\section{Introduction}
\begin{remark}
This paper consists of part of the author's paper \cite{PuCI}. This was done due to advice of  some of my colleagues. The other parts of
  \cite{PuCI} will be published later in a separate paper.
\end{remark}
Let $(A,\mathfrak{m} )$ be a Noetherian   local ring of dimension $d$ and let $\mathfrak{a} $ be  an $\mathfrak{m} $-primary ideal in $A$. We consider multiplicative $\mathfrak{a} $-stable filtration's i.e.,
$\EuScript{F} = \{ \mathfrak{a} _n \}_{n\geq 0}$ such that
\begin{enumerate}[\rm (i)]
  \item $\mathfrak{a} _0 = A$.
  \item $\mathfrak{a}  \subseteq \mathfrak{a} _1$ and $\mathfrak{a} _1 \neq A$.
  \item $\mathfrak{a} _n \mathfrak{a} _m \subseteq \mathfrak{a} _{n + m}$ for all $n, m \geq 0$.
  \item $\mathfrak{a}  \mathfrak{a} _n = \mathfrak{a} _{n + 1}$ for all $n \gg 0$.
\end{enumerate}
Let $\EuScript{S}_{\mathfrak{a}}(A)  = \bigoplus_{n\geq 0} \mathfrak{a} ^n$ be the Rees ring of $A$ with respect to \ $\mathfrak{a} $. Let $\EuScript{S}_{\EuScript{F}}(A) =  \bigoplus_{n\geq 0} \mathfrak{a} _n$ be the Rees ring of $A$ with respect to \ $\EuScript{F}$. We have an inclusion of rings $\EuScript{S}_{\mathfrak{a}}(A)  \hookrightarrow \EuScript{S}_{\EuScript{F}}(A)$ and $\EuScript{S}_{\EuScript{F}}(A)$ is a finite $\EuScript{S}_{\mathfrak{a}}(A) $-module. We  give some examples of multiplicative $\EuScript{S}_{\mathfrak{a}}(A) $-stable filtration's:

$(0)$: Of course the first example of $\EuScript{S}_{\mathfrak{a}}(A) $-stable filtration is the $\mathfrak{a} $-adic filtration $\{ \mathfrak{a} ^n \}_{n \geq 0}$.

$(1)$ Assume $\operatorname{depth} A > 0$. Consider
$$\widetilde{\mathfrak{a} ^n} = \bigcup_{r \geq 1} (\mathfrak{a} ^{n+r} \colon \mathfrak{a} ^r).$$
Then $\{ \widetilde{\mathfrak{a} ^n} \}_{n \geq 0}$ is the well-known Ratliff-Rush filtration of $A$ with respect to \ $\mathfrak{a} $. It is well-known that $\widetilde{\mathfrak{a} ^n} = \mathfrak{a} ^n$ for all $n\gg 0$. Furthermore it is easily verified that $\{ \widetilde{\mathfrak{a} ^n} \}_{n \geq 0}$ is an $\mathfrak{a} $-stable filtration, see \cite{RR}.

$(2)$ Assume $A$ is analytically unramified, i.e., the completion of $A$ is reduced. In this case there are two $\mathfrak{a} $-stable filtration which is of interest
\begin{enumerate}[\rm (i)]
\item
If $I$ is an ideal in $A$ then by $I^*$ we denote the integral closure of $I$ in $A$.
The integral closure filtration of  $\mathfrak{a} $ is $\{ (\mathfrak{a} ^n)^* \}_{n \geq 0}$.  As $A$ is analytically unramified the integral closure filtration of $\mathfrak{a} $ is $\mathfrak{a} $-stable,  see \cite{Rees} (also see \cite[9.1.2]{Hun-Sw}).  The integral closure filtration of $\mathfrak{m} $-primary ideals  has inspired plenty of research, see for instance \cite[Appendix 5]{ZS}, \cite{Rees-2}, \cite{Hun}, \cite{It}, \cite{ItN}, \cite{Masuti-1}, \cite{Masuti-2}
\cite{G}, \cite{GHM}, \cite{CPR} and \cite{Ver}.
\item
If $A$ contains a field of characteristic $p >0$ then the tight closure filtration of $\mathfrak{a} $ has been  of some  interest recently,  see \cite{GVM}.
\end{enumerate}

Let $G_{\mathfrak{a}}(A)  = \bigoplus_{n \geq 0} \mathfrak{a} ^n/\mathfrak{a} ^{n+1}$ be the associated graded ring of $A$ with respect to \ $\mathfrak{a} $. Let $\EuScript{F} = \{ \mathfrak{a} _n \}_{n \geq 0}$ be an $\mathfrak{a} $-stable filtration. Consider
$G_{\EuScript{F}}(A)  = \bigoplus_{n \geq 0} \mathfrak{a} _n/\mathfrak{a} _{n+1}$ be the associated graded ring of $A$ with respect to \ $\EuScript{F}$. Let $\ell(M)$ denote the length of an $A$-module $M$.  Clearly $G_{\EuScript{F}}(A)  $ is a finite $G_{\mathfrak{a}}(A) $-module. It follows the function
$H_{\EuScript{F}}(n) = \ell(A/\mathfrak{a} _{n +1})$  is of polynomial type, i.e., there exists a polynomial $P_\EuScript{F}(X) \in \mathbb{Q}[X]$ of degree $d$ such that $P_{\EuScript{F}}(n) = H_{\EuScript{F}}(n)$ for all $n \gg 0$. We write
$$P_\EuScript{F}(X) = e_0^\EuScript{F}(A)\binom{X+d}{d} -  e_1^\EuScript{F}(A)\binom{X+d -1}{d -1} + \cdots + (-1)^d e_d^\EuScript{F}(A). $$
If $\EuScript{F}$ is the $\mathfrak{a} $-adic then we set $e_i^\mathfrak{a} (-)= e_\EuScript{F}^i(-)$. Also if $\EuScript{F}$ is the $\mathfrak{a} $-integral closure filtration then we set ${e_i^\mathfrak{a} }^*(-)= e_\EuScript{F}^i(-)$.
Clearly $e_0^\EuScript{F}(A) = e_0^\mathfrak{a} (A) > 0$, for instance see \cite[11.4]{AM}.

\emph{For the rest of the paper we will assume that $A$ is a Cohen-Macaulay \ local ring with residue field $k$}\\
Northcott showed that $e_1^\mathfrak{a} (A) \geq 0$. Furthermore if $k$ is infinite then $e_1^\mathfrak{a} (A) = 0$ if and only if $\mathfrak{a} $ is a parameter ideal, see \cite{North}.
It is well-known  that for any $\mathfrak{a} $-stable  multiplicative filtration we have $e_1^\EuScript{F}(A) \geq 0$.
Narita showed that $e_2^\mathfrak{a} (A) \geq 0$.  Furthermore if $k$ is infinite and $d = 2$ then $e_2^\mathfrak{a} (A) = 0$ if and only if reduction number of $\mathfrak{a} ^n$ is one for all $n \gg 0$, see \cite{Nar}. Narita also gave an example of a ring with $e_3^\mathfrak{a} (A) < 0$.
We now consider the integral closure filtration of $\mathfrak{a} $. In this case ${e_2^\mathfrak{a} }^*(A) \geq 0$. Itoh proved that ${e_3^\mathfrak{a} }^*(A) \geq 0$, see \cite[Theorem 3]{ItN}. He conjectured that if $A$ is Gorenstein and
${e_3^\mathfrak{a} }^*(A) = 0$ then $G_{\mathfrak{a}}(A) ^* = \bigoplus_{n \geq 0} (\mathfrak{a} ^n)^*/(\mathfrak{a} ^{n+1})^*$ is Cohen-Macaulay. He proved the result when
${\mathfrak{a} }^* = \mathfrak{m} $, see \cite[Theorem 3]{ItN} (also see \cite[3.3]{CPR}). See \cite{KM} for a geometric view-point of Itoh's conjecture.

Recall an ideal $I$ is normal if $I^n$ is integrally closed for all $n \geq 1$. In this paper we solve Itoh's conjecture for normal $\mathfrak{m} $-primary ideals. Note if $A$ contains an
$\mathfrak{m} $-primary normal ideal then it is automatically analytically unramified, (cf., \cite[p.\ 153]{HM} and \cite[9.1.1]{Hun-Sw}).
In this paper we prove a significant case of Itoh's conjecture.
\begin{theorem}\label{main-new}
Let $(A,\mathfrak{m} )$ be a Cohen-Macaulay \ local ring and let $\mathfrak{a} $ be an $\mathfrak{m} $-primary normal ideal with $e_3^\mathfrak{a} (A) = 0$. Then $G_{\mathfrak{a}}(A) $ is Cohen-Macaulay.
\end{theorem}
\emph{Note}: we are not assuming $A$ is Gorenstein in Theorem \ref{main-new}.

The proof of Theorem \ref{main-new} requires three techniques, two of which is introduced in this paper  and one which was introduced in an earlier paper of the author. We call the first step \textit{complete intersection approximation}. This step enables us to reduce to the
case when $A$, $G_{\mathfrak{a}}(A) $ are complete intersections and $M$ is a maximal Cohen-Macaulay \ (MCM) $A$-module.\textit{ This reduction  surprised a few and attracted some
skepticism}. Nevertheless it is true.
The second step consists in studying the filtration
 $\{ \operatorname{Hom}_\mathfrak{a} (M,\mathfrak{a} ^n) \}_{n \in \mathbb{Z} }$ of the dual $\operatorname{Hom}_A(M,A)$.  This filtration is
classical cf.  \cite[p.\ 12]{Ser}.  However it has not been used before in the study of blow-up algebra's (modules) of Cohen-Macaulay \ rings (modules).
The third step consists of investigating local cohomology of an infinitely generated $\EuScript{S}_{\mathfrak{a}}(A) $-module $L^\mathfrak{a} (A) = \bigoplus_{n\geq 0}A/\mathfrak{a} ^{n+1}$.

\emph{First Step:}\\
Let $\mathcal{P}$ be a property of  Noetherian rings. For instance $\mathcal{P}  =$ regular, complete intersection (CI), Gorenstein, Cohen-Macaulay \ etc.
\begin{defn}
Let $(A,\mathfrak{m} )$ be a Noetherian local ring and let $\mathfrak{a} $ be a proper ideal
in $A$. We say $A$ admits a \emph{$\mathcal{P} $-approximation} with respect to \ $\mathfrak{a} $ if there exists
a  local ring $(B,\mathfrak{n} )$, an ideal $\mathfrak{b} $ of $B$  and $\psi \colon B \rightarrow A$,  a local ring homomorphism
with $\psi(\mathfrak{b} )A =  \mathfrak{a} $ such that the following \emph{three}
properties hold:
 \begin{enumerate}
\item
$A$ is a finitely generated $B$ module ( via $\psi$).
\item
$\dim A = \dim B $.
\item
$B$ and $G_{\mathfrak{b}}(B) $  have property $\mathcal{P} $.
\end{enumerate}
 If the above conditions hold we say $[B,\mathfrak{n} ,\mathfrak{b} ,\psi]$ is a \emph{$\mathcal{P} $-approximation}
of $[A,\mathfrak{m} ,\mathfrak{a} ]$.
\end{defn}

\begin{remarkC}

\noindent $\bullet$  Regular approximations \textit{seems} to be  rare while Cohen-Macaulay approximations \emph{don't seem} to  have many applications.

  \noindent $\bullet$   For applications we will often insist that $\psi$, $\mathfrak{b} $ satisfy some additional properties.
\end{remarkC}

When $A$ is complete we prove the following general result(see \ref{equiTT}).  Let $\mathfrak{s} (\mathfrak{a} )$ denote the analytic spread of $\mathfrak{a} $, see \ref{equimultiple-defn}.

\begin{thm}\label{introGAPP}
Let $(A,\mathfrak{m} )$ be  a complete  Noetherian local ring and let $\mathfrak{a} $ be a proper ideal
 in $A$ such that $ \mathfrak{s} (\mathfrak{a} ) + \dim A/\mathfrak{a}  = \dim A$. Then
$A$  admits a  CI-approximation.
with respect to \ $\mathfrak{a} $.
\end{thm}

Notice the hypothesis of Theorem \ref{introGAPP} is satisfied  if $\mathfrak{a} $ is  $\mathfrak{m} $-primary.
Another significant case when Theorem \ref{introGAPP} holds is when $A$ is equidimensional and $\mathfrak{a} $ is equimultiple; see \cite[2.6]{HUO}.  For definition of an equimultiple ideal see \ref{equimultiple-defn}.

\emph{Second Step:}\\
Before we state our second step we need some preliminaries.
Let $M$ be a finitely generated $A$-module.
Recall an $\mathfrak{a} $-\textit{filtration} $\EuScript{G} = {\{M_n\}}_{n \in \mathbb{Z} }$ on $M$ is a
collection of submodules of $M$ with the properties
\begin{enumerate}
\item
$M_n \supset M_{n+1}$ for all $n \in \mathbb{Z} $.
\item
$M_n = M$ for all $n \ll 0$.
\item
$\mathfrak{a}  M_n \subseteq M_{n+1}$ for all $n \in \mathbb{Z} $.
\end{enumerate}
If
$\mathfrak{a}  M_n = M_{n+1}$ for all $n \gg 0$ then we say $\EuScript{G}$ is $\mathfrak{a} $-\emph{stable}.



Let $\EuScript{R}_{\mathfrak{a}}(A)  = \bigoplus_{n\in \mathbb{Z} } \mathfrak{a} ^n$ be the extended Rees algebra of $A$ with respect to \ $A$.
If $\EuScript{G} = \{M_n\}_{n \in \mathbb{Z} }$ is an $\mathfrak{a} $-stable filtration on $M$ then  set
$\EuScript{R}(\EuScript{G},M) = \bigoplus_{n \in \mathbb{Z} }M_nt^n$ the \emph{extended Rees-module} of $M$
\emph{with respect to }\ $\EuScript{G}$.
 Notice that $\EuScript{R}(\EuScript{G},M)$ is a finitely generated graded $\EuScript{R}_{\mathfrak{a}}(A) $-module.


