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\vskip 4ex
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\centerline{\bf  IDEALS HAVING A ONE-DIMENSIONAL}
\vskip 3ex
\centerline{\bf  FIBER CONE}
\vskip 4ex
\centerline{ Marco D'Anna, Anna Guerrieri and William Heinzer}

\vskip 3ex
\centerline{Dedicated to Jim Huckaba on the occasion of his retirement}
\vskip 1cm

\document
\baselineskip=19pt

\noindent
{\bf Abstract.} 
For a regular ideal $I$ having a principal reduction
in a Noetherian local ring $(R,\m)$ we consider 
properties of the powers of $I$ as reflected in
the fiber cone $F(I)$ and 
the associated graded ring $G(I)$ of $I$. In particular,
we examine the postulation number of $F(I)$ and compare it
with the reduction number of $I$, and the postulation
number of $G(I)$ when the latter is meaningful. We discuss
a sufficient condition for $F(I)$ to be Cohen--Macaulay and 
consider for a fixed $R$ what is possible for the reduction
number $r(I)$ of $I$ and the multiplicity of $F(I)$.


\vskip 0.4cm

\section{Introduction.}

\vskip 0.4cm

Given an ideal $I$ in a Noetherian local ring $(R,\m)$, 
information on properties of $I^n$ as $n$ grows 
is encoded in various graded rings related to $I$. These
graded rings are built by using and manipulating the 
$I$-adic filtration, $\{I^n\}_{n\ge 0}$ on $R$. Among these
rings are:

\noindent
(i)  
the {\it Rees algebra} of $I$,
$\de R[It] = \oplus_{n \ge 0}I^n$,

\noindent
(ii)  
the {\it associated graded ring} of $I$, 
$\de G(I) = \oplus_{n \ge 0}I^n/I^{n+1} = \oplus_{n \ge 0}G_n 
\cong R[It]/IR[It]$, 
and

\noindent
(iii)
the {\it fiber cone} of $I$, 
$\de F(I)  = \oplus_{n \ge 0}I^n/\m I^n = \oplus_{n \ge 0}F_n 
\cong  R[It]/\m R[It]$. 
 
These are a few of the  {\it blowup algebras} of $I$,
(see \cite{V1}),
where by this term one refers to those algebraic objects that 
are related to the concept of blowing up a variety along a subvariety.

The graded rings $G(I)$ and $F(I)$ are both {\it homogeneous, }or
{\it standard, } 
in the sense that they are generated by their forms of
degree one over their subring of elements of degree zero.
If $I$ is $\m$-primary, the Hilbert function giving the length,
$\la (I^n/I^{n+1})$,  of $I^n/I^{n+1}$ as an $R$-module
is $H_G(n)$, where $H_G(X)$ is  the Hilbert function 
of the associated graded ring  $G(I)$.
For an arbitrary ideal $I \subseteq \m$ of $(R,\m)$,
the fiber cone $F(I)$ has the attractive property of 
being a finitely generated graded ring over the 
residue field $K := R/\m$.
It is well known in this setting that 
the Hilbert function $H_F(n)$ 
giving the dimension of $I^n / \m I^n$ as a vector space
over $K$ is defined for $n$ sufficiently large by
a polynomial $h_F(X) \in \Q[X]$, the {\it Hilbert polynomial}
of $F(I)$  \cite{Mat, Corollary, page 95}, \cite{AM, Corollary 11.2}.
A simple application of Nakayama's lemma,
\cite{Mat, Theorem~2.2}, shows that 
the cardinality of a minimal set of generators of $I^n$,
$\mu(I^n)$, is equal to $\lambda(I^n/\m I^n)$, the Hilbert function
$H_F(n)$ of $F(I)$. For these reasons $G(I)$ and $F(I)$ are good 
objects to analyze and compare when studying the asymptotic 
properties of $I$.

If $I$ is $\m$-primary it is natural
to ask about the relationship of
the Hilbert function $H_G(X)$ and 
Hilbert polynomial $h_G(X)$ of $G(I)$
with the Hilbert function $H_F(X)$ and 
Hilbert polynomial $h_F(X)$ of $F(I)$. We begin
such an investigation here in the one-dimensional case.
We also consider,  with no restriction 
on the dimension of $R$,   properties of the fiber cone $F(I)$ of 
ideals $I$ that have principal reductions in $R$.

