}~M}}\frak{m}_{i}$ by proof of Proposition 3.3. Let $\frak{p}\in\textmd{Cos}_{R}~M$, and $x_{1},x_{2},\cdots,x_{t}$ be a maximal $M$ co-regular sequence contained in $\frak{p}$, then there exists $\frak{m}\in\textmd{Supp}_{R}~M$ such that $x_{i}\in \frak{m}$ for
$i=1,2,\cdots,t$. Otherwise, for any $1\leq k\leq s$, there exists
$x_{i_{k}}\not\in\frak{m}_{k}$ for some $1\leq i_{k}\leq t$. Then
$$(0:_{M}(x_{1},x_{2},\cdots,x_{t}))=$$ $$(0:_{M_{1}}(x_{1},x_{2},\cdots,x_{t}))
\oplus(0:_{M_{2}}(x_{1},x_{2},\cdots,x_{t}))\oplus\cdots\oplus(0:_{M_{s}}(x_{1},x_{2},\cdots,x_{t}))=0.$$
This is a contradiction. Hence there exists $\frak{m}\in\textmd{Supp}_{R}~M$ such that
$$\textmd{Cograde}_{R}(\frak{p},M)\leq\textmd{Cograde}_{R}(\frak{m},M).$$
\end{pf}

The following Theorem is the main result of this section. We give a characterization of Co-Cohen Macaulay modules by vanishing
properties of Dual Bass numbers.

\begin{Theorem}Let $R$ be a Noetherian ring, $M$ an Artinian
$R$-module such that $\textmd{Ann}_{R}~(0:_{M}\frak{p})=\frak{p}$
for any $\frak{p}\in\textmd{V}(\textmd{Ann}_{R}~M)$. Let $\textmd{Max}~\textmd{Cos}_{R}~M$ be the maximal elements of $\textmd{Cos}_{R}~M$. Then the
following conditions are equivalent:

$(1)$. $M$ is a Co-Cohen Macaulay $R$-module.

$(2)$. $\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)=\textmd{ht}_{\tiny{M}}\frak{p}$ for any $\frak{p}\in\textmd{Cos}_{R}~M$, and $\textmd{ht}_{M}\frak{m}=\textmd{Co-dim}_{R}~M$ for any $\frak{m}\in\textmd{Supp}_{R}~M$.

$(3)$. $\pi_{i}(\frak{p},M)=0$ for any $0\leq i<\textmd{ht}_{M}\frak{p}$ and for any ${\frak{p}}\in\textmd{Cos}_{R}~M$, and
$\textmd{ht}_{M}\frak{m}=\textmd{Co-dim}_{R}~M$ for any $\frak{m}\in\textmd{Supp}_{R}~M$.

$(4)$. $\textmd{Cograde}_{R}(\frak{m},M)=\textmd{ht}_{M}\frak{m}=\textmd{Co-dim}_{R}~M$ for any $\frak{m}\in\textmd{Supp}_{R}~M$.

$(5)$. $\pi_{i}(\frak{m},M)=0$ for any $0\leq i<\textmd{ht}_{M}\frak{m}$, and $\textmd{ht}_{M}\frak{m}=\textmd{Co-dim}_{R}~M$ for any
$\frak{m}\in\textmd{Supp}_{R}~M$.
\end{Theorem}

\begin{pf} Notice that for any $\frak{p}\in\textmd{Cos}_{R}~M$, $\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)<\infty$ by Lemma 2.4(1). Moreover,
$\textmd{Co-dim}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)=\textmd{ht}_{M}\frak{p}$, and $$\textmd{Cograde}_{R}(\frak{m},M)=\textmd{Cograde}_{R_{\frak{m}}}\textmd{Hom}_{R}(R_{\frak{m}},M)$$
for any $\frak{m}\in\textmd{Supp}_{R}~M$. Hence, by Lemma 2.4(2), we have
(2)$\Longleftrightarrow$(3) and (4)$\Longleftrightarrow$(5). On the other hand, (2)$\Longrightarrow$(4)$\Longrightarrow$(1) is obvious.

(1)$\Longrightarrow$(2). From Proposition 3.5 we get
$\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)
=\textmd{ht}_{\tiny{M}}\frak{p}$ for any $\frak{p}\in\textmd{Cos}_{R}~M$. Since
$\textmd{Cograde}_{R}(\frak{m},M)=\textmd{Cograde}_{R_{\frak{m}}}\textmd{Hom}_{R}(R_{\frak{m}},M)\leq\textmd{ht}_{M}\frak{m}$
for any $\frak{m}\in\textmd{Supp}_{R}~M$, we have
\begin{eqnarray*}
\textmd{Cograde}_{R}(\textmd{J}(M),M)&=&\textmd{min}\{\textmd{Cograde}_{R}(\frak{m},M)~|~\frak{m}\in\textmd{Supp}_{R}~M\}\\
&\leq&\textmd{max}\{\textmd{ht}_{M}\frak{m}~|~\frak{m}\in\textmd{Supp}_{R}~M\}\\
&\leq&\textmd{max}\{\textmd{ht}_{M}\frak{m}~|~\frak{m}\in\textmd{Cos}_{R}~M\}\\
&=&\textmd{Co-dim}_{R}~M,
\end{eqnarray*}
where the third inequality follows from Lemma 3.6. Hence, by definition of Co-Cohen Macaulay $R$-module,
we get $\textmd{ht}_{M}\frak{m}=\textmd{Co-dim}_{R}~M$ for any $\frak{m}\in\textmd{Supp}_{R}~M$.
\end{pf}

\begin{Proposition}Let $R$ be a $U$ ring, $M$ a Co-Cohen Macaulay $R$-module.
If $\frak{p},\frak{q}\in\textmd{Cos}_{R}~M$, with $\frak{q}\subseteq\frak{p}$. Then all saturated chains of prime ideals starting from $\frak{q}$ and ending at $\frak{p}$ have the same finite length equal to $\textmd{ht}(\frak{p}/\frak{q})$.
\end{Proposition}

\begin{pf} Suppose that $\textmd{ht}(\frak{p}/\frak{q})=s,~\textmd{ht}_{M}\frak{q}=n$. Let $\frak{q}=\frak{p}_{0}\subsetneq\frak{p}_{1}\subsetneq\cdots\subsetneq\frak{p}_{r}=\frak{q}$ be a saturated chain of prime ideals. By Theorem 3.8 we have
$\pi_{n}(\frak{q},M)>0$, and $\pi_{n+r}(\frak{p},M)>0$ follows
from Lemma 2.5. Moreover, since$$n+r=\textmd{ht}_{M}\frak{q}+r\leq\textmd{ht}_{M}\frak{p},$$ then by Theorem 3.7 we have
$n+r=\textmd{ht}_{M}\frak{p}$. On the other hand, $r\leq s$, and
$n+s\leq\textmd{ht}_{M}\frak{p}$. Hence $r=s$.
\end{pf}
\\

Proposition 3.8 show that the Co-Support of Co-Cohen Macaulay
modules over $U$ rings have catenary property.


\section{Acknowledgments}

I am grateful to Professor Z. Tang for helpful discussion and suggestion, and thank Professor N. T. Cuong for pointing out Proposition 2.1 to me. I would also like to thank Professor X. Sun and Dr. Y. Chen for help and encouragement.