Set $M^* = \operatorname{Hom}_A(M, A)$. Set $M^* = \operatorname{Hom}_A(M, A)$. For $n \in \mathbb{Z} $ set
$$  M_n^* = \{ f \in M^* \mid    f(M) \subseteq \mathfrak{a} ^n \}.      $$
It is well-known that $M_{n+1}^* = \mathfrak{a}  M_n^*$ for all $n \gg 0$.
Note $M_n^* = M^*$ for $n \leq 0$. Set $\EuScript{F} = \{M_n^*\}_{n \in \mathbb{Z} }$. We call this filtration \emph{ the dual filtration } of $M$ with respect to \  $\mathfrak{a} $.
 \emph{We construct  an explicit isomorphism (see \ref{main}):}
$$\Psi_M \colon \EuScript{R}(\EuScript{F},M^{*}) \quad \longrightarrow \operatorname{\ ^* Hom}_\mathcal{R} (\EuScript{R}(\mathfrak{a} , M),\EuScript{R}_{\mathfrak{a}}(A)  ). $$
It is an important observation that dual filtration's behaves well  with respect to \ the Veronese functor, see \ref{dual-Ver-prop}.
\begin{remark}
The reader might wonder why we are working with modules when the main problem is for a ring. The point is that even if we start with an
$\mathfrak{a} $-adic filtration on a Cohen-Macaulay \ ring (with a canonical module $\omega_A$), then it induces an $\mathfrak{a} $-stable filtration on $\omega_A$. Even if $A$ is Gorenstein then the corresponding filtration is  no longer multiplicative, only $\mathfrak{a} $-stable.
\end{remark}


\emph{Third Step:}\\
In \cite{Pu5} we introduced $L^\mathfrak{a} (M) = \bigoplus_{n \geq 0}M/\mathfrak{a} ^{n+1}M$ where $M$ is a Cohen-Macaulay \ $A$-module. Note $\EuScript{S}_{\mathfrak{a}}(A) $ is a subring of $A[t]$.
Let  $\EuScript{S}_{\mathfrak{a}}(M)  = \bigoplus_{n \geq 0} \mathfrak{a} ^nM$ be the Rees module of $M$ with respect to \ $\mathfrak{a} $. Note $\EuScript{S}_{\mathfrak{a}}(M)  $ is a finitely generated $\EuScript{S}_{\mathfrak{a}}(A) $-module.
We have an exact sequence of $\EuScript{S}_{\mathfrak{a}}(A) $-modules
\[
0 \rightarrow \EuScript{S}_{\mathfrak{a}}(M)  \rightarrow M[t]    \rightarrow  L^\mathfrak{a} (M)[-1]   \rightarrow 0. \tag{$\dagger$}
\]
This yields a $\EuScript{S}_{\mathfrak{a}}(A) $-module structure on $L^\mathfrak{a} (M)(-1)$ and so on $L^\mathfrak{a} (M)$.  Notice $L^\mathfrak{a} (M)$ is not finitely generated as a $\EuScript{S}_{\mathfrak{a}}(A) $-module.
Let $\mathfrak{M}  = (\mathfrak{m} , \EuScript{S}_{\mathfrak{a}}(A) _+)$ be the $*$-maximal ideal of $\EuScript{S}_{\mathfrak{a}}(A) $. Assume $r = \dim M \geq 1$. In \cite{Pu5} we proved that for $i = 0, \cdots, r-1$ the local cohomology modules $H^i_\mathfrak{M} (L^\mathfrak{a} (M))$ satisfy
some nice properties, see section \ref{Lprop} for a summary of these properties. Note that using $(\dagger)$ we get an inclusion of graded $\EuScript{S}_{\mathfrak{a}}(A) $-modules
$$ 0 \rightarrow H^{r-1}_\mathfrak{M} (L^I(M))[-1] \rightarrow H^r_\mathfrak{M}  (\EuScript{S}_{\mathfrak{a}}(M) )$$
Note it is possible that $H^r_\mathfrak{M}  (\EuScript{S}_{\mathfrak{a}}(M) )_n $ can have infinite length for some $n$. However $H^{r-1}_\mathfrak{M} (L^I(M))_n $ has finite length for all $n \in \mathbb{Z} $.

An ideal $I$ is said to be asymptotically normal if $I^n$ is integrally closed for all $n \gg 0$. Itoh's conjecture easily follows from the following result:
\begin{lemma}\label{crucial-itoh-2-introduction}
Let $(A,\mathfrak{m} )$ be a Cohen-Macaulay \ local ring of dimension $d \geq 3$ and let $\mathfrak{a} $ be an asymptotically  normal $\mathfrak{m} $-primary ideal with $e_3^\mathfrak{a} (A) = 0$.  Then the local cohomology modules $H^i(L^\mathfrak{a} (A))$ vanish for $1 \leq i \leq d -1$.
\end{lemma}





















Here is an overview of the contents of this paper. In section two we introduce notation and discuss a few
preliminary facts that we need. In section 3 we prove the result on complete intersection approximations  when $\mathfrak{a}  = \mathfrak{m} $ and $A$ is a quotient of a regular local ring.
We also prove it when
$A$ is complete, $\mathfrak{a} $ is $\mathfrak{m} $-primary  and contains a field.
In section four we discuss dual filtration's.
In section five we prove our result on complete intersection approximation of equimultiple ideals.
In section six we discuss some preliminaries on $L^\mathfrak{a} (A)$ that we need to prove Itoh's conjecture.
Finally in our last section we prove Itoh's conjecture for normal ideals.


\section{preliminaries}

In this paper all rings are commutative Noetherian and all modules  (unless stated otherwise) are assumed to
be finitely generated. Let $A$ be a ring, $\mathfrak{a} $ an ideal in $A$ and let $M$ be an $A$-module

\s Let $(A,\mathfrak{m} )$ be a Noetherian   local ring of dimension $d$,
 $\mathfrak{a} $  an $\mathfrak{m} $-primary ideal in $A$ and let $M$ be a Cohen-Macaulay \ (CM)  module of dimension $r$. Set $G_{\mathfrak{a} }(A) = \bigoplus_{n \ge 0} \mathfrak{a} ^n/\mathfrak{a} ^{n+1} $;  the
\textit{associated graded ring} of $A$ with respect to \ $\mathfrak{a} $   and let $G_\mathfrak{a} (M)= \bigoplus_{n \ge 0} \mathfrak{a} ^nM/\mathfrak{a} ^{n+1} M$ be the \textit{associated graded module} of $M$ with respect to \ $\mathfrak{a} $.

The Hilbert-Samuel function of $M$ with respect to $\mathfrak{a} $ is the function
\[
n \mapsto \ell(M/\mathfrak{a} ^{n+1}M)\quad \text{for all} \ n \geq 0.
\]

It is well known that for large values of $n$ it is given by a polynomial $p^{I}_{M}(n)$ of
degree $r = \dim M$, called the Hilbert-Samuel polynomial of $M$ with respect to $I$.
It can be written in the form
\[
p^{\mathfrak{a} }_{M}(x) = \sum_{i=0}^{r}(-1)^ie_{i}^{\mathfrak{a} }(M)\binom{X + r - i}{r-i}
\]

The integers $e_{0}^{\mathfrak{a} }(M), e_{1}^{\mathfrak{a} }(M), ..., e_{r}^{\mathfrak{a} }(M)$ are
called the \emph{ Hilbert coefficients} of $M$ with
respect to $\mathfrak{a} $. The number $e_{0}^{\mathfrak{a} }(M)$ is also called the \emph{multiplicity} of $M$ with respect to $\mathfrak{a} $. The existence of the Hilbert-Samuel polynomial is equivalent to the fact that
the formal power series $\sum_{n \geq 0}\ell(M/\mathfrak{a} ^{n+1}M)z^n$
represents a rational function of a
special type:
\begin{align*}
\sum_{n \geq 0}\ell(M/\mathfrak{a} ^{n+1}M)z^n &= \frac{h^{\mathfrak{a} }_{M}(z)}{(1-z)^{r+1}}\quad \text{where}
\ r = \dim M \ \text{and}\\
h_{M}^{\mathfrak{a} }(z) &= h_{0}^{\mathfrak{a} }(M) + h_{1}^{\mathfrak{a} }(M)z + \cdots + h_{s}^{\mathfrak{a} }(M)z^s \in
\mathbb{Z}[z]
\end{align*}
\s \label{mod-N}
Let $\EuScript{G} = {\{M_n\}}_{n \in \mathbb{Z} }$ be an $\mathfrak{a} $-filtration on $M$ and let $N$ be a submodule of $M$. By the \emph{quotient filtration} on $M/N$ we mean the filtration   $\overline{\EuScript{G}} = \{ (M_n + N)/N \}_{n \in \mathbb{Z} }$.
If $\EuScript{G}$ is an $\mathfrak{a} $-stable filtration on
$M$ then $\overline{\EuScript{G}}$ is an $\mathfrak{a} $-stable filtration on
$M/N$.\textit{ Usually} for us $N =\mathbf{x} M$ for some $\mathbf{x} = x_1,\ldots,x_s \in \mathfrak{a} $.

\s If $\EuScript{G} = \{M_n\}_{n \in \mathbb{Z} }$ is an $\mathfrak{a} $ stable filtration on $M$ then  set
$\EuScript{R}(\EuScript{G},M) = \bigoplus_{n \in \mathbb{Z} }M_nt^n$ the \emph{extended Rees-module} of $M$
\emph{with respect to }\ $\EuScript{G}$.
 Notice that $\EuScript{R}(\EuScript{G},M)$ is a finitely generated graded $\EuScript{R}_{\mathfrak{a}}(A) $-module. If $\EuScript{G}$ is the usual $\mathfrak{a} $-adic filtration then
set \\  $ \EuScript{R}(\mathfrak{a} , M) = \EuScript{R}(\EuScript{G},M)$.


Set $G(\EuScript{G},M)=\bigoplus_{n \in \mathbb{Z} } M_n/M_{n+1}$, the \emph{associated graded
module }of $M$ \emph{with respect to} \ $\EuScript{G}$. Notice $G(\EuScript{G},M)$ is a finitely generated graded module over
 $G_{\mathfrak{a} }(A)$.
 Furthermore $\EuScript{R}(\EuScript{G},M) /t^{-1}\EuScript{R}(\EuScript{G},M) = G(\EuScript{G}, M)$.

\s \label{shift-filt}
Let $\EuScript{G} = {\{M_n\}}_{n \in \mathbb{Z} }$ be an $\mathfrak{a} $-filtration on $M$ and let $s \in \mathbb{Z} $. By the $s$-\emph{th shift} of $\EuScript{G}$, denoted by $\EuScript{G}(s)$ we mean the filtration ${\{ \EuScript{G}(s)_n\}}_{n \in \mathbb{Z} }$ where
$\EuScript{G}(s)_n = \EuScript{G}_{n+s}$. Clearly
$$G(\EuScript{G}(s), M) = G(\EuScript{G}, M)(s) \quad \text{and} \quad \mathcal{R} (\EuScript{G}(s), M) = \mathcal{R} (\EuScript{G}, M)(s). $$



\s All   filtration's in this paper $\EuScript{G} = \{M_n\}_{n \in \mathbb{Z} }$ will be \textit{separated} i.e.,
 $\bigcap_{n \in \mathbb{Z} }M_n = \{0\}$.
This is automatic if $A$ is local,  $\mathfrak{a}  \neq A$ and $\EuScript{G}$ is $\mathfrak{a} $-stable.
If $m$ is a non-zero element of $M$ and if $j$ is the largest integer such that
$ m \in M_j$,
then we let $m^{*}_{\EuScript{G}}$ denote the image of $m $ in $M_j  \setminus M_{j+ 1}$ and we call it the \textit{initial form} of $m$ with respect to \ $\EuScript{G}$. If $\EuScript{G}$ is clear from the context then we drop the subscript $\EuScript{G}$.





\s \emph{Veronese functor:}. Let $\EuScript{F} = \{M_n \}_{n\in \mathbb{Z} }$ be an $\mathfrak{a} $- stable filtration. For $l \geq 1$ set $\EuScript{F}^{<l>}= \{M_{nl} \}_{n \in \mathbb{Z} }$. Notice $\EuScript{F}^{<l>}$ is an $\mathfrak{a} ^l$-stable filtration. We also have an $\EuScript{R}_{\mathfrak{a}^l}(A) $-isomorphism $\EuScript{R}(\EuScript{F},M)^{<l>} \cong \EuScript{R}(\EuScript{F}^{<l>}, M)$.

\s For definition and  basic properties of superficial sequences see \cite[p.\ 86-87]{Pu1}.

\s\label{AtoA'} \textbf{Flat Base Change:} In our paper we do many flat changes of rings.
 The general set up we consider is
as follows:


 Let $\phi \colon (A,\mathfrak{m} ) \rightarrow (A',\mathfrak{m} ')$ be a flat local ring homomorphism
 with $\mathfrak{m}  A' = \mathfrak{m} '$. Set $\mathfrak{a} ' = \mathfrak{a}  A'$ and if
 $N$ is an $A$-module set $N' = N\otimes A'$. Set $k = A/\mathfrak{m} $ and $k' = A'/\mathfrak{m} '$.


 \textbf{Properties preserved during our flat base-changes:}

\begin{enumerate}[\rm (1)]
\item
$\ell_A(N) = \ell_{A'}(N')$. So
 $H_{\mathfrak{a} }(M,n) = H_{\mathfrak{a} '}(M',n)$ for all $n \geq 0$.
\item
$\dim M = \dim M'$ and  $\operatorname{grade}(K,M) = \operatorname{grade}(KA',M')$ for any ideal $K$ of $A$.
\item
$\operatorname{depth} G_{\mathfrak{a} }(M) = \operatorname{depth} G_{\mathfrak{a} '}(M')$.
\end{enumerate}

\textbf{Specific flat Base-changes:}


\begin{enumerate}[\rm (a)]
\item
$A' = A[X]_S$ where $S =  A[X]\setminus \mathfrak{m}  A[X]$.
The maximal ideal of $A'$ is $\mathfrak{n}  = \mathfrak{m}  A'$.
The residue
field of $A'$ is $k' = k(X)$. Notice that $k'$ is infinite.
\item
$A' =  \widehat{A}$  the completion of $A$ with respect to \ the maximal ideal.
\item
Applying (a) and (b) successively ensures that $A'$ is complete with $k'$ infinite.
\item
$A' = A[X_1,\ldots,X_n]_S$ where $S =  A[X_1,\ldots,X_n]\setminus \mathfrak{m}  A[X_1,\ldots,X_n]$.
The maximal ideal of $A'$ is $\mathfrak{n}  = \mathfrak{m}  A'$.
The residue
field of $A'$ is $l = k(X_1,\ldots,X_n)$. Notice that if $I$ is integrally closed then $I'$ is
also integrally closed. Recall an ideal $K$ is said to be asymptotically normal if  $K^n$ is integrally closed for all $n \gg 0$.
When $\dim A \geq 2$, $A$ is  Cohen-Macaulay \ and $I$ is asymptotically normal, say $I = (a_1,\cdots,a_n)$ then in \cite[Corollary 2]{Ciu} it is proved that
the element $y = \sum_{i = 1}^{n} a_iX_i$ in $A'$ is $I'$-superficial and
the $A'/(y)$ ideal $J = I'/(y)$ is asymptotically normal. We call $A'$ a general extension of $A$.
\end{enumerate}



\s \label{lying-above} We will need the following result. Let $\phi \colon (B,\mathfrak{n} ) \rightarrow (A.\mathfrak{m} )$ be a local map such that $A$ is finitely generated $B$-module (via $\phi$). Consider
$R = B[X_1,\cdots,X_n]$ and $S = A[X_1,\cdots. X_n]$ and let $\psi \colon R \rightarrow S$ be the map induced by $\phi$. Note $S$ is a finite $R$-module via $\psi$.
Let $P = \mathfrak{n}  R$ and $Q = \mathfrak{m}  S$. Then $S_P = S_Q$. To see this note that $Q$ is the only prime in $S$ lying over $P$. The result follows from an exercise problem
in \cite[Exercise 9.1]{Ma}.