Thus, in certain aspects, this paper  is a continuation 
of our work in \cite{DGH}. If
$I$ is a regular ideal having a principal reduction, 
we analyze in \cite{DGH}
the mutual relations
among the following, where $\N$ denotes the nonnegative integers:

\roster
\item 
$r = r(I)
= \text{min}\{ n \in \N \ | \ I^{n+1}=xI^n \text{ for some x } \in I\}$, 
the {\it reduction number} of $I$,
\item
$k = k(I)
= \text{min}\{ n \in \N \ | \ \I= (I^{n+1}: I^n )\}$, 
the {\it Ratliff--Rush number} of $I$,
\item
$h = h(I)  =
\text{min}\{ n \in \N \ | \ \widetilde {I^m}
= I^m\ \text{for all}\ m\ge n\}$,
the {\it asymptotic Ratliff--Rush number} of $I$.
\endroster
In defining $k$ and $h$, we are using the
{\it Ratliff--Rush closure} of $I$ and of its powers,
namely
$ \de\widetilde{I^k} := \cup_{n \in \N}(I^{n+k}:_R I^n) $. 
This concept was introduced by L. J. Ratliff and D. E. Rush in \cite{RR}
where it was also observed that
$\widetilde {I^m} = I^m$ 
for all sufficiently large integers $m$ \cite{RR, Remark~(2.3)}. 
This motivates our
definition of $h$. We have $h(I) = 0$ if and only if 
$G(I)$ contains a regular homogeneous element of 
positive degree. If $h(I) > 0$, then $h(I) \ge 2$.

It is shown in \cite{RV} in the case of the maximal ideal of a 
one-dimensional local Noetherian ring and in a more general setting
in \cite{DGH, (2.1) and (2.2)} that if $I$ has a 
principal reduction, then $h \le r$ and $k \le \max\{0,r-1\}$. 
If $R$ is a reduced Noetherian ring with
total ring of fractions $Q(R)$ and if the integral
closure $\overline R$ of $R$ in $Q(R)$ is a finitely generated $R$-module,
it is shown in \cite{DGH, (3.2) and (3.10)} that if $I$ is contained
in the conductor of $\overline R$ into $R$, then 
$k = \max\{0, r-1\}$,  and, if $I \ne \widetilde{I}$,
then $h=r$. 

%With the same hypotheses, one also obtains 
%$\widetilde{I^{n+1}} \cap I^n = I^n \I$, for all $n \ge 0$
%see \cite{DGH, (3.6)}. 
%This gives an analogue for the Ratliff--Rush filtration 
%to the equalities shown by Itoh, \cite{I},
%and by Huneke, \cite{H1}, for the filtration of the integral closures 
%of the powers of $I$.
%Further, it is shown in \cite{DGH, (3.9)} that, in this context,
%the reduction number of the
%filtration $\{\widetilde{I^n}\}_{n \ge 0}$ is less than or 
%equal to $1$.

Suppose $I$ is an $\m$-primary ideal of a Noetherian 
local ring $(R,\m)$. The {\it postulation 
number} $n(I)$  of $I$ is the largest integer $n$ such that 
$H_G(n) \ne h_G(n)$, where $H_G(X)$ and $h_G(X)$ are,
respectively, the Hilbert function and Hilbert polynomial
of $G(I)$. Thus $H_G(n)$ is the length of $I^n / I^{n+1}$ as an
$R$-module. T. Marley shows in \cite{Mar, Theorem 2} 
that if $(R,\m)$ is a $d$-dimensional Cohen--Macaulay local ring
with infinite residue field and $I$ is an $\m$-primary ideal
such that $G(I)$ has depth at least $d-1$, then
$r(I) = n(I) + d$. In the case where $I = \m$ 
this had been  shown by J. Sally  \cite{Sa, Proposition~3}. 
Note that the condition on the depth of 
$G(I)$ is vacuous if $d =1$, a case also considered by
A. Ooishi in \cite{O, Proposition 4.10}. Thus if $I$ is an
$\m$-primary ideal in a one-dimensional Cohen--Macaulay local
ring $(R,\m)$ with $R/\m$ infinite, then $n(I) = r(I) - 1$.