\s\label{equimultiple-defn}  The \emph{fiber cone} of $\mathfrak{a} $ is the
$k$-algebra
$F(\mathfrak{a} ) = \bigoplus_{n\geq 0}\mathfrak{a} ^n/\mathfrak{m}  \mathfrak{a} ^{n}.$
Set $\mathfrak{s} (\mathfrak{a} ) = \dim F(\mathfrak{a} )$,
the \emph{analytic spread} of $\mathfrak{a} $.
 Recall $\dim A \geq \mathfrak{s} (\mathfrak{a} ) \geq \operatorname{ht} \mathfrak{a} $.
We say $\mathfrak{a} $ is\textit{ equimultiple} if
$\mathfrak{s} (\mathfrak{a} ) = \operatorname{ht} \mathfrak{a} $. Clearly $\mathfrak{m} $-primary ideals are equimultiple.



\s \label{redNO} We assume $k = A/\mathfrak{m} $ is infinite. Let $\mathfrak{c}  = (x_1,\ldots,x_l)$ be  a minimal reduction of
$\mathfrak{a} $ with respect to \ $M$.  We denote by  $\operatorname{red}_\mathfrak{c} (\mathfrak{a} , M) := \min \{ n  \mid \  \mathfrak{c}  \mathfrak{a} ^n M= \mathfrak{a} ^{n+1}M \} $
the \textit{reduction number} of $\mathfrak{a} $ with respect to \ $\mathfrak{c} $ and $M$. Let
 $$\operatorname{red}(\mathfrak{a} , M) = \min \{ \operatorname{red}_\mathfrak{c} (\mathfrak{a} , M) \mid \mathfrak{c} 
\text{\ is  a minimal reduction of \ }  \mathfrak{a}  \}$$
 be the \textit{reduction number } of $M$ with respect to \ $\mathfrak{a} $. Set
$\operatorname{red}_\mathfrak{a} (A) = \operatorname{red}(\mathfrak{a} , A)$.


\s \label{gen-cm} Let $R= \bigoplus_{n\geq 0}R_n$ be a standard graded algebra over an Artin local ring $(R_0,\mathfrak{m} _0)$. Set $\mathfrak{M}  = \mathfrak{m} _0 \oplus R_+$. We say that $R$ is generalized Cohen-Macaulay \ if $H^i_{\mathfrak{M} }(R)$ has finite length for $i < \dim R$.

\section{CI-Approximation: Some special cases}\label{Gorapp-special}
In this section we prove that if $(A,\mathfrak{m} )$ is a quotient of a regular local ring then
$[A,\mathfrak{m} ,\mathfrak{m} ]$ has a CI-approximation $[B,\mathfrak{n} ,\mathfrak{n} , \phi]$ with $\phi$ onto. In Theorem \ref{equi} we  state our main result regarding CI-approximations. This will be proved in section \ref{sectionGAPP} . Finally in Theorem \ref{field} we prove a result regarding CI-approximation of an $\mathfrak{m} $-primary ideal $\mathfrak{a} $ in a complete  equicharacteristic
local ring $(A,\mathfrak{m} )$.
The following Lemma  regarding annihilators is crucial.

\begin{lemma}
\label{annihilator}
Let $M$ be an $A$-module and let $\mathfrak{a} $ be an ideal. Set $G = G_{\mathfrak{a}}(A) $.
 Suppose there exists
 $\xi_1,\ldots,\xi_s \in \mathfrak{a} $ such that $\xi_{i}^{*} \in \operatorname{ann}_{G} G_{\mathfrak{a} }(M)$
for each $i$. Also assume that
 $\xi_{1}^{*},\ldots,\xi_{s}^{*}$ is a $G$-regular sequence.
 Then there exists $u_1,\ldots,u_s \in \mathfrak{a} $ such that
\begin{enumerate}[\rm (1)]
\item
$u_1,\ldots,u_s$ is an $A$-regular sequence.
\item
$u_i \in \operatorname{ann} M$ for each $i = 1,\ldots,s$.
\item
For each $i \in \{1,\ldots,s \}$ there exists $n_i \geq 1$ such
 that $u_{i}^{*} = (\xi_{i}^{*})^{n_i}$.
\item
 $u_{1}^{*},\ldots,u_{s}^{*}  \in  \operatorname{ann}_G G_\mathfrak{a} (M).$
\item
 $u_{1}^{*},\ldots,u_{s}^{*} $ is a  $G$-regular sequence.
\end{enumerate}
\end{lemma}
\begin{proof}
Suppose we have constructed $u_1,\ldots,u_s$ satisfying (2) and (3) then
 $u_1,\ldots,u_s$ satisfy all the remaining properties. This can be seen as follows:

(4) follows from (3); since $\xi_{i}^{*} \in \operatorname{ann}_{G} G_{\mathfrak{a} }(M)$ for each $i$.

(1) and (5). As  $\xi_{1}^{*},\ldots,\xi_{s}^{*}$ is a $G$-regular sequence
 we also get
$(\xi_{1}^{*})^{n_1},\ldots,(\xi_{s}^{*})^{n_s}$ is a $G$-regular sequence
\cite[16.1]{Ma}.
Thus $u_{1}^{*},\ldots,u_{s}^{*} $ is a  $G$-regular sequence. It follows from   \cite[2.4]{VV} that
$u_1,\ldots,u_s$ is an $A$-regular sequence.
Thus it suffices to show there exists $u_1,\ldots,u_s$ satisfying (2) and (3).

Fix $i \in \{1,\ldots,s \}$.
Set $\xi = \xi_i$.
  Say $\xi \in \mathfrak{a} ^r \setminus \mathfrak{a} ^{r+1}$ for some $r \geq 1$.
Since $\xi^* \in \operatorname{ann}_G G_{\mathfrak{a} }(M)$ we have $\xi M \subseteq \mathfrak{a} ^{r+1}M$.
Set $\mathfrak{q}  = \mathfrak{a} ^{r+1}$.
By the \emph{determinant trick}, \cite[2.1]{Ma}, there exists a monic polynomial $f(X) \in A[X]$ such that
\begin{align*}
f(X) &= X^{n} + a_1X^{n-1} + \ldots + a_{n-1}X + a_n \quad \text{with} \quad a_i \in \mathfrak{q} ^i, \text{\ for} \ i = 1,\ldots n \\
\text{and} \ u = f(\xi) &= \xi^{n} + a_1\xi^{n-1}+ \ldots +  a_{n-1}\xi + a_n \in \operatorname{ann} M.
\end{align*}
As $\xi^*$ is $G$-regular we have $\xi^n \in \mathfrak{a} ^{nr} \setminus \mathfrak{a} ^{nr+1}$.
 However  for each $i \geq 1$ we
 have
\[
a_i\xi^{n-i} \in \mathfrak{q} ^i\mathfrak{a} ^{(n-i)r} = \mathfrak{a} ^{i(r+1)}\mathfrak{a} ^{nr - ir} = \mathfrak{a} ^{nr + i} \subseteq
 \mathfrak{a} ^{nr +1}.
\]
Thus $u^* = (\xi^{n})^{*} = (\xi^*)^n$ (since $\xi^*$ is $G$-regular).
Set $u_i = u$ and $n_i = n$.
\end{proof}

The following Corollary is useful.
\begin{corollary}\label{redTOmaxDIM}
 Let $(A,\mathfrak{m} )$ be Cohen-Macaulay \ local ring, $\mathfrak{a} $ an $\mathfrak{m} $-primary ideal and let $M$ be an $A$-module. Set
$c = \dim A - \dim M$.  If $G_{\mathfrak{a}}(A) $ is Cohen-Macaulay \ then there exists $u_1,\ldots, u_c \in \mathfrak{a} $ such that
\begin{enumerate}[\rm (1)]
 \item
$ u_1,\ldots, u_c \in \operatorname{ann}_A M $.
 \item
 $ u_{1}^*,\ldots, u_{c}^* \in \operatorname{ann}_{G_{\mathfrak{a}}(A) } G_\mathfrak{a} (M)$.
 \item
 $ u_1,\ldots, u_c $ is an $A$-regular sequence.
\item
$ u_{1}^*,\ldots, u_{c}^*$ is a $G_{\mathfrak{a}}(A) $-regular sequence.
\item
 $u_1 t,\ldots, u_c t \in \operatorname{ann}_{\EuScript{R}_{\mathfrak{a}}(A)  } \mathcal{R} (M).$
\item
$u_1 t,\ldots, u_c t$ is a $\EuScript{R}_{\mathfrak{a}}(A) $-regular sequence.
\end{enumerate}
\end{corollary}
\begin{proof}
 Since $G_{\mathfrak{a}}(A) $ is Cohen-Macaulay, we have that
\begin{align*}
  \operatorname{grade}  \operatorname{ann}_{G_{\mathfrak{a}}(A) } G_\mathfrak{a} (M)  &= \operatorname{ht} \operatorname{ann}_{G_{\mathfrak{a}}(A) } G_\mathfrak{a} (M) \\
                           &=  \dim G_{\mathfrak{a}}(A)   - \dim G_{\mathfrak{a} }(M),  \quad \text{since $G_{\mathfrak{a}}(A)  $ is *-local} \\
                           &= \dim A - \dim M .
\end{align*}
Therefore (1), (2), (3) and (4) follow from Lemma \ref{annihilator}.
The assertion (5) is clear.

(6). Since $t^{-1}$ is $\EuScript{R}_{\mathfrak{a}}(A) $-regular and $u_1^*,\ldots, u_c^*$ is $G_{\mathfrak{a}}(A)  = \EuScript{R}_{\mathfrak{a}}(A) /t^{-1}\EuScript{R}_{\mathfrak{a}}(A) $-regular sequence it follows that $t^{-1}, u_1t,\ldots, u_ct$ is  a $\EuScript{R}_{\mathfrak{a}}(A) $-regular sequence. Notice
$\EuScript{R}_{\mathfrak{a}}(A) $ is a *-local. It follows that $ u_1t,\ldots, u_ct$ is a $\EuScript{R}_{\mathfrak{a}}(A) $-regular sequence.
\end{proof}

We state our general result regarding CI-Approximations
\begin{theorem}\label{equi}
Let $(A,\mathfrak{m} )$ be a complete
 local ring and let $\mathfrak{a} $ be a proper  ideal in $A$ with $\dim A/\mathfrak{a}  + \mathfrak{s} (\mathfrak{a} )
 = \dim A$. Then
$A$ has a CI-approximation with respect to \ $\mathfrak{a} $. Furthermore if $\mathfrak{a}  = (x_1,\ldots,x_m)$ then we may
choose
a CI-approximation $[B,\mathfrak{n} ,\mathfrak{b} ,\varphi]$ of $[A,\mathfrak{m} ,\mathfrak{a} ]$ such that there exists $v_1,\ldots,v_m \in \mathfrak{b} $ with $\varphi(v_i) = x_i$ for  $i = 1,\ldots, m$ and $\mathfrak{b}  = ( v_1,\ldots,v_m)$.
\end{theorem}
Theorem \ref{equi} is proved in section \ref{sectionGAPP}.
\begin{remark}\label{equiREM}
 The hypothesis of the theorem holds  when $\mathfrak{a} $ is $\mathfrak{m} $-primary. It is also satisfied when $\mathfrak{a} $ is equimultiple and
$A$ is equidimensional \cite[34.5]{Nag}.  Our hypothesis ensures that $G_{\mathfrak{a}}(A) $ has a homogeneous system of parameters,
  \cite[2.6]{HUO}.
\end{remark}

For the case when $\mathfrak{a}  = \mathfrak{m} $ and $A$ is a quotient of a regular local ring $T$ we have:
\begin{theorem}\label{GorApprox-REG-QT}
 Let $(A,\mathfrak{m} )$  be a local ring such that $A$ is a quotient of a regular local ring $(T,\mathfrak{t} )$. Then
$[A,\mathfrak{m} ,\mathfrak{m} ]$ has a CI-approximation $[B,\mathfrak{n} ,\mathfrak{n} , \phi]$ with $\phi$-onto.
\end{theorem}
\begin{proof}
 Set $k = A/\mathfrak{m}  = T/\mathfrak{t} $. Let $\psi \colon T \rightarrow A$ be the quotient map. We consider $A$ as  a
$T$-module. So $ G_{\mathfrak{a}}(A)  = G_{\mathfrak{t} }(A)$. Set $c = \dim T - \dim A = \dim G_{\mathfrak{t}}(T)  - \dim G_{\mathfrak{a}}(A) $.
Its well-known  $G_{\mathfrak{t}}(T)  \cong k[X_1,\ldots,X_n]$; where $n = \dim T$. In particular $G_{\mathfrak{t}}(T) $ is
Cohen-Macaulay. Let $u_1,\ldots u_c$ be as in Corollary \ref{redTOmaxDIM}.

Set $(B,\mathfrak{n} ) = (T/(\mathbf{u}),\mathfrak{t} /(\mathbf{u}))$ and let $\phi \colon B \rightarrow A$ be the map induced by
$\psi$. So $\phi$ is onto. Clearly
$[B,\mathfrak{n} ,\mathfrak{n} , \phi]$ is a CI-approximation  of $[A,\mathfrak{m} ,\mathfrak{m} ]$.
\end{proof}


\noindent For  $\mathfrak{m} $-primary ideals in a complete  equicharacteristic complete local ring we prove:
\begin{theorem}\label{field}
 Let $(A,\mathfrak{m} )$ be a complete
equicharacteristic local ring and let $\mathfrak{a}$ be an
$\mathfrak{m} $-primary ideal.  Then $[A,\mathfrak{m} ,\mathfrak{a}]$ has a CI-approximation $[B,\mathfrak{n} ,\mathfrak{b},\phi]$ with $\mathfrak{b}  = \mathfrak{n} $ and $B/\mathfrak{n}  \cong A/\mathfrak{m} $ and  $\mu(\mathfrak{n} ) =
\mu(\mathfrak{a} )$.
\end{theorem}
\begin{proof}
  $A$  contains its residue field $k = A/\mathfrak{m} .$; see \cite[28.3]{Ma}.  Let
$\mathfrak{a} = (x_1, \ldots, x_n).$  Set $S = k [|X_1,\ldots X_n|]$
and let $\psi : S \rightarrow A$ be the natural map which sends
$X_i$ to $x_i$ for each $i.$
Since $A/(\underline{X}) A = A/\mathfrak{a}$ has finite
length (as an $S$-module) it follows that $A$ is a finitely
generated as an S-module, see \cite[8.4]{Ma}.  Set $\eta = (X_1, \ldots
X_n).$ Notice $\psi (\eta)A = \mathfrak{a}$ and $G_{\eta} (S) = k
[X^{*}_{1}, \ldots X^{*}_{n}]$ the polynomial ring in $n$-variables.