We define the {\it fiber postulation
number} $fp(I)$ of $I$ to be the largest integer $n$ such
that $H_F(n) \ne h_F(n)$, where $H_F(X)$ and $h_F(X)$ are, respectively,
the Hilbert function and Hilbert polynomial of the fiber cone $F(I)$. 
Suppose $I$ is a nonprincipal regular  ideal having a principal reduction
in a Noetherian local ring $(R,\m)$. We observe in 
Proposition \FPnoRedno \me that $fp(I) \le r(I) - 1$,
where $r(I)$ is the reduction number of $I$. Thus
if $I$ is a nonprincipal $\m$-primary ideal of a one-dimensional
Cohen--Macaulay local ring, then the fiber postulation number
$fp(I)$ is less than or equal to the postulation number $n(I)$ of $I$. 

For each integer $n \ge 3$, we exhibit  
in Example \FPnoIneq \me  the
existence of a
one-dimensional Cohen--Macaulay local domain
$R_n$ and an ideal $I_n$ primary for the
maximal ideal of $R_n$ such that $fp(I_n) = 0$
while $r(I_n) = n - 1$.
This shows that the difference $r(I) - fp(I)$ can
be arbitrarily large for ideals $I$ and rings 
$R$ as in Propositon \FPnoRedno. \me For the
ideals $I_n$ of Example \FPnoIneq \me the
fiber postulation number $fp(I_n)$ is 
strictly smaller than the postulation number $n(I)$.
It follows that  the fiber cone
$F(I_n)$ is not Cohen--Macaulay.
Indeed,  as we observe in Proposition~\FPnoeq :\me 
if $F(I)$ is Cohen--Macaulay, then $fp(I) = r(I) - 1$.
This  and a higher dimensional analogue stating that
if $F(I)$ is Cohen--Macaulay, then 
$fp(I) = r(I) - \ell$,  where $\ell$ is the analytic spread of $I$, 
are consequences of results of T. Cortadellas and S. Zarzuela \cite{CZ}
and C. D'Cruz, K. N. Raghavan and J. Verma \cite{DRV} on the structure of 
the Hilbert function of a Cohen--Macaulay fiber cone $F(I)$. 
One can use  \cite{BH, Proposition 4.1.12} to obtain the 
asserted relation between the 
reduction number and fiber postulation number.

In Example \negative, \me we exhibit a 3-dimensional 
Cohen--Macaulay local domain $(R,\m)$ of multiplicity 3
having,  for each positive integer $k$,  an ideal $I_k$
such that: (i) $I_k$ has a principal reduction, (ii) $I_k$ is 
minimally generated by $k + 3$ elements, and (iii) the 
fiber cone $F(I_k)$ has multiplicity 3 and is not Cohen--Macaulay.


We observe in Remark \tangent \me that 
the  ideal $I_3$ of Example \FPnoIneq \me
provides an example of a one-dimensional 
Cohen--Macaulay local domain $(R,\m)$ 
and an $\m$-primary ideal $I$ such that 
the associated graded ring 
$G(I)$ is Cohen--Macaulay, while  the fiber cone
$F(I)$ is not Cohen--Macaulay.

In Example \highmult \me we exhibit a 3-dimensional
Cohen--Macaulay local domain $(R,\m)$ of multiplicity 2
with the following property. For
each positive integer $k$, the ring $R$ has  a 
stable  ideal $I_k$ having  a principal reduction
such that the fiber cone $F(I_k)$ 
of $I_k$ is Cohen--Macaulay and has multiplicity $k + 2$. 
Therefore, in contrast with the case where $\dim R = 1$, for
$R$ of higher dimension there may exist no upper
bound on the multiplicity of $F(I)$ as $I$ varies 
over the ideals of $R$ that have a principal reduction.


In \S 5 we consider a question raised by S. Huckaba 
\cite{Hu1,Question 2.6} as to whether the multiplicity 
of a quasi-unmixed analytically unramified Noetherian
local ring containing an infinite field 
of characteristic different from 2 is strictly larger than the 
reduction number of each regular ideal of 
analytic spread one of the ring. Using an interesting
observation shown to us by Craig Huneke 
(Proposition \huneke \me) we obtain a positive answer
to Huckaba's question in several special cases. 
For example, in Corollary \analytictwodim \me  we confirm a 
positive answer to the question for the
case of a 2-dimensional Noetherian analytically irreducible 
local domain. However, the general case of Huckaba's 
question remains open. 

\vskip 3ex
The authors thank Craig Huneke for interesting and helpful conversations
on the topic described in \S 5. 

\vskip 3ex

\section{The postulation number  of a one-dimensional fiber cone.}

\vskip 3ex

Suppose $I$ is a regular ideal of a Noetherian local ring $(R,\m)$.
If  there exists $x \in I$ and an integer $n \ge 0$ such
that $xI^n = I^{n+1}$, then $xR$ is said to be a 
{\it principal reduction of } $I$. In terms of the
fiber cone $F(I)$ this is reflected in the fact that if
$\overline x \in I/\m I$ denotes the image of $x$, then
$F(I)$ is a finitely generated integral extension of
its polynomial subring $K[\overline x]$, where 
$K := R/\m$. Indeed, if $K$ is infinite, the
converse also holds: if  the
fiber cone $F(I)$ is one-dimensional, then using
Noether normalization, there exists $\overline x \in I/\m I$
such that $F(I)$ is integral over $K[\overline x]$.  If 
$x \in I$ is a preimage of $\overline x$, then $xR$ is a
principal reduction of $I$.

D. G. Northcott and D. Rees in their famous paper \cite{NR}
introduce reductions of ideals and associate to an
ideal $I$ a homogeneous ideal in a polynomial ring 
over $K$, the {\it null form ideal of } $I$. 
The null form ideal of $I$ is the kernel in a 
presentation of the fiber cone $F(I)$ as a homomorphic
image of a polynomial ring over $K$.
K.  Shah in \cite{S} coined the term `fiber cone' for
this ring.

 
If  $I$ is a regular ideal having a principal 
reduction, the reduction number
$r(I)$ is known to be independent of the principal
reduction $x$ \cite{Hu1, page~504}. In this 
situation, $r(I)$ is also equal to the reduction
number of the algebra $F(I)$.  If 
$m_1, \dots,m_s$ is a minimal set of homogeneous module
generators for $F(I)$ over $(R/\m)[\overline x]$, then 
$r(I) = \max\{\deg m_i \}$, see \cite{V2}.

The Hilbert function $H_F(X)$ defined so that 
$H_F(n) = \la (I^n /mI^n) $
is given for all sufficiently large $n$ by a polynomial
$h_F(X)$. Since, in this case, $\de F(I) = \oplus_{n \ge 0}I^n /mI^n$ is a
one-dimensional homogeneous graded ring over a field,
$h_F(X)$ is a constant $f_0$. Here
$f_0$ is a positive integer. It is the multiplicity 
of the graded ring $F(I)$  and the number of 
elements in a minimal
generating set  of $I^n$ for all sufficiently large $n$.

It is of interest to also consider 
the first iterated Hilbert function of $F(I)$. This is 
the function 
$\de H^1_F(n) = \sum_{j=0}^n H_F(j) = \sum_{j=0}^n \la(I^j/ \m I^j)$.
It is a polynomial function for $n > fp(I)$. 
The associated  Hilbert polynomial is
$h^1_F(X) = f_0 (X+1) - f_1= f_0 X - (f_1 - f_0)$.
In analogy with the Hilbert coefficients of $G(I)$ in
the case where $I$ is $\m$-primary, it is of interest  
to have sufficient conditions in order that  
$H^1_F(0) - h^1_F(0) = 1 - f_0 + f_1  \ge 0$. In other words, 
sufficient conditions in order that $f_1 \ge f_0 - 1$.
Notice that for $n\ge r-1$, where $r=r(I)$ one has 
$$
H^1_F(n) = \sum_{j=0}^n \la(I^j/ \m I^j) =
H^1_F(r-1) + (n-(r-1)) f_0= nf_0 - ((r-1)f_0 - H^1_F(r-1)).
$$
It follows that $f_1 - f_0 = (r-1)f_0 - H^1_F(r-1)$
and $f_1 = rf_0 - H^1_F(r-1)$. We observe in
Example \negative \me below that this  last 
equality implies  that $f_1$ can be negative 
and that for a fixed ring $R$ there may be no
lower bound on the value of $f_1$ as we vary over
ideals of $R$ having a principal reduction. 
In the case, however, where the fiber cone $F(I)$
is Cohen--Macaulay, as we observe in Remark \northcott, \me
it follows from classical results
that $f_1 \ge f_0 -1$.