If $\dim A = \dim S$ then set $[B,\mathfrak{n} ,\mathfrak{b} ,\phi] = [S,\eta,\eta,\psi].$
Otherwise set $c = \dim S-\dim A = \dim G_n (S) - \dim
G_{\mathfrak{a}} (A).$
Let $u_1,\ldots u_c$ be as in Corollary \ref{redTOmaxDIM}.
Set $B = S/(\mathbf{u})$, $\mathfrak{n}  = \eta/(\mathbf{u})$ and $\mathfrak{b}  = \mathfrak{n} $. Clearly $G_{\mathfrak{n} } (B) = G_{\eta}
(S)/(\mathbf{u}^*)$ is CI.
 The map $\psi$ induces $\phi : B \rightarrow A.$  It can
be easily checked that $[B,\mathfrak{n} ,\mathfrak{n} ,\phi]$ is a CI-approximation of $[A,\mathfrak{m} ,\mathfrak{a}]$.
Finally by our construction its clear that $\mu(\mathfrak{n} ) = \mu(\mathfrak{a} )$.
\end{proof}


\section{A classical filtration of the dual}\label{section-classical}
Let $M$ be  an $A$-module.
The following  filtration of the dual of $M$, i.e, $M^{*} = \operatorname{Hom}_A(M, A)$ is classical cf.  \cite[p.\ 12]{Ser}.
Set
\[
M_{n}^{*} = \{ f \in M^{*} \mid f(M) \subseteq \mathfrak{a} ^n \} \cong \operatorname{Hom}_A(M,\mathfrak{a} ^n).
\]
It is easily verified that   $\EuScript{F}_M = \{ M_{n}^{*} \}_{n \in \mathbb{Z} }$ is an $\mathfrak{a} $-stable filtration on $M^{*}$.
We call $\EuScript{F}_M$ to be the \emph{dual-filtration} of $M$ with respect to  \ $\mathfrak{a} $.
Set $\mathcal{R}  = \mathcal{R} (\mathfrak{a} ) = \bigoplus_{n \in \mathbb{Z} }\mathfrak{a} ^nt^n$  and
$\mathcal{R} (M) = \mathcal{R} (\mathfrak{a} ,M) = \bigoplus_{n \in \mathbb{Z} }\mathfrak{a} ^nMt^n$. Let $f \in  M_{n}^{*}$.  We show that $f$
induces a\textit{ homogeneous} $\mathcal{R} $-\textit{linear map} $\hat{f}$ \textit{of degree} $n$ \textit{from} $\mathcal{R} (M)$ to $\mathcal{R} $; (see \ref{mapAUG}).
Theorem  \ref{main}  shows that $\Psi_M$ is a $\mathcal{R} $-linear isomorphism.
\begin{align*}
\Psi_M \colon \mathcal{R} (\EuScript{F},M^{*}) &\longrightarrow \operatorname{\ ^* Hom}_\mathcal{R} (\mathcal{R} (M),\mathcal{R}  ) \\
f &\mapsto \hat{f} 
\end{align*}
It is easily seen,  see \ref{initialobs},  that $\Psi_M$ induces a natural map
$$\Phi_M \colon G(\EuScript{F},M^{*}) \longrightarrow \operatorname{\ ^* Hom}_{G_{\mathfrak{a}}(A) }(G_{\mathfrak{a} }(M),G_{\mathfrak{a}}(A) ). $$
We prove $\Phi_M$ is injective.
In Corollary \ref{corGor} we give a sufficient condition for $\Phi_M$ to be an isomorphism.

The following result is well-known.
\begin{proposition}
\label{stablefilt}
Let $(A,\mathfrak{m} )$ be local, $\mathfrak{a} $ an ideal in $A$ and let $M$ an  $A$-module.
Then $\mathfrak{a}  \operatorname{Hom}_{A}(M, \mathfrak{a} ^n) = \operatorname{Hom}_A(M,\mathfrak{a} ^{n+1})$ for all $n \gg 0$. \qed
\end{proposition}






\s \label{mapAUG} Let $f \in M_{n}^{*}$.

\noindent\textbf{Claim:} $f$ induces a homogeneous $\mathcal{R} $-linear map $\hat{f}$ of degree $n$ from $\mathcal{R} (M)$ to $\mathcal{R} $.

Since $f(M) \subseteq \mathfrak{a} ^n$, for each $j \geq 0 $ we have $f(\mathfrak{a} ^jM) \subseteq \mathfrak{a} ^{n+j}$. Furthermore for $j < 0 $ as $\mathfrak{a} ^jM = M$ and notice that $f(M) \subset \mathfrak{a} ^n \subset \mathfrak{a} ^{n+j}$.
Thus
\begin{equation*}
f(\mathfrak{a} ^jM) \subseteq \mathfrak{a} ^{n+j} \quad \text{for each $j \in \mathbb{Z} $}. \tag{$*$}
\end{equation*}
  This enables us to define
\begin{align*}
\hat{f} \colon \mathcal{R} (M) &\longrightarrow \mathcal{R}  \\
\sum_{j \in \mathbb{Z} } m_jt^j &\mapsto \sum_{j \in \mathbb{Z} } f(m_j)t^{n+j}.
\end{align*}

Next we prove that $\hat{f}$ is $\mathcal{R} $-linear.
Clearly $\hat{f}$ is $A$-linear. Notice
\begin{align*}
 \hat{f} \left( (x_jt^j)\bullet(m_it^i) \right) &= \hat{f} (x_jm_it^{i+j}) = f(x_jm_i)t^{i+j+n} = x_jt^jf(m_i)t^{i+n}  \\
(x_jt^j)\bullet \hat{f} (m_it^i) &= x_jt^jf(m_i)t^{i+n}.
\end{align*}
Thus $\hat{f} $ is $\mathcal{R} $-linear.

\s \label{main-1} We define a map
\begin{align*}
\Psi_M \colon \mathcal{R} (\EuScript{F},M^{*}) &\longrightarrow \operatorname{\ ^* Hom}_\mathcal{R} (\mathcal{R} (M),\mathcal{R}  ) \\
f &\mapsto \hat{f} 
\end{align*}


\begin{theorem}\label{main}
$\Psi_M \colon \mathcal{R} (\EuScript{F},M^{*}) \longrightarrow  ^*\operatorname{Hom}_{\mathcal{R} }(\mathcal{R} (M), \mathcal{R} )$ is a $\mathcal{R} $-linear isomorphism.
\end{theorem}
\begin{proof}
If $a \in \mathfrak{a} ^j$ and $f \in M_{i}^{*}$ then
$af \in M^{*}_{i+j}$. Check  that $(at^i)\bullet \hat{f} = \hat{af}$.
Thus $\Psi_M$ is $\mathcal{R} $-linear.
Clearly  if $\hat{f} = 0$ then $f = 0$. Thus $\Psi_M$ is injective.

We show $\Psi_M$ is surjective. Let $g \in \operatorname{\ ^* Hom}_\mathcal{R} (\mathcal{R} (M),\mathcal{R}  )_n$. We write $g$ as
\[
g\left(\sum_{j \in \mathbb{Z} }m_jt^j\right)  = \sum_{j \in \mathbb{Z} }g_j(m_j)t^{n+j}.
\]
For each $j \in \mathbb{Z} $ the map $g_j \colon \mathfrak{a} ^jM \rightarrow \mathfrak{a} ^{n+j}$ is $A$-linear.
Define $f = i\circ g_0$ where $i \colon \mathfrak{a} ^n \rightarrow A$ is the inclusion map. Clearly $f \in M_{n}^{*}$.

\textit{Claim:} $\hat{f} = g$.

Let $m_jt^j \in \mathfrak{a} ^jMt^j$.

Case 1. $j = 0$.

Set $m = m_0$. Notice
$$g(m t^0) = g_0(m)t^n = \hat{f}(m t^0).$$
The last equality above holds since $\hat{f}$ is $\mathcal{R} $-linear.

\noindent\textit{In the next two cases we use Case 1 and the fact that  $\hat{f}$ is $\mathcal{R} $-linear.}

Case 2. $j< 0$.

Notice $m_jt^j =  t^j \bullet m_jt^{0}$. So
$$g(m_jt^j) = t^j\bullet g(m_jt^0) =  t^j\bullet \hat{f}(m_jt^0) = \hat{f}(m_jt^j).$$

Case 3. $j >0$.

Set $m_j = \sum_{l = 1}^{s}u_{jl}n_l$ where $u_{jl} \in \mathfrak{a} ^j$ and $n_l \in M$.
Fix $l$.
Notice
$$ u_{jl}n_lt^j = u_{jl}t^j\bullet n_lt^0.$$
Therefore for $l = 1,\ldots, s$,
$$g(u_{jl}n_lt^j) = u_{jl}t^j\bullet g(n_l t^0) = u_{jl}t^j \bullet \hat{f}(n_lt^0) = \hat{f}(u_{jl}n_lt^j).$$
 Note the last equality above is since $\hat{f}$ is $\mathcal{R} $-linear. So we have
$$ g(m_jt^j) = g\left(\sum_{l = 1}^{s}u_{jl}n_l\right)
           = \sum_{l =1}^{s}g(u_{jl}n_l)
           = \sum_{l = 1}^{s}\hat{f}(u_{jl}n_l)
           = \hat{f}(m_jt^j). $$
  Again   the last equality above holds since $\hat{f}$ is $\mathcal{R} $-linear.
\end{proof}


\begin{observation}\label{initialobs}
$f \in M_{n}^{*}$ induces $\hat{f} \colon\mathcal{R} (M) \rightarrow \mathcal{R} $ which is homogeneous of degree  $n$. So $\hat{f}$ induces a map $\tilde{f}\colon G_{\mathfrak{a} }(M) \rightarrow G_{\mathfrak{a}}(A) $ which is also
homogeneous of degree $n$. Clearly $\tilde{f} = 0$ \text{if and only if} \
$f \in M_{n+1}^{*}$. So we have
\begin{align*}
\Phi_M \colon G(\EuScript{F},M^{*}) &\longrightarrow \operatorname{\ ^* Hom}_{G_{\mathfrak{a}}(A) }(G_{\mathfrak{a} }(M),G_{\mathfrak{a}}(A) ) \\
f + M_{n+1}^{*} &\mapsto \tilde{f}. \\
\text{Clearly} \quad \Phi_M &= \Psi_M\otimes \frac{\mathcal{R} }{t^{-1}\mathcal{R} }.
\end{align*}
\end{observation}

\begin{corollary}\label{basicCor}
Set $G = G_{\mathfrak{a}}(A) $.
There is an exact sequence
\[
0 \longrightarrow G(\EuScript{F},M^{*}) \xrightarrow{\Phi_M} \operatorname{\ ^* Hom}_{G}(G_{\mathfrak{a} }(M),G) \longrightarrow \operatorname{\ ^* Ext}^{1}_{\mathcal{R} }(\mathcal{R} (M),\mathcal{R}  )(+1)
\]
\end{corollary}
\begin{proof}
The map $ 0\rightarrow \mathcal{R} (+1) \xrightarrow{t^{-1}} \mathcal{R}  \rightarrow G \rightarrow 0$ induces the exact sequence
\begin{align*}
0&\longrightarrow \operatorname{\ ^* Hom}_{\mathcal{R} }(\mathcal{R} (M),\mathcal{R} )(+1) \xrightarrow{t^{-1}} \operatorname{\ ^* Hom}_{\mathcal{R} }(\mathcal{R} (M),\mathcal{R} ) \longrightarrow \operatorname{\ ^* Hom}_{\mathcal{R} }(\mathcal{R} (M),G)\\
\ &\longrightarrow \operatorname{\ ^* Ext}_{\mathcal{R} }^{1}(\mathcal{R} (M),\mathcal{R} )(+1)
\end{align*}
Note that $\operatorname{\ ^* Hom}_{\mathcal{R} }(\mathcal{R} (M),G) \cong \operatorname{\ ^* Hom}_{G}(G_{\mathfrak{a} }(M),G)$. The result follows by using
Theorem \ref{main} and Observation \ref{initialobs}.
\end{proof}

An important consequence of  Corollary \ref{basicCor} is the following:
\begin{corollary}\label{corGor}
Let $(A,\mathfrak{m} )$ be a Gorenstein local ring and let $\mathfrak{a} $ be an  ideal such that $G_{\mathfrak{a}}(A) $ is  Gorenstein. Let $M$ be a maximal Cohen-Macaulay \ $A$-module with $G_{\mathfrak{a} }(M)$-Cohen-Macaulay. Then
$$ G(\EuScript{F},M^{*}) \cong \operatorname{\ ^* Hom}_{G_{\mathfrak{a}}(A) }(G_{\mathfrak{a} }(M),G_{\mathfrak{a}}(A) ).$$
\end{corollary}
\begin{proof}
Notice $\mathcal{R} $ is also a Gorenstein ring and $\mathcal{R} (M)$ is maximal Cohen-Macaulay. It follows that
$\operatorname{\ ^* Ext}_{\mathcal{R} }^{1}(\mathcal{R} (M),\mathcal{R} ) = 0$. The result follows from Corollary \ref{basicCor}.
\end{proof}




We now discuss dual filtration's and the Veronese functor. We show
\begin{proposition}
  \label{dual-Ver-prop}
  Let $(A,\mathfrak{m} )$ be  local, $\mathfrak{a} $ an ideal in $A$ and let $M$ be an $A$-module. Let $\EuScript{F}$ be the dual filtration of $M$ with respect to $\mathfrak{a} $. Then for all $l\geq 1$;  $\EuScript{F}^{<l>}$ is the dual filtration of $M$ with respect to \ $\mathfrak{a} ^l$.
\end{proposition}
\begin{proof}
Fix $l \geq 1$. Let $\EuScript{G}$ be the dual filtration of $M$ with respect to  \ $\mathfrak{a} ^l$.
  We note that $\EuScript{F}_{nl} = \operatorname{Hom}_A(M, \mathfrak{a} ^{nl}) = \EuScript{G}_{n}$. The result follows.
\end{proof}

\section{CI approximation: The general case} \label{sectionGAPP}
In this section we prove  our general result regarding  complete intersection (CI)
approximation; see  \ref{equiTT}. In  \ref{field} we showed that every $\mathfrak{m} $-primary ideal $\mathfrak{a} $ in an equicharacteristic
local ring $A$ has a
CI-approximation. Even if we are interested only in the case of $\mathfrak{m} $-primary ideals; we have to take some care for
dealing with the case of local rings with mixed characteristics. \emph{The essential point is to show existence of
homogeneous regular sequences in certain graded ideals.}


This section is divided into two subsections. In the first subsection $R = \bigoplus_{n\geq 0}R_n$ be  a standard graded algebra
over a local ring $(R_0,\mathfrak{m} _0)$. When $R$ is Cohen-Macaulay \ and $M =\bigoplus_{n\geq 0}M_n$ is a finitely generated \  graded $R$-module generated by elements in $M_0$,
we give conditions to ensure a regular sequence $\xi_1,\ldots,\xi_c \in R_+ \cap\operatorname{ann}_R M$
where $c= \dim R - \dim M$ (see Theorem \ref{hann}).
In the second subsection we prove Theorem \ref{equiTT}.

\textbf{Homogeneous regular sequence }

Let  $R = \bigoplus_{n\geq 0}R_n$ be  a standard  algebra
over a local ring $(R_0,\mathfrak{m} _0)$ and let $M =\bigoplus_{n\geq 0}M_n$ be a finitely generated \  graded $R$-module.
 In general a graded module $M$
need not have homogeneous  regular sequence $\xi_1,\ldots,\xi_c \in R_+ \cap\operatorname{ann}_R M$ where $c = \operatorname{grade} M$ (see \ref{eqEx}).
 When $R$ is Cohen-Macaulay \ and $M =\bigoplus_{n\geq 0}M_n$ is a finitely generated \  \ graded $R$-module generated by elements in $M_0$,
we give conditions to ensure a regular sequence $\xi_1,\ldots,\xi_c \in R_+ \cap\operatorname{ann}_R M$
where $c= \dim R - \dim M$.

We adapt an example from \cite[p.\ 34]{BH} to show that a homogeneous  regular sequence of length
$\operatorname{grade} M$
in $R_+ \cap\operatorname{ann}_R M$ need not exist.

\begin{example}\label{eqEx}
 Let $A = k[[X]]$ and $T = A[Y]$. Set $R = T/(XY)$. Notice that $R_0 = A$. Set $\mathfrak{M}  = (X,Y)R$ the unique graded maximal ideal of $R$ and let $M = R/\mathfrak{M}  = k$. Clearly $\operatorname{grade} M = \operatorname{grade}(\mathfrak{M}  , R) = 1$. It can be easily checked that
every homogeneous element in $R_+$ is a zero-divisor.
\end{example}












The following result regarding existence of homogeneous regular sequence in $(\operatorname{ann} M) \cap R_+$ (under certain conditions) is crucial in our proof of Theorem \ref{equiTT}.
\begin{theorem}
\label{hann}
Let $(R_0,\mathfrak{m} _0)$ be a  \ local ring and let $R = \bigoplus_{n\geq 0}R_n$ be a
standard graded $R_0$-algebra and let $M = \bigoplus_{n\geq 0}M_n$ be a finitely generated
$R$-module. Assume $R$ has a homogeneous s.o.p. Set $c = \dim R - \dim M$.
Assume
\begin{enumerate}[ \rm (1)]
\item
$R_0$ is Cohen-Macaulay \ and $R$ is Cohen-Macaulay.
\item
$M$ is generated as an $R$-module by some elements in $M_0$.
\item
There exists
$y_1,\ldots,y_s \in R_0$  a system of parameters for both $M_0$ and $R_0$ and a  part of a homogeneous system of parameters for both $M$ and $R$.
\end{enumerate}
Then there exists a homogeneous regular sequence
  $\xi_1,\ldots,\xi_c \in R_+$ such that $\xi_i \in \operatorname{ann}_R M$ for each $i$.
\end{theorem}
\begin{proof}
We prove this by induction on $s = \dim R_0$.
When $s = 0$,  $R_0$ is Artinian. So $\operatorname{grade}(\operatorname{ann} M,R) = \operatorname{grade}(\operatorname{ann} M \cap R_+, R)$. As $R$ is CM \ we have $\operatorname{grade}(\operatorname{ann} M, R) = c$.
 The result  follows from \cite[1.5.11]{BH}.

We  prove the assertion for $s = r +1$
assuming it to be valid for $s=r$. Set $S = R/y_1R = \bigoplus_{n\geq 0} S_n$ and $N = M/y_1M$. Note that $N$ is also generated in degree $0$.  Let $\overline{y_i} = y_i \mod (y_1)$ with $2 \leq i \leq s$. Then $\overline{y_2} \ldots \overline{y_s}$
is a system of parameters for both $S_0$ and $N_0 = M_0/y_1M$. Notice $c = \dim R - \dim M = \dim S - \dim N$. Also note that $S$ has h.s.o.p. By induction hypothesis
there exists a $S$-regular sequence
  $\overline{\xi_1},\ldots,\overline{\xi_c} \in S_+$ such that $\overline{\xi_i} \in \operatorname{ann}_S N$ for each $i$.

$R$ is CM. So $y_1$ is  $R$-regular.
Therefore $y_1,\xi_1,\xi_2\ldots,\xi_c$ is an $R$-regular sequence.
Set $\mathfrak{q}  = y_1R_+$. Let $M$ be generated as an $R$-module by $u_1,\ldots,u_m \in M_0$.
Fix $i \in \{1,\dots,c \}$ and set $\xi = \xi_i$. By construction $\xi u_j \in y_1 M$ for each
 $j$. However $\xi \in R_+$ and $y_1 \in R_0$ . So $\xi u_j \in \mathfrak{q}  M$ for each
 $j$. By the determinant trick (observing that each $u_j$ has  degree zero)
 there exists a \textit{ homogeneous} polynomial
\[
\Delta(\xi) = \xi^{n} + a_{1}\xi^{n-1} + \ldots +a_{n-1}\xi + a_n \in \operatorname{ann}_R M \quad \text{and} \ a_i \in \mathfrak{q} ^i \ \text{for} \ i =1,\dots,n.
\]
Clearly $\Delta(\xi) \in R_+$.
Notice that $\Delta(\xi) = \xi^{n} \mod (y_1)$. So
$y_1,\Delta(\xi_1),\ldots,\Delta(\xi_c)$ is a regular sequence, \cite[16.1]{Ma}.
Therefore $ \Delta(\xi_1),\Delta(\xi_2)\ldots,\Delta(\xi_c)$
is an $R$-regular sequence. This proves the assertion when $s =r+1$.
\end{proof}


\textbf{CI approximation}

In this subsection we prove our general result regarding  CI-approximation.



\begin{theorem}\label{equiTT}
Let $(A,\mathfrak{m} )$ be a complete
 local ring and let $\mathfrak{a} $ be a proper  ideal in $A$ with $\dim A/\mathfrak{a}  + \mathfrak{s} (\mathfrak{a} )
 = \dim A$. Then
$A$ has a CI-approximation with respect to \ $\mathfrak{a} $. Furthermore if $\mathfrak{a}  = (x_1,\ldots,x_m)$ then we may
choose
a CI-approximation $[B,\mathfrak{n} ,\mathfrak{b} ,\varphi]$ of $[A,\mathfrak{m} ,\mathfrak{a} ]$ such that  $\mathfrak{b}  = (v_1,\ldots,v_m)$ with $\varphi(v_i) = x_i$ for  $i = 1,\ldots, m$.
\end{theorem}
\begin{proof}
Notice $G_{\mathfrak{a}}(A) $ has a homogeneous s.o.p (see proof of Proposition 2.6 in \cite{HUO}).
 Let $ y_1,\ldots,y_s \in A$ such that $ \overline{y_1},\ldots,\overline{y_s}$ (their images in $A/\mathfrak{a} $ ) is an
\emph{s.o.p} of $A/\mathfrak{a} $.

\textbf{Case 1.} \emph{A contains a field:}
 As $A$ is a complete,
$A$ contains  $k = A/\mathfrak{m} $.
  Set $S = k[[ Y_1,\ldots, Y_s, X_1,\ldots,X_n ]]$. Define
$\phi \colon S \rightarrow A$ which maps $Y_i$ to $y_i$ and $X_j$ to $x_j$ for $i= 1,\ldots, s$
and $j = 1,\ldots, n$.
 Set $\mathfrak{q}  = (Y_1,\ldots, Y_s)$.
Notice
\[
\frac{A}{(\mathfrak{q} , X_1,\ldots,X_n)A} = \frac{A}{(\mathfrak{a} , y_1,\ldots,y_s)} \quad \text{has finite length.}
\]
As $S$ is complete we get that $A$ is a finitely generated $S$-module, cf. \cite[8.4]{Ma}. Since
$\psi((\mathbf{X} ))A = \mathfrak{a} $,  we get  $G_{\mathfrak{a}}(A)  = G_{(\mathbf{X} )}(A)$ is a finitely generated
$G_{(\mathbf{X} )}(S)$-module.

Notice $G_{(\mathbf{X} )}(S) \cong
S/(\mathbf{X} )[X_{1}^{*},\ldots, X_{n}^{*}]$ and $S/(\mathbf{X} ) \cong k[| Y_1,\ldots, Y_s|]$.
Furthermore
 $G_{(\mathbf{X} )}(A) = G_{\mathfrak{a}}(A) $ is generated in degree zero as a $G_{(\mathbf{X} )}(S)$-module.
Set $c =  \dim S - \dim A  = \dim G_{(\mathbf{X} )}(S)  - \dim G_{\mathfrak{a}}(A)  $.  Notice  that $ Y_1,\ldots, Y_s$ is a s.o.p of  $G_{(\mathbf{X} )}(S)_0$ and $G_{\mathfrak{a}}(A) _0$.
We apply Theorem \ref{hann} to get $\xi_{1},\ldots,\xi_{c} \in (\mathbf{X} )  $ such that
 $\xi_{1}^{*}, \ldots, \xi_{c}^{*} \in G_{(\mathbf{X} )}(S)_+  $ is a \emph{regular sequence}   in  $G_{(\mathbf{X} )}(S)$ and
$\xi_{i}^{*} \in \operatorname{ann}_{G_{(\mathbf{X} )}(S)} G_{\mathfrak{a}}(A) $ for all $i$. We now apply  Lemma
\ref{annihilator} to get $u_1,\ldots, u_c  \in (\mathbf{X} )$ an $S$-regular sequence in
$\operatorname{ann}_S (A)$ such that $u_{1}^{*},\ldots, u_{c}^{*}$ is a $G_{(\mathbf{X} )}(S)$-regular
 sequence in $\operatorname{ann}_{G_{(\mathbf{X} )}(S)} G_{\mathfrak{a}}(A)  $.  Set $\mathbf{u} = (u_1,\ldots, u_c) $,   $B = S/\mathbf{u} $, $\mathfrak{b}  = (\mathbf{X}  +\mathbf{u})/\mathbf{u}$.  Let $\mathfrak{n} $ be the maximal ideal in $B$ and let $\overline{\phi} \colon B \rightarrow A$
be the map induced by $\phi \colon  S \rightarrow A$. Then it is clear that
$(B,\mathfrak{n} ,\mathfrak{b} , \overline{\phi})$ is a CI approximation of $A$ with respect to \ $\mathfrak{a} $. Set $v_i = \overline{X_i}$. Then $\mathfrak{b}  = (v_1,\ldots,v_m)$ with $\varphi(v_i) = x_i$ for  $i = 1,\ldots, m$.

\textbf{Case 2.} \textit{$A$ does not contain a field:} There exists a DVR $(D,\pi)$ and a local
ring homomorphism $\eta \colon D \rightarrow A$ such that $\eta$ induces an isomorphism
 $D/(\pi) \rightarrow A/\mathfrak{m} $.  Set $S = D[| Y_1,\ldots, Y_s, X_1,\ldots,X_n|]$. Define
$\phi \colon S \rightarrow A$, with  $\phi(d) =  \eta(d)$ for each $d \in D$ and maps  $Y_i$ to $y_i$ and $X_j$ to $x_j$ for $i= 1,\ldots, s$
and $j = 1,\ldots, n$.
 Set $\mathfrak{q}  = (Y_1,\ldots, Y_s)$.
Notice
\[
\frac{A}{(\mathfrak{q} , X_1,\ldots,X_n)A} = \frac{A}{(\mathfrak{a} , y_1,\ldots,y_s)} \quad \text{has finite length.}
\]
As $S$ is complete we get that $A$ is a finitely generated $S$-module.

Set $T = S/(\mathbf{X} ) = G_{\mathbf{X} }(S)_0$. Notice $\dim T$ is one more than that of $A/\mathfrak{a}  = G_{\mathfrak{a}}(A) _0$.
To deal with this situation we  use an argument from \cite{PuEuler}, which we repeat for the convenience of the reader. Let $\mathfrak{t}$ be the maximal ideal of $T$. Set $\overline{Y_i} = $ image of $Y_i$ in $T$. Set $V= A/\mathfrak{a} $.
Note that
\[
\mathfrak{t} = \sqrt{\operatorname{ann}_T(V/\overline{\mathbf{Y} } V)} = \sqrt{\operatorname{ann}_T(V) + (\overline{\mathbf{Y} })}.
\]
So there exists $\overline{f} \in \operatorname{ann}_T(V) \setminus (\overline{\mathbf{Y} })$ such
that $\overline{f},\overline{Y_1},\ldots,\overline{Y_s}$ is an s.o.p. of $T$. Since $T$ is CM; $f,\overline{Y_1},\ldots,\overline{Y_s}$ is a $T$-regular
sequence.
So $X_1,\ldots,X_n,f,Y_1,\ldots,Y_s$ is a $S$-regular
sequence. Since $\overline{f} \in \operatorname{ann}_T(A/\mathfrak{a} )$  we get $f A \subseteq \mathfrak{a}  A = (\mathbf{X} )A$. Using the
determinant trick
 there exists
\[
\Delta = f^m + \alpha_{1}f^{m-1} + \ldots + \alpha_{m-1}f + \alpha_{m} \in \operatorname{ann}_S A \quad \& \alpha_i \in (\mathbf{X} )^i; \ 1\leq i \leq m.
\]
 Notice that $\Delta = f^m \ \text{mod}( \mathbf{X} )$. Thus
 $X_1,\ldots,X_n, \Delta,Y_1,\ldots,Y_s$ is a $S$-regular
sequence. So $\Delta$ is $S$-regular. Set $U = S/( \Delta)$ and $\mathfrak{c}  = ((\mathbf{X} ) + ( \Delta))/( \Delta)$.
 Notice that $A$ is a finitely generated $U$-module and $G_{\mathfrak{c} }(A) = G_{\mathfrak{a}}(A) $. Furthermore
$G_{\mathfrak{c} }(U) \cong T/(f^m)[X_{1}^{*},\ldots, X_{n}^{*}]$ is  CI.

 Note that as a
 $G_{\mathfrak{c} }(U)$ module $G_{\mathfrak{a}}(A)  $ is generated in degree zero. Furthermore $\dim G_{\mathfrak{c} }(U)_0 = \dim G_{\mathfrak{a}}(A) _0$. Set $c =  \dim U - \dim A  = \dim G_{\mathfrak{c} }(U)  - \dim G_{\mathfrak{a}}(A)  $.
The subsequent argument is similar to  that of Case 1.
\end{proof}
\begin{remark}\label{high-red} It follows from the proof of Theorem \ref{equiTT} that if $\mathfrak{a} $ is $\mathfrak{m} $-primary there exists CI-approximations with arbitrary high reduction numbers. To see this note that
there exist a complete intersection $R$ (with $G(R)$ a complete intersection) and a   map $R \rightarrow A$ and elements $f_1,\cdots, f_m \in  \operatorname{ann}_R A$ such that $f_1^*,\cdots, f_m^* \in \operatorname{ann}_{G(R)} G_{\mathfrak{a}}(A) $ and $f_1^*,\cdots, f_m^*$ is $G(R)$-regular. Furthermore $B = R/(f_1,\cdots, f_m)$. Also note that $\deg f_i^* > 0$ for all $i$. We may simply take $B'$ to be
$R/(f_1^l,\cdots, f_m^l)$ for large $l$.