\vskip 2ex

\rmk
\label{\northcott}
Suppose $(R,\m)$ is a Noetherian
local ring and $I$ is a regular ideal of $R$ having
a principal reduction. If the fiber cone $F(I)$ is
Cohen--Macaulay, then a result of Northcott 
implies that $f_1 \ge f_0 -1$. For the one-dimensional 
homogeneous graded ring $F(I)$ has the same Hilbert function
as the local ring  $(S,\n)$ obtained by localizing $F(I)$ at its 
homogeneous maximal ideal. If $F(I)$ is Cohen--Macaulay,
then $(S,\n)$ is a one-dimensional Cohen--Macaulay local ring.
Northcott defines normalized Hilbert coefficients $e_0$ and $e_1$
of  $\n$ such that $\lambda(S/\n^{n+1}) = e_0(n+1) - e_1$ for 
$n >> 0$, 
and proves \cite{No, page~211} that $e_1 \ge e_0 - 1$.
Since $\lambda(S/\n^{n+1}) = H^1_F(n) = f_0(n+1) - f_1$ 
for $n >> 0$, we have $e_0 = f_0, e_1 = f_1$ and thus 
$f_1 \ge f_0 - 1$.


\vskip 2ex


We are interested in comparing the fiber postulation 
number $fp(I)$ of $I$  with other  invariants associated to 
$I$.  We start by observing in  Proposition \FPnoRedno \me  an upper
bound for $fp(I)$ in terms of the reduction number of $I$.

\prop
\label{\FPnoRedno} 
Suppose $(R,\m)$ is a Noetherian
local ring and $I$ is a nonprincipal regular ideal of $R$ having
a principal reduction $xR$. Then $fp(I) \le r(I) - 1$,
where $r(I) = r$ is the reduction number of $I$.
\endb

\proof 
Notice that $I^{s}/ \m I^{s} = x^{s-r}I^r/\m x^{s-r}I^r 
\cong I^r/\m I^r$ for all $s \ge r = r(I)$. Thus 
$H_F(n) = \la(I^r/\m I^r)$ for every $n \ge r$. 
It follows that $h_F(X) = f_0 = \la(I^r/\m I^r)$
and that $H_F(n) = h_F(n)$ for every $n \ge r$.
Therefore   $fp(I) \le r(I) - 1$.
\qed

\vskip 3ex

With notation as in the proof of Proposition \FPnoRedno, \me observe 
that $fp(I) < r(I)-1$ if and only if 
$h_F(r-1) = H_F(r-1) = f_0$. Thus
$fp(I) < r(I) - 1$  if and only if 
$I^{r-1}$ is minimally generated by $f_0$ elements.
We use this observation in order to construct in Example~\FPnoIneq \me
for each integer $n \ge 3$  an example of a   
one-dimensional Cohen--Macaulay local domain
$R_n$ and an ideal $I_n$ primary for the
maximal ideal of $R_n$ such that $fp(I_n) = 0$
while $r(I_n) = n - 1$.


As is the case for our examples in \S 4 of 
\cite{DGH}, the rings $R_n$ 
in Example \FPnoIneq \me  are 
complete one-dimensional local Cohen--Macaulay domains of the form
$R_n = K[\![t^s : s \in S_n]\!]$, i.e. formal power
series in the indeterminate $t$ with coefficients in
a field $K$ and exponents from an additive submonoid $S_n$ of
the nonnegative integers that contains all sufficiently large
integers. The formal power series ring
$K[\![t]\!]$ is a finitely generated $R_n$-module having the
same fraction field as $R_n$ and is thus the integral closure
of $R_n$. To establish the asserted properties of the
rings $R_n$ and ideals $I_n$ in Example \FPnoIneq \me we
work directly with the additive monoid $S_n$ and the 
corresponding semigroup ideal $I_n$ of $S_n$.