\end{remark}


\section{Some Preliminaries to prove Itoh's conjecture}\label{Lprop}
In \cite{Pu5} we introduced a new technique to investigate problems relating to associated graded modules.
In this section we collect all the relevant results which we proved in \cite{Pu5}. Throughout thus section
$(A,\mathfrak{m} )$ is a Noetherian local ring with infinite residue field, $M$ is a \emph{Cohen-Macaulay }\ module of dimension
$r \geq 1$ and $I$ is an $\mathfrak{m} $-primary ideal.

\s \label{mod-struc} Set $\mathcal{S}  = A[It]$;  the Rees Algebra of $I$. Set
$L^{I}(M) = \bigoplus_{n\geq 0}M/I^{n+1}M$. We note that $L^I(M)(-1) =  M[t]/\mathcal{R} (I, M)$.  So  $L^{I}(M)$ is a $\mathcal{S} $-module. Note that $L^I(M)$ is \emph{not} a finitely generated $\mathcal{S} $-module.

 \s Set $\mathfrak{M}  = \mathfrak{m} \oplus \mathcal{S} _+$. Let $H^{i}(-) = H^{i}_{\mathfrak{M} }$ denote the $i^{th}$-local cohomology functor with respect to \ $\mathfrak{M} $. Recall a graded $\mathcal{S} $-module $L$ is said to be
*-Artinian if
every descending chain of graded submodules of $L$ terminates. For example if $E$ is a finitely generated $\mathcal{S} $-module then $H^{i}(E)$ is *-Artinian for all
$i \geq 0$.

\begin{definition}(\cite[sect. 6]{HeL})
   Consider the following chain of submodules of
$M$:
\[
IM \subseteq (I^2M\colon_M I) \subseteq (I^3M\colon_M I^2) \subseteq \ldots \subseteq(I^{n+1}M \colon_M I^n)\subseteq \ldots
\]
As $M$ is Noetherian this chain stabilizes.
The stable value is denoted as $\widetilde{IM}$ and is called
 the \emph{Ratliff-Rush submodule of $M$ with respect to \ $I$}.
 The filtration
$\{\widetilde {I^nM}\}_{n \geq 1}$ is called the \textit{Ratliff-Rush filtration} of $M$
with respect to \ $I$.
\end{definition}

\s \label{RRdefn} It can be shown that if $\operatorname{depth} M >0$ then $\widetilde {I^nM} = I^nM$ for all $n \gg 0$.
Furthermore if $M = A$ then $\widetilde {I} \subseteq \overline{I}$,  where $\overline{I}$ is the integral closure of $I$. We say an ideal $I$ is Ratliff-Rush if $\widetilde {I} = I$. Note if $I$ is $\mathfrak{m} $-primary and depth of $A$ is positive then $\operatorname{depth} G_I(A) \geq 1$ if and only if  $I^n$ is Ratliff-Rush for all $n \geq 1$.

\s \label{zero-lc} In \cite[4.7]{Pu5} we proved that
\[
H^{0}(L^I(M)) = \bigoplus_{n\geq 0} \frac{\widetilde {I^{n+1}M}}{I^{n+1}M}.
\]
\s \label{Artin}
For $L^I(M)$ we proved that for $0 \leq i \leq  r - 1$
\begin{enumerate}[\rm (a)]
\item
$H^{i}(L^I(M))$ are  *-Artinian; see \cite[4.4]{Pu5}.
\item
$H^{i}(L^I(M))_n = 0$ for all $n \gg 0$; see \cite[1.10 ]{Pu5}.
\item
 $H^{i}(L^I(M))_n$  has finite length
for all $n \in \mathbb{Z}$; see \cite[6.4]{Pu5}.
\item
$\ell(H^{i}(L^I(M))_n)$  coincides with a polynomial for all $n \ll 0$; see \cite[6.4]{Pu5}.
\end{enumerate}

\s \label{I-FES} The natural maps $0\rightarrow I^nM/I^{n+1}M \rightarrow M/I^{n+1}M \rightarrow M/I^nM \rightarrow 0 $ induce an exact
sequence of $\mathcal{S} $-modules
\begin{equation}
\label{dag}
0 \longrightarrow G_{I}(M) \longrightarrow L^I(M) \xrightarrow{\Pi} L^I(M)(-1) \longrightarrow 0.
\end{equation}
We call (\ref{dag}) \emph{the first fundamental exact sequence}.  We use (\ref{dag}) also to relate the local cohomology of $G_I(M)$ and $L^I(M)$.

\s \label{II-FES} Let $x$ be  $M$-superficial with respect to \ $I$ and set  $N = M/xM$ and $u =xt \in \mathcal{S} _1$. Notice $L^I(M)/u L^I(M) = L^I(N)$.
For each $n \geq 1$ we have the following exact sequence of $A$-modules:
\begin{align*}
0 \longrightarrow \frac{I^{n+1}M\colon x}{I^nM} \longrightarrow \frac{M}{I^nM} &\xrightarrow{\psi_n} \frac{M}{I^{n+1}M} \longrightarrow \frac{N}{I^{n+1}N} \longrightarrow 0, \\
\text{where} \quad \psi_n(m + I^nM) &= xm + I^{n+1}M.
\end{align*}
This sequence induces the following  exact sequence of $\mathcal{S} $-modules:
\begin{equation}
\label{dagg}
0 \longrightarrow \mathcal{B} ^{I}(x,M) \longrightarrow L^{I}(M)(-1)\xrightarrow{\Psi_u} L^{I}(M) \xrightarrow{\rho^x}  L^{I}(N)\longrightarrow 0,
\end{equation}
where $\Psi_u$ is left multiplication by $u$ and
\[
\mathcal{B} ^{I}(x,M) = \bigoplus_{n \geq 0}\frac{(I^{n+1}M\colon_M x)}{I^nM}.
\]
We call (\ref{dagg}) the \emph{second fundamental exact sequence. }

\s \label{long-mod} Notice  $\ell\left(\mathcal{B} ^{I}(x,M) \right) < \infty$. A standard trick yields the following long exact sequence connecting
the local cohomology of $L^I(M)$ and
$L^I(N)$:
\begin{equation}
\label{longH}
\begin{split}
0 \longrightarrow \mathcal{B} ^{I}(x,M) &\longrightarrow H^{0}(L^{I}(M))(-1) \longrightarrow H^{0}(L^{I}(M)) \longrightarrow H^{0}(L^{I}(N)) \\
                  &\longrightarrow H^{1}(L^{I}(M))(-1) \longrightarrow H^{1}(L^{I}(M)) \longrightarrow H^{1}(L^{I}(N)) \\
                 & \cdots \cdots \\
               \end{split}
\end{equation}

\s \label{Artin-vanish} We will use the following well-known result regarding *-Artinian modules quite often:

Let $V$ be a *-Artinian $\mathcal{S} $-module.
\begin{enumerate}[\rm (a)]
\item
$V_n = 0$ for all $n \gg 0$
\item
If $\psi \colon V(-1) \rightarrow V$ is a monomorphism then $V = 0$.
\item
If $\phi \colon V \rightarrow V(-1)$ is a monomorphism then $V = 0$.
\end{enumerate}



 \s \label{power-of-I} Set $L = L^I(M)$,
 \begin{align*}
\xi_I(M) &:= \underset{0 \leq i \leq r-1}\min\{ \ i \ \mid H^{i}(L)_{-1} \neq 0 \ \text{or} \ \ell(H^{i}(L)) = \infty \}.\\
\operatorname{amp}_I(M) &:= \max\{\ |n| \ \mid H^{i}(L)_{n-1} \neq 0 \  \text{for} \ i = 0,\ldots, \xi_I(M) - 1 \}.
\end{align*}
In \cite[7.5]{Pu5} we showed that
\[
 \operatorname{depth} G_{I^l}(M) = \xi_I(M) \ \text{for all} \ l > \operatorname{amp}_I(M).
\]

\s \label{normal}
Recall an ideal $I$ is said to be asymptotically normal if $I^n$ is integrally closed for all $n \gg 0$.
If $I$ is a asymptotically normal  $\mathfrak{m} $-primary ideal  and $\dim A \geq 2$ then by a result of Huckaba and Huneke \cite[3.8]{HH}, $\operatorname{depth} G_{I^l}(A) \geq 2$
for all $l \gg 0$(also see \cite[7.3]{Pu5}). It  also follows from \cite[9.2]{Pu5} that in this case $H^1(L)_n = 0$ for $n < 0$. In particular $\ell(H^1(L)) < \infty$ (here $L = L^I(M)$).

\s \label{asympCM} We will also need the following fact for general filtration's (and was proved for $I$-adic filtration's in \cite[9.2]{Pu5}). Let $\EuScript{F} = \{ M_n \}_{n \in \mathbb{Z} }$ be an $I$-stable filtration with $M_n = M$ for $n \leq 0$. If the associated graded module of the Veronese $G(\EuScript{F}^{< c >}, M)$ has depth $\geq 2$ for some $c \geq 1$ then
$H^1(G(\EuScript{F}, M))_n = 0$ for $n < 0$.
To see this set $L(\EuScript{F}, M)= \bigoplus_{n \geq 0} M/M_{n+1}$. Then as before $L(\EuScript{F}, M)$ is a $\mathcal{S} $-module.
We note that for all $l \geq 1$ we have
\[
\left(L(\EuScript{F},M)(-1)\right)^{<l>} = \bigoplus_{n \geq 0} M/M_{nl} =  L(\EuScript{F}^{<l>},M)(-1)
\]
As $\operatorname{depth} G(\EuScript{F}^{< c >}, M) \geq 2$ it follows from an analogue of \ref{Artin}; \ref{dag} for filtration's and \ref{Artin-vanish} that
$H^i(L(\EuScript{F}^{<c>},M)) = 0$ for $i = 0,1$. As the Veronese functor commutes with local cohomology we get that
$H^i(L(\EuScript{F}, M))_{nc-1} = 0$ for all $n \in \mathbb{Z} $. In particular $H^1(L(\EuScript{F}, M))_{-1} = 0$. Let $x$ be $M$-superficial with respect to \ $\EuScript{F}$.
Set $N = M/xM$. Then by an analogue of \ref{longH} to filtration's we have  an exact sequence for all $n \in \mathbb{Z} $
\[
H^0(L(\overline{\EuScript{F}}, N))_n \rightarrow H^1(L(\EuScript{F}, M))_{n-1} \rightarrow H^1(L(\EuScript{F}, M))_{n}
\]
As $H^0(L(\overline{\EuScript{F}}, N))_n = 0$ for $n < 0$ and $H^1(L(\EuScript{F}, M))_{-1} = 0$;  an easy induction yield's $H^1(L(\EuScript{F}, M))_{n} = 0$ for $n < 0$.
By the exact sequence \ref{dag} and as $H^0(L(\EuScript{F}, M))_n = 0$ for $n < 0$ we get that $H^1(G(\EuScript{F}, M))_n = 0$ for $n < 0$.
\section{ Proof of Itoh's-conjecture for normal ideals}
In this section we give a proof of Itoh's conjecture for normal ideals.
\begin{theorem}\label{itoh}
 Let $(A,\mathfrak{m} )$ be a Cohen-Macaulay \  local ring of dimension $d \geq 1$ and let $\mathfrak{a} $ be a normal $\mathfrak{m} $-primary ideal.
 If $e_3^\mathfrak{a} (A) = 0$ then $G_\mathfrak{a} (A)$ is Cohen-Macaulay.
\end{theorem}

\begin{remark}\label{r1-itoh}
The following assertion is well-known.
 If $\mathfrak{a} ^n$ is integrally closed for all $n$ then $\mathfrak{a} ^n$ is Ratliff-Rush for all $n \geq 1$.
 It follows that $\operatorname{depth} G_\mathfrak{a} (A) \geq 1$, see \ref{RRdefn}.
\end{remark}

We first show that
\begin{lemma}\label{small}
 Theorem \ref{itoh} holds  if $\dim A = 1, 2$.
 \end{lemma}
 This Lemma is certainly well-known to the experts. However we provide a proof
 for the convenience of the reader.
\begin{proof}[Proof of Lemma \ref{small}]
 If $d = 1$ then by \ref{r1-itoh} the result holds.

 Now assume $d = 2$. Note that we may assume that $A$ has an infinite residue field. Let $\mathfrak{c} $ be a minimal reduction of $\mathfrak{a} $.
 Set $\sigma_i = \ell(\mathfrak{a} ^{i+1}/\mathfrak{c}  \mathfrak{a} ^i)$. Then  as $\operatorname{depth} G_\mathfrak{a} (A)$ is positive, it follows from a result of Huneke \cite[2.4]{Hun}
 that the $h$-polynomial of $A$ with respect to \ $\mathfrak{a} $ is
 \[
  h(z) =  \ell(A/\mathfrak{a} ) + (\sigma_0 - \sigma_1)z + (\sigma_1 - \sigma_2)z^2 + \cdots + (\sigma_{s-2} - \sigma_{s-1})z^{s-1} + \sigma_{s-1} z^s.
 \]
It follows that
\begin{align*}
 e_1^\mathfrak{a} (A) &= \sum_{i \geq 0} \sigma_i, \\
 e_2^\mathfrak{a} (A) &= \sum_{i \geq 1} i \sigma_i, \text{and} \\
 e_3^\mathfrak{a} (A) &= \sum_{i \geq 2} \binom{i}{2} \sigma_i.
 \end{align*}
As $e_3^\mathfrak{a} (A) = 0$ we have $\sigma_2 = 0$. So we have $\mathfrak{a} ^3 = \mathfrak{c}  \mathfrak{a} ^2$. As $\mathfrak{a} $ is integrally closed  we also have
$\mathfrak{a} ^2 \cap \mathfrak{c}  = \mathfrak{c}  \mathfrak{a} $, see \cite[Theorem 4.7 ]{Hun} (for rings containing a field) and \cite[Theorem 1]{It}. The result follows from Valabrega-Valla Theorem
\cite[2.3]{VV}.
\end{proof}
\s Recall that an ideal $\mathfrak{a} $ is said to be \textit{asymptotically normal} if $\mathfrak{a} ^n$ is integrally closed for all $n \gg 0$.
The crucial result to prove Itoh's conjecture is the following:
\begin{lemma}\label{crucial-itoh}
Let $(A,\mathfrak{m} )$ be a Cohen-Macaulay \    local ring of dimension $3$ and let $\mathfrak{a} $ be an asymptotically  normal $\mathfrak{m} $-primary ideal with $e_3^\mathfrak{a} (A) = 0$. Set $L^\mathfrak{a} (A) = \bigoplus_{n \geq 0}A/\mathfrak{a} ^{n+1}$ considered as a  module over the Rees algebra $\mathcal{S}  = A[\mathfrak{a}  t]$. Let $\mathfrak{M}  = \mathfrak{m} \oplus \mathcal{S} _+$ be the maximal homogeneous ideal of $\mathcal{S} $. Then the local cohomology modules $H^i_\mathfrak{M} (L^\mathfrak{a} (A))$ vanish for $i = 1, 2$
\end{lemma}
We will also need to extend Lemma \ref{crucial-itoh} to dimensions $d \geq 4$.
\s \emph{Remark and a Convention:} Note that all the relevant graded modules considered upto Lemma \ref{rachel-itoh}  below are modules over the Rees algebra $\mathcal{S}  = A[\mathfrak{a}  t]$. Also all local cohomology will
taken over $\mathfrak{M}  = \mathfrak{m} \oplus \mathcal{S} _+$  the maximal homogeneous ideal of $\mathcal{S} $.
We also note that if $x$ is $A$-superficial with respect to \ $A$ then the Rees algebra $\mathcal{S} ^\prime  = A/(x)[\mathfrak{a} /(x)t]$ is a quotient of $\mathcal{S} $. As we are only interested in vanishing of certain local-cohomology modules, by the independence theorem of local cohomology
it does not matter if we take local cohomology of an $\mathcal{S} ^\prime$-module with respect to \ $\mathfrak{M} ^\prime$ or over $\mathfrak{M} $ (and considering the module in question as an $\mathcal{S} $-module). So throughout we will only write $H^i(-)$ to mean $H^i_\mathfrak{M} (-)$.

\begin{lemma}\label{crucial-itoh-2}
Let $(A,\mathfrak{m} )$ be a Cohen-Macaulay \ local ring of dimension $d \geq 3$ and let $\mathfrak{a} $ be an asymptotically  normal $\mathfrak{m} $-primary ideal with $e_3^\mathfrak{a} (A) = 0$.  Then the local cohomology modules $H^i(L^\mathfrak{a} (A))$ vanish for $1 \leq i \leq d -1$.
\end{lemma}
\begin{proof}
We prove the result by induction on $d \geq 3$.
For $d = 3$ the result follows from Lemma \ref{crucial-itoh}. Now assume $d \geq 4$ and the result has been proved for $d-1$.
After passing through a general extension (see \ref{AtoA'}(d)) we may choose $x$, an $A$-superficial element
with respect to  \ $\mathfrak{a} $ such that in the ring $B = A/(x)$ the ideal $\mathfrak{b}  = \mathfrak{a} /(x)$ is asymptotically normal. Furthermore note that $e_3^\mathfrak{b} (B) = 0$.
Set $L = L^\mathfrak{a} (A)$ and $\overline{L} = L^\mathfrak{b} (B)$. By induction hypothesis we have $H^i(\overline{L}) = 0$ for $1 \leq i \leq d-2$. By \ref{longH} we get a surjective map
$H^1(L)(-1) \rightarrow H^1(L)$ and for $2 \leq i \leq d-1$ injections
$H^i(L)(-1) \rightarrow H^i(L)$. By \ref{Artin-vanish} it follows that $H^i(L) = 0$ for $i = 2,\cdots d - 1$. By \ref{normal} we get that $H^1(L)$ has finite length. So the surjection
$H^1(L)(-1) \rightarrow H^1(L)$ induces an isomorphism $H^1(L)(-1) \cong H^1(L)$ and this forces $H^1(L) = 0$.
\end{proof}

We now give a proof of Theorem \ref{itoh} assuming Lemma \ref{crucial-itoh}.
\begin{proof}[Proof of Theorem \ref{itoh}]
If $d = \dim A = 1,2$, the result follows from Lemma \ref{small}. If $d \geq 3$
then by Lemma \ref{crucial-itoh-2} we get $H^i(L^\mathfrak{a} (A)) = 0$  for $1 \leq i \leq d -1$. Also as $\mathfrak{a} $ is normal,  in-particular $\mathfrak{a} ^n$ is Ratliff-Rush for all
$n \geq 1$. So by \ref{zero-lc} we get $H^0(L^\mathfrak{a} (A)) = 0$. By taking cohomology of the first fundamental sequence \ref{dag} we get that $H^i(G_\mathfrak{a} (A)) = 0$
for  $0 \leq i \leq d -1$. Thus $G_\mathfrak{a} (A)$ is Cohen-Macaulay.
\end{proof}

\s \label{red-itoh} Thus to prove Itoh's conjecture all we have to do is to prove Lemma \ref{crucial-itoh}. This requires several preparatory results.
\emph{For the rest of the section we will assume $\dim A = 3$ and $\mathfrak{a} $ is an asymptotically normal ideal with $e_3^\mathfrak{a} (A) = 0$}. We will also assume that the residue field of $A$ is infinite.
We first show
\begin{lemma}\label{asymp-itoh}
(with hypotheses as in \ref{red-itoh}.) Then $G_{\mathfrak{a} ^n}(A)$ is Cohen-Macaulay \ for $n \gg 0$
and $\operatorname{red}(\mathfrak{a} ^n) \leq 2$ for $n \gg 0$.
\end{lemma}
\begin{remark}
Lemma \ref{asymp-itoh} is certainly known to the experts. We provide a proof for the convenience of the reader.
\end{remark}
\begin{proof}[Proof of Lemma \ref{asymp-itoh}]
Let $\mathfrak{c}  = (x_1, x_2, x_3)$ be a minimal reduction of $\mathfrak{a} $.
Notice that for all $n \geq 1$ we have $e_3^{\mathfrak{a} ^n}(A) = e_3^\mathfrak{a} (A) = 0$. Furthermore $\mathfrak{c} ^{[n]} = (x_1^n, x_2^n, x_3^n)$ is  a minimal reduction of $\mathfrak{a} ^n$.
By a result of Huckaba and Huneke \cite[3.8]{HH}  we have $\operatorname{depth} G_{\mathfrak{a} ^n}(A) \geq 2$
for all $n \gg 0$, say $n \geq n_0$. As $\mathfrak{a} $ is asymptotically normal we may also assume $\mathfrak{a} ^n$ is integrally closed for all $n \geq n_0$

Fix $n \geq n_0$.  Set $\sigma_i = \ell( (\mathfrak{a} ^n)^{i+1}/ \mathfrak{c} ^{[n]} (\mathfrak{a} ^n)^{i})$. Then by a result of Marley \cite[3.9]{Mar}, it follows that
that the $h$-polynomial of $A$ with respect to \ $\mathfrak{a} ^n$ is
 \[
  h(z) =  \ell(A/\mathfrak{a} ^n) + (\sigma_0 - \sigma_1)z + (\sigma_1 - \sigma_2)z^2 + \cdots + (\sigma_{s-2} - \sigma_{s-1})z^{s-1} + \sigma_{s-1} z^s.
 \]
It follows that
\begin{align*}
 e_1^{\mathfrak{a} ^n}(A) &= \sum_{i \geq 0} \sigma_i, \\
 e_2^{\mathfrak{a} ^n}(A) &= \sum_{i \geq 1} i \sigma_i, \text{and} \\
 e_3^{\mathfrak{a} ^n}(A) &= \sum_{i \geq 2} \binom{i}{2} \sigma_i.
 \end{align*}
As $e_3^{\mathfrak{a} ^n}(A) = 0$ we have $\sigma_2 = 0$. So we have $(\mathfrak{a} ^n)^3 = \mathfrak{c} ^{[n]} (\mathfrak{a} ^n)^2$. As $\mathfrak{a} ^n$ is integrally closed  we also have
$(\mathfrak{a} ^n)^2 \cap \mathfrak{c} ^{[n]} = \mathfrak{c} ^{[n]} \mathfrak{a} ^n$.  So by Valabrega-Valla Theorem \cite[2.3]{VV} we have that $G_{\mathfrak{a} ^n}(A)$ is Cohen-Macaulay.
\end{proof}
Next we show
\begin{lemma}\label{rachel-itoh}
(with hypotheses as in \ref{red-itoh}.) We have
\begin{enumerate}[ \rm(1)]
\item
$a(G_\mathfrak{a} (A)) < 0$.
\item
$H^2(L^\mathfrak{a} (A)) = 0$.
\item
$\sum_{i = 0}^{2}(-1)^i\ell(H^i(G_\mathfrak{a} (A)))  = 0.$
\item
For $i = 0, 1, 2$ we have $H^i(G_\mathfrak{a} (A))_n = 0$ for $n < 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
 (1) We have $\operatorname{red}_{\mathfrak{c} ^{[n]}}(\mathfrak{a} ^n) \leq 2$ for all $n \gg 0$. By a result of Hoa, \cite[2.6]{Hoa}, the result follows.

 (2) Set $G = G_\mathfrak{a} (A)$ and $L = L^\mathfrak{a} (A)$. Also let $S = A[\mathfrak{a}  t]$ be the Rees algebra of $A$ with respect to \ $\mathfrak{a} $ and let $\mathfrak{M} $ be its
 maximal homogeneous ideal. Throughout let $H^i(-) = H^i_\mathfrak{M} (-)$.
 The first fundamental exact sequence $0 \rightarrow G \rightarrow L \rightarrow L(-1) \rightarrow 0$ yields an exact sequence in cohomology
 \[
 H^2(L)_n \rightarrow H^2(L)_{n-1} \rightarrow H^3(G)_n.
 \]
As $a(G) < 0$ we get for $n \geq 0$, surjections $H^2(L)_{n} \rightarrow H^2(L)_{n-1}$. As $H^2(L)_n = 0$ for $n \gg 0$ we get
$H^2(L)_n = 0$ for $n \geq -1$.

After passing through a general extension we may choose $x$, an $A$-superficial element
with respect to  \ $\mathfrak{a} $ such that in the ring $B = A/(x)$ the ideal $\mathfrak{b}  = \mathfrak{a} /(x)$ is asymptotically normal. Also notice $\dim B = 2$.
Set $\overline{L} = L^\mathfrak{b} (B)$. By \cite[9.2]{Pu5} we get that $H^1(\overline{L})_n = 0$ for $n < 0$.
By \ref{longH} we have an exact sequence
\[
 H^1(\overline{L})_n \rightarrow H^2(L)_{n-1} \rightarrow H^2(L)_n
\]
By setting $n = -1$ we get $H^2(L)_{-2} = 0$. Iterating we get $H^2(L)_n = 0$ for all $n \leq -2$.

It follows that $H^2(L) = 0$.

(3) We first note that as $G_{\mathfrak{a} ^n}(A)$ is Cohen-Macaulay \ for all $n \gg 0$ the ring $G_\mathfrak{a} (A)$ is generalized Cohen-Macaulay, see \cite[7.8]{Pu5}. We also have $H^0(L)$
and $H^1(L)$ have finite length (see \ref{zero-lc} and \ref{normal}).

The first fundamental exact sequence $0 \rightarrow G \rightarrow L \rightarrow L(-1) \rightarrow 0$ yields an exact sequence in cohomology
 \begin{align*}
  0 &\rightarrow H^0(G) \rightarrow H^0(L) \rightarrow H^0(L)(-1) \\
  &\rightarrow H^1(G) \rightarrow H^1(L) \rightarrow H^1(L)(-1) \\
  &\rightarrow H^2(G) \rightarrow H^2(L) = 0.
 \end{align*}
 Taking lengths, the result follows.

 (4) As $\mathfrak{a} $ is asymptotically normal we have that $H^1(L)_n = 0$ for $n < 0$ (see \cite[9.2]{Pu5}). Also by \ref{zero-lc} we get $H^0(L)_n = 0$
 for $n < 0$. The result follows from the above exact sequence in cohomology.
 \end{proof}

\begin{remark}\label{apply-gor-approx}
 Till now we have not used our theory of complete intersection approximation. We do it now.
 We first complete $A$. Let $\mathfrak{a}  = (a_1,\ldots, a_n)$.  We take a general extension $A' = A[X_1,\ldots, X_n]_{\mathfrak{m}  A[X_1,\ldots, X_n]}$. Let $y = \sum_{i=1}^{n}a_iX_i$.
 Then the ideal $J = \mathfrak{a} '/(y)$ is asymptotically normal. Set $B = A'/(y)$. By a result of Huckaba and Huneke we get that there exists $c$ such that $G_{J^n}(B)$ is Cohen-Macaulay \ for $n \geq c$.  We take a CI-approximation $(R,\mathfrak{n} ,\mathfrak{b} ,\psi)$ of $A$ with respect to \ $\mathfrak{a} $ (note \emph{not of} $A'$). By \ref{high-red} we may assume $\operatorname{red}(\mathfrak{b} , R) \geq c+2$. By construction note that $\mathfrak{b} $ is generated by $b_1,\ldots,b_n$ and $\psi(b_i) = a_i$ for all $i$. Now set $R' = R[X_1,\ldots, X_n]_{\mathfrak{n}  R[X_1,\ldots, X_n]}$. By \ref{lying-above} we get that $A' = A\otimes_R R'$. We note that $(R',\mathfrak{n} ',\mathfrak{b} ',\psi')$ is a CI-approximation of $(A',\mathfrak{m} ',\mathfrak{a} ')$. Set $z = \sum_{i=1}^{n}b_iX_i$. Then note $\psi'(z) = y$.
Also note that $z$ is $R'$-superficial with respect to \ $\mathfrak{b} '$, see \cite[2.6]{Ciu} . We now complete $R'$ with respect to \ $\mathfrak{n} '$.
 Thus we may assume that our ring $(A,\mathfrak{m} , \mathfrak{a} )$ has a CI-approximation $[T,\mathfrak{t} ,\mathfrak{q} ,\psi]$ such that
 \begin{enumerate}
 \item Both $A$ and $T$ are complete.
   \item reduction number of $\mathfrak{q} $ is $\geq c + 2$.
   \item there exists $z \in \mathfrak{q} $ which is $A$-superficial with respect to \ $\mathfrak{a} $ such that \\ $G_{\mathfrak{a} ^n}(A/zA)$ is Cohen-Macaulay \ for all $n \geq c$. Furthermore $z$ is $G_\mathfrak{q} (T)$-regular.
   \item $A$ has a canonical module $\omega_A$.
   \end{enumerate}

   Let $\mathcal{R} _\mathfrak{q} (T)$ be the extended Rees-algebra of $T$ with respect to \ $\mathfrak{q} $.  Let $\mathfrak{N} $ be the $*$-maximal ideal of $\mathcal{R} _\mathfrak{q} (T)$.
\end{remark}
We first show
\begin{lemma}\label{gcm}
 (with hypotheses as in \ref{apply-gor-approx}.) We have
 \begin{enumerate}[\rm (1)]
  \item For any prime $P$ in $\mathcal{R} _\mathfrak{q} (T)$ with $\operatorname{ht} P \leq 3$ we have $\mathcal{R} _\mathfrak{a} (A)_P$ is Cohen-Macaulay.
  \item
  $H^i_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))$ has finite length for $0 \leq i \leq 3$.
 \end{enumerate}
\end{lemma}
\begin{proof}
 As $\mathcal{R} _\mathfrak{q} (T)$ is a Gorenstein ring the assertion (2) follows from (1) by local duality.

 (1) We first consider the case $t^{-1} \notin P$. Then $(\mathcal{R} _\mathfrak{a} (A))_P$ is a localization of
 $(R_\mathfrak{a} (A))_{t^{-1}} = A[t,t^{-1}]$ which is Cohen-Macaulay.