\vskip 3ex

\ex
\label{\FPnoIneq}
Fix an integer  $n \geq 3$, and consider the 
numerical semigroup $S_n$ generated by the
elements $a := 2n$, $b := 4n-1$, $d := n(2n-1)$,
and $c_h := (n+h)(2n-1) + 1$, where $h = 3 \ldots n$.
Thus $S_n = <a, b, d, c_3, \ldots, c_n>$. We prove 
that these $n+1$ elements are the minimal generating
set for $S_n$. In considering the general case, it
is useful to keep in mind the first few examples,
$S_3 = < 6, 11, 15, 31>$, $S_4 = <8, 15, 28, 50, 57>$
and $S_5 = <10, 19, 45, 73, 82, 91>$.

\vskip 2ex

\noindent
{\bf Claim 2.3.1: } The monoid   $S_n$ is minimally generated by
$a$, $b$, $d$, $c_3$, $c_4, \dots, c_n$.
\proof
We have, modulo $2n$,  that  $b \equiv -1 \equiv 2n-1$, 
$d \equiv -n \equiv n$ and $c_h \equiv -n-h+1 \equiv n-(h-1)$.
The smallest integer $x$ in $<a,b>$ such that $x \equiv n$
is $x=nb= 4n^2-n$, which is larger than $d$; hence $d \notin <a,b>$
and $a,b,d$ are minimal generators of the submonoid $<a,b,d>$.

Now we consider $c_h$.
If $h > 3$ and if $3 \le i <h$, then $c_h-c_i \notin <a,b,d>$.
In fact,  we have $c_h-c_i =(h-i)(2n-1) <d$;
hence 
$c_h-c_i \notin <a,b,d> \iff  c_h-c_i \notin <a,b>$.
But $c_h-c_i \equiv -h+i$ (mod $2n$) \  and
the smallest integer $x$ in $<a,b>$ such that $x \equiv -h+i$
is $x=(h-i)b = (h-i)(4n-1)$ which is larger than $c_h-c_i=(h-i)(2n-1)$.
It follows that \  $c_h-c_i \notin <a,b,d>$ \ for every $i =3,\dots,h-1$.
Thus, for every $h > 3$, \  $c_h \notin <a,b,d,c_3,\dots, c_{h-1}>$
\ if and only if \  $c_h \notin <a,b,d>$.

Therefore, in order to prove that $S_n$ is minimally generated 
by $a,b,d,c_3,c_4,\dots,c_n$, it suffices  to show that 
$c_h \notin <a,b,d>$, for every $h =3,\dots,n$. 
Since $c_h \equiv n-(h-1)$ (mod $2n$), we compute the smallest 
element in $<a,b,d>$ that is congruent to $n - (h-1)$ modulo $2n$ 
and observe that  it is larger than $c_h$.
We have that $a\alpha+b\beta+d\delta \equiv -\beta+n\delta$ (mod $2n$).
Hence we want to compute the minimum of the set 
$H=\{b\beta+d\delta \ | \ \beta, \delta \geq 0,
 \ -\beta+n\delta \equiv n-(h-1)\}$.

We claim that $\min H=(h-1)b+d$ (i.e. we have the minimum for 
$\beta=(h-1)$ and  $\delta = 1$).
Assume, by way of contradiction, that  there exists an
integer $b\beta+d\delta \equiv n-(h-1)$ 
 such that $b\beta+d\delta <  (h-1)b+d$.
This implies $\beta < (h-1)$ or $\delta =0$.

If $\delta=0$, then $b\beta \equiv -\beta \equiv n-(h-1)$, that is 
$\beta \equiv -n+(h-1) \equiv n+(h-1)$; this implies 
$\beta = n+h-1 +m(2n)$, with $m \in \Bbb N$. But $nb > d$,
thus $ (n+h-1 +m(2n))b > (h-1)b+d$, a  contradiction.

If $\beta < (h-1)$, then $b\beta+d\delta \equiv -\beta+n\delta
\equiv n-(h-1)$. But this implies that 
$n(\delta-1) \equiv \beta-(h-1)$, which is a contradiction,
since $n(\delta-1) \equiv n$ or $n(\delta-1) \equiv 0$
and $-n <\beta-(h-1)<0$.

We conclude that  $(h-1)b+d$ is the minimum element $x \in 
<a,b,d>$ such that $x \equiv n-(h-1)$. Since  for every $h=3,\dots,n$,
we have
$(h-1)b+d= (h-1)(4n-1)+n(2n-1) = 2n^2+n(4(h-1)-1)-h+1
>2n^2+n(2(h-1)+1)-h+1=c_h$, it follows that 
$c_h \notin <a,b,d>$. Therefore  $S_n$ is minimally
generated by $a$, $b$, $d$, $c_3, \dots,c_n$. \qed


\medskip
The {\it Frobenius number } $g(S_n)$ of $S_n$ is by definition
the largest integer not belonging to $S_n$. 
We next compute $g(S_n)$ :
since $c_n$ is a generator of $S_n$, we have $c_n -a \notin S_n$.
Moreover all the elements of the set $\{c_n-a+1, c_n-a+2, \dots , c_n \}$
belong to $S_n$: 
\roster
\item  $c_n -a+1=(2n-1)(2n-1)+1 = c_{n-1}$;
\item  for all $m =2, \dots n-3$,
$c_n-a+m= 2n(2n-1)+1-2n+m \equiv m+1 \equiv
n-(n-m)+1 \equiv c_{n-m}$ (mod $a$) \ and \ 
$c_n-a+m > c_{n-m}$;
\item  $c_n-a+n-2= 2n(2n-1)-n-1 \equiv b+d$ (mod $a$) \ and \ 
$c_n-a+n-2 >b+d$;
\item   $c_n-a+n-1= 2n(2n-1)-n \equiv d$ (mod $a$) \ and
\ $c_n-a+n-2 >d$;
\item  for every $m =0,\dots , n-2$, $c_n-a+n+m = 2n(2n-1)+1-n+m
= (2m+1))a+(n-m-1)b$;
\item  $c_n-a+2n-1= 2n(2n-1)=2d$;
\item  $c_n-a+2n=c_n$.
\endroster
Therefore $S_n$ contains all integers greater than or equal to
$c_n-a+1 =c_{n-1}$ and $g(S_n) = c_n-a$.

\vskip 2ex
Consider the semigroup ideal $I_n : = <a, b, c_3, \ldots, c_h>$ of 
$S_n$.
 
\vskip 2ex

\noindent
{\bf Claim 2.3.2: } For every integer $m \ge 1$,
the ideal $mI_n$ is minimally generated by $n$ elements.
\proof
Since the elements $a, b, c_3, \dots, c_n$ are part of the 
minimal set of generators of $S_n$, it follows at once that
$I_n$ is minimally generated by $a$, $b$, $c_3,\dots c_n$.
We prove that $2I_n$ is minimally generated by
$2a$, $a+b$, $2b$, $a+c_4,\dots, a+c_n$.
We have the following relations:
$$ 
a+c_3=2b+d   \tag{$R_1$}   
$$ 
and, for every $h=4, \dots n$, 
$$
 a+c_h=b+c_{h-1}. \tag{$R_2$}
$$
Moreover, since $b+c_n -2a >c_n -a =g(S_n)$, for some $s \in S_n$ we have 
$$
 b+c_n =2a+s.  \tag{$R_3$}
$$
The same is true for $c_h+c_k$, for all $h,k =3,\dots, n$,
since $c_h+c_k \geq 2c_3 >b+c_n$: for some $t_{h,k} \in S_n$,
$$
 c_h+c_k = 2a+t_{h,k}.  \tag{$R_4$}
$$
It follows that $2I_n =<2a, a+b, 2b, a+c_4,\dots, a+c_n>$.
We check that this is the  minimal set of generators
(i.e. there are not other relations).
Clearly we have $a+b-2a=2b-(a+b) =b-a \notin S_n$.
Notice that $y(b-a) \notin S_n$ for every $y <n$: in fact, 
$y(b-a) < d$ and $y(b-a) \notin <a,b>$,
since $y(b-a) <yb$ which is the smallest integer $x$ in $<a,b>$
such that $x \equiv -y$ (mod $a$).
It follows that $2b-2a \notin S_n$. 
Moreover, for every $h=4,\dots, n$,
since $c_h$ is part of the minimal
set of generators of $I_n$, we have   $a+c_h - 2a= c_h-a \notin S_n$,
 $a+c_h-(a+b)=c_h -b \notin S_n$,
$a+c_h-2b=b+c_{h-1}-2b = c_{h-1}-b \notin S_n$ and,
 for $k<h$,  $a+c_h-(a+c_k)= c_h-c_k \notin S_n$.
Therefore  $2I_n$ is minimally generated by 
$2a$, $a+b$, $2b$, $a+c_4,\dots, a+c_n$.