 Next consider the case when $t^{-1} \in P$. Set $Q = P/(t^{-1})$ a prime ideal of height $\leq 2$ in $G_\mathfrak{q} (T)$. As
 $G_\mathfrak{a} (A)$ is generalized Cohen-Macaulay \ and since $G_\mathfrak{q} (T)$ is Gorenstein, by local duality we have that $(G_\mathfrak{a} (A))_Q$ is Cohen-Macaulay.
 It follows that $(\mathcal{R} _\mathfrak{a} (A))_P$ is Cohen-Macaulay.
\end{proof}
As a consequence we get
\begin{corollary}\label{exact-loc}
(with hypotheses as in \ref{apply-gor-approx}.) We have
\begin{enumerate}[\rm (1)]
  \item $H^3_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A)) = 0$
  \item an exact sequence
\[
 0 \rightarrow H^3_\mathfrak{N} (G_\mathfrak{a} (A)) \rightarrow H^4_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))(+1) \xrightarrow{t^{-1}} H^4_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A)) \rightarrow 0.
\]
  \item $\operatorname{Ext}^1_{\mathcal{R} _\mathfrak{q} (T)}(\mathcal{R} _\mathfrak{a} (A),\mathcal{R} _\mathfrak{q} (T)) = 0$.
\end{enumerate}
\end{corollary}
\begin{proof}
 The short exact sequence $0 \rightarrow \mathcal{R} _\mathfrak{a} (+1)  \xrightarrow{t^{-1}} H^4_\mathfrak{N} (\mathcal{R} _\mathfrak{a} )  \rightarrow G_\mathfrak{a} (A) \rightarrow 0$, yields an exact sequence
 of local cohomology modules
 \begin{align*}
  0 &\rightarrow H^0_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))(+1) \xrightarrow{t^{-1}} H^0_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A)) \rightarrow  H^0_\mathfrak{N} (G_\mathfrak{a} (A)) \\
  &\rightarrow H^1_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))(+1) \xrightarrow{t^{-1}} H^1_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A)) \rightarrow  H^1_\mathfrak{N} (G_\mathfrak{a} (A)) \\
   &\rightarrow H^2_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))(+1) \xrightarrow{t^{-1}} H^2_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A)) \xrightarrow{\pi}  H^2_\mathfrak{N} (G_\mathfrak{a} (A)) \\
   &\rightarrow H^3_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))(+1) \xrightarrow{t^{-1}} H^3_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))
 \end{align*}
Let $K$ be the cokernel of $\pi$. We note that $H^i_\mathfrak{N} (G_\mathfrak{a} (A)) \cong H^i(G_\mathfrak{a} (A))$. As all the module in the above exact sequence have finite length
we have
\[
 \ell(K) = \sum_{i = 0}^{2} (-1)^i\ell(H^i(G_\mathfrak{a} (A))  = 0 \quad \text{by Lemma \ref{rachel-itoh}(3)}.
\]
Thus we have an inclusion
\[
 H^3_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))(+1) \xrightarrow{t^{-1}} H^3_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))
\]
As $\ell(H^3_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))) $ is finite we have an isomorphism
$H^3_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))(+1) \cong H^3_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))$. Again as $\ell(H^3_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))) $ is finite this implies
that $H^3_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A)) = 0$.  So (1), (2) follows. Also by local duality we have (3).
\end{proof}
The following result is a crucial ingredient in proving Lemma \ref{crucial-itoh}.
\begin{theorem}\label{ing}(with hypotheses as in \ref{apply-gor-approx}.)
 Let $E_G$ be the injective hull of $k$ considered as a
 $G_\mathfrak{q} (T)$-module. Set $W = \operatorname{Hom}_{G_\mathfrak{q} (T)}(H^3(G_\mathfrak{a} (A)), E_G)$. Then
 \begin{enumerate}[\rm (1)]
  \item
 there exists an $\mathfrak{a} $-stable filtration $\EuScript{F}$ on
 $\omega_A$ such that $G_\EuScript{F}(\omega_A) \cong W[1]$
 \item
 $ \operatorname{depth} G_\EuScript{F}(\omega_A) \geq 2$. Furthermore $z$ is $G_\EuScript{F}(\omega_A)$-regular.
 \item
 $h_\EuScript{F}(z) = a_0 + a_1z + a_2 z^2$.
 \item
 Set $B = A/zA$. Let $\overline{\EuScript{F}}$ be the quotient filtration on $\omega_B = \omega_A/z\omega_A$. Then $H^1(G(\overline{\EuScript{F}},\omega_B)_n = 0$ for $n < 0$.
 \end{enumerate}
\end{theorem}
\begin{proof}
(1) Let $E_\mathcal{R} $ be the injective hull of $k$ considered as a $\mathcal{R} _\mathfrak{q} (T)$-module.  Set $(-)^\vee = \operatorname{Hom}_{\mathcal{R} _q(T)}(-, E_\mathcal{R} )$.
Dualizing the exact sequence in Lemma \ref{exact-loc} we get a sequence of $\mathcal{R} _\mathfrak{q} (T)$-modules
\[
 0 \rightarrow H^4_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))^\vee \xrightarrow{t^{-1}}  H^4_\mathfrak{N} (\mathcal{R} _\mathfrak{a} (A))^\vee(+1) \rightarrow W \rightarrow 0.
\]
As $R_\mathfrak{q} (T)$ is Gorenstein then by local duality we have an exact sequence
\[
 0 \rightarrow \operatorname{Hom}_{\mathcal{R} _q(T)}(\mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (T)) \xrightarrow{t^{-1}} \operatorname{Hom}_{\mathcal{R} _q(T)}(\mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (T))(+1) \rightarrow X \rightarrow 0
\]
where $X \cong W$ upto shift.
By Theorem \ref{main} there exists an $\mathfrak{q} $-stable filtration $\EuScript{F}$ on $\omega_A \cong \operatorname{Hom}_T(A,T)$ such that
\[
 \mathcal{R} (\EuScript{F}, \omega_A) \cong \operatorname{Hom}_{\mathcal{R} _q(T)}(\mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (T))
\]
Also notice that as $\mathfrak{q}  A = \mathfrak{a} $ it follows that $\EuScript{F}$ is in fact an $\mathfrak{a} $-stable  filtration on $\omega_A$. Note after a shift $G(\EuScript{F})= W[1]$.


(2)
We note that upto shift $W \cong \operatorname{Hom}_{G_\mathfrak{q} (T)}(G_\mathfrak{a} (A), G_\mathfrak{q} (T))$. The result follows.

(3) To prove this result we first \\
\emph{Claim:} The Hilbert series of $W = \bigoplus_{n \in \mathbb{Z} }W_n$ is
\[
H_W(z) = \sum_{n \in \mathbb{Z} }\ell(W_n)  = \frac{a_0z + a_1z^2 + a_2z^3}{(1-z)^3}.
\]
 Set $G = G_\mathfrak{a} (A)$.
Let $V = H^3(G)$. As $a(G) < 0$ we have that $V_n = 0$ for $n\geq 0$. So $W_m = 0$
for $m \leq 0$.
By Grothendieck-Serre formula we have
\[
H_\mathfrak{a} (n) - P_\mathfrak{a} (n) = \sum_{i = 0}^{3}(-1)^i \ell(H^i(G)_n).
\]
Where $H_\mathfrak{a} $ (and $P_\mathfrak{a} (z)$) are Hilbert function(and Hilbert polynomial) of $A$ with respect to \ $\mathfrak{a} $. As $H^i(G)_n = 0$ for $n < 0$ for $i = 0,1,2$ we get that
\[
\ell(V_n) = P_\mathfrak{a} (n) \quad \text{for all} \ n < 0.
\]
It follows that $\ell(W_m) = P_\mathfrak{a} (-m)$ for all $m \geq 1$. So $P_\mathfrak{a} (-z)$ is the Hilbert polynomial of $W$ (it should be remarked that $\deg(P_\mathfrak{a} (z)) = 2$).
As postulation number of $W$ is one  (and as $W_0 = 0$)  it follows from \cite[4.1.12]{BH} the Hilbert series of
$W$ is
\[
 \frac{a_0z + a_1z^2 + a_2z^3}{(1-z)^3}
\]
As $G_\EuScript{F}(A) \cong W[1]$ the result follows.

(4) We have an exact sequence
\[
0 \rightarrow \mathcal{R} _\mathfrak{q} (T)(-1) \xrightarrow{zt} \mathcal{R} _\mathfrak{q} (T) \rightarrow   \mathcal{R} _\mathfrak{q} (\overline{T}) \rightarrow 0.
\]
This yields an exact sequence
\begin{align*}
  0 \rightarrow \operatorname{Hom}_{\mathcal{R} _\mathfrak{q} (T)}(\mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (T))(-1) &\xrightarrow{zt} \operatorname{Hom}_{\mathcal{R} _\mathfrak{q} (T)}(\mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (T)) \rightarrow  \\
  \operatorname{Hom}_{\mathcal{R} _\mathfrak{q} (T)}(\mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (\overline{T})) &\rightarrow \operatorname{Ext}^1_{\mathcal{R} _\mathfrak{q} (T)}(\mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (T))(-1) \rightarrow.
\end{align*}
By \ref{exact-loc} we get that $\operatorname{Ext}^1_{\mathcal{R} _\mathfrak{q} (T)}(\mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (T)) = 0$.
We note that
\[
\operatorname{Hom}_{\mathcal{R} _\mathfrak{q} (T)}(\mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (\overline{T})) = \operatorname{Hom}_{\mathcal{R} _\mathfrak{q} (T)}(\mathcal{R} _\mathfrak{a} (A)/zt \mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (\overline{T})).
\]
Set  $\mathfrak{b}  = \mathfrak{a} /z A$. We have an exact sequence
\[
0 \rightarrow F = \bigoplus_{n \geq 1}\frac{\mathfrak{a} ^n \cap (z) }{ z\mathfrak{a} ^{n-1}} \rightarrow \mathcal{R} _\mathfrak{a} (A)/zt \mathcal{R} _\mathfrak{a} (A) \rightarrow \mathcal{R} _\mathfrak{b} (B) \rightarrow 0.
\]
As $z$ is $A$-superficial with respect to \ $\mathfrak{a} $  we get that $F$ has finite length. As $\mathcal{R} _\mathfrak{q} (\overline{T})$ is Cohen-Macaulay \  and has dimension $3$ we get
\[
\operatorname{Hom}_{\mathcal{R} _\mathfrak{q} (T)}(\mathcal{R} _\mathfrak{a} (A)/zt \mathcal{R} _\mathfrak{a} (A), \mathcal{R} _\mathfrak{q} (\overline{T})) \cong \operatorname{Hom}_{\mathcal{R} _\mathfrak{q} (T)}(\mathcal{R} _\mathfrak{b} (B), \mathcal{R} _\mathfrak{q} (\overline{T})).
\]
Thus we have an exact sequence
\[
0 \rightarrow \mathcal{R} (\EuScript{F}, \omega_A)(-1)\xrightarrow{zt} \mathcal{R} (\EuScript{F},\omega_A) \rightarrow \mathcal{R} (\EuScript{G},\omega_B) \rightarrow 0;
\]
where $\EuScript{G}$ is the dual filtration on $\omega_B$. Now by our construction $\mathcal{R} _{\mathfrak{b} ^n}(B)$ is Cohen-Macaulay \ for $n \geq c$. As $s = \operatorname{red}(\mathfrak{q} , T) = \operatorname{red}(\overline{\mathfrak{q} }, \overline{T}) \geq c + 2$
and $G_{\overline{\mathfrak{q} }}(\overline{T})$ is Gorenstein we get by \cite[Theorem 1]{Ooishi-gor-powers}  that $G_{\overline{\mathfrak{q} }^{s-1}}(\overline{T})$ is Gorenstein. It follows that $\mathcal{R} _{\mathfrak{q} ^{s-1}}(\overline{T})$ is Gorenstein.
It follows that $\mathcal{R} (\EuScript{G}^{<s-1>},\omega_B)$ is Cohen-Macaulay. The result now follows from \ref{asympCM}.
\end{proof}

We now give
\begin{proof}[Proof of Lemma \ref{crucial-itoh}]
By  Lemma \ref{rachel-itoh}(2) we get $H^2(L^\mathfrak{a} (A)) = 0$. By \ref{dag} we have an exact sequence
\[
H^1(L^\mathfrak{a} (A)) \rightarrow H^1(L^\mathfrak{a} (A))(-1) \rightarrow H^2(G_\mathfrak{a} (A)) \rightarrow H^2(L^\mathfrak{a} (A)) = 0
\]
As $H^1(L^\mathfrak{a} (A))_n = 0$ for $n < 0$ it follows that $H^2(G_\mathfrak{a} (A))_n = 0$ for $n \leq 0$. Note that to prove $ H^1(L^\mathfrak{a} (A)) = 0$ it suffices to show $H^2(G_\mathfrak{a} (A)) = 0$ (because
$H^1(L^\mathfrak{a} (A))$ has finite length).
Set $G = G_\mathfrak{a} (A)$ and $\overline{G} = G/ztG$. We have a exact sequence
\[
0 \rightarrow K \rightarrow G(-1) \xrightarrow{zt} G \rightarrow \overline{G} \rightarrow 0;
\]
where $K$ has finite length. Taking local cohomology we get an exact sequence
\begin{align*}
  H^2(G)(-1)&\xrightarrow{zt} H^2(G) \rightarrow H^2(\overline{G}) \\
  H^3(G)(-1)&\xrightarrow{zt} H^3(G) \rightarrow 0.
\end{align*}
Set $Y = H^2(G)/ztH^2(G)$. Note $Y_n = 0$ for $n \leq 0$. Taking Matlis dual's  we get an exact sequence
\[
0 \rightarrow H^3(G)^\vee \xrightarrow{zt} H^3(G)^\vee(1) \rightarrow H^2(\overline{G})^\vee \rightarrow Y^\vee \rightarrow 0.
\]
By \ref{ing} we get an exact sequence
\[
0 \rightarrow G(\overline{\EuScript{F}}, \omega_B)  \rightarrow H^2(\overline{G})^\vee \rightarrow Y^\vee \rightarrow 0.
\]
We note that $\operatorname{depth} H^2(\overline{G})^\vee \geq 2$ and $H^1(G(\overline{\EuScript{F}}, \omega_B))_n = 0$ for $n < 0$. So $Y^\vee_n = 0$ for $n < 0$.
It follows that $Y_n = 0$ for $n > 0$. As $Y_n = 0$ for $n \leq 0$ we get $Y = 0$. So $H^2(G)/zt H^2(G) = 0$. By graded Nakayama Lemma we get $H^2(G) = 0$. As discussed before this implies that $H^1(L^\mathfrak{a} (A)) = 0$.
\end{proof}













\emph{Acknowledgments:}
I thank  Lucho Avramov, Juergen Herzog, Srikanth Iyengar  and Jugal Verma for
 many useful discussions on the subject of this paper.

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