Proceeding by induction on $m$, we assume
that  $2 \leq m \leq n-2$ and that 
for every $k$ with  $2 \leq k \leq m$, the
semigroup ideal $kI_n$ is minimally 
generated by $ka$, $(k-1)a+b$, \dots, $kb$, $(k-1)a+c_{k+2},\dots, 
(k-1)a+c_n$. We prove that 
$(m+1)I_n$ is minimally generated by
$(m+1)a$, $ma+b$, \dots, $(m+1)b$, $ma+c_{m+3},\dots, 
ma+c_n$ (in the case $m=n-2$, the inequality $m+3 >n$ means that 
$(m+1)I_n =(n-1)I_n$ is minimally generated by 
$(n-1)a$, $(n-2)a+b$, \dots $(n-1)b$).

Using relations $(R_1)$ and $(R_2)$, 
notice that $a+(m-1)a+c_{m+2}= a+(m-1)b+c_3 = (m+1)b+d$.
Moreover, for every $h= m+2, \dots, n-1$, we have 
$b+(m-1)a+c_h= ma+c_{h+1}$ (using $(R_2)$) and $b+(m-1)a+c_n=(m+1)a+s$
(using $(R_3)$).

Now we consider the elements $c_k+ra+sb$ and $c_k+(m-1)a+c_h$
(where $k=3,\dots n$, $r+s=m$ and $s>0$).
We easily get, by $(R_4)$, that $c_k+(m-1)a+c_h= (m+1)a+t_{k,h}$; 
moreover, if $k+s \leq n$, using $(R_2)$, we get $c_k +ra+sb= c_{k+s}+ma$,
while, if $k+s > n$, using $(R_2)$ and $(R_3)$, 
$c_k+ra+sb = c_n+(r+n-k)a+(s-(n-k))b= 
(r+n-k+2)a+(s-(n-k)-1)b+s$.
Hence $(m+1)I_n =<ma, (m-1)a+b, \dots, mb, (m-1)a+c_{m+2},\dots, 
(m-1)a+c_n>$.

In order to prove that $(m+1)I_n$  is minimally generated by these
elements, we show that there are no other relations.
If $r+s=u+v=m+1$ and $s>v$, the difference $ra+sb-(ua+vb)=
(s-v)(b-a) \notin S_n$, since $s-v <n$.
Moreover, for every $h\geq m+3$ and 
$r+s =m+1$, $ma+c_h-(ra+sb) = (m-r)a+c_h -(m-r+1)b \notin S_n$,
since $(m-r)a+c_h$ is part of the minimal set of generators of 
$(m-r+1)I_n$.  Analogously, if $k<h$,
$ma+c_h-(ma+c_k) = c_h-c_k \notin S_n$, since $c_h$ is part of 
a minimal set of generators of $I_n$.


It follows that, for every $m=2, \dots, n-1$, $mI_n$ is minimally 
generated by $ma$, $(m-1)a+b$, \dots, $mb$, $(m-1)a+c_{m+2},\dots, 
(m-1)a+c_n$.
In particular $(n-1)I_n = <(n-1)a, (n-2)a+b, \dots, (n-1)b>$.
Now, since 
$$
nb=na+d,  \tag{$R_5$}
$$ 
we have, with similar arguments as above, that
$nI_n$ is minimally generated by $na$, $(n-1)a+b$, \dots, $nb$,
that is $nI_n= (n-1)I_n+a$.
Hence, for every $m \geq n$, also $mI_n$ is minimally $n$-generated;
moreover the reduction number of $I_n$ is $r(I_n)=n-1$.
This completes the proof of Claim 2.3.2 and
the presentation of Example \FPnoIneq.  \qed


\vskip 2ex
In Example \negative, \me we exhibit a 3-dimensional 
Cohen--Macaulay local domain $(R,\m)$ of multiplicity 3
having,  for each positive integer $k$,  an ideal $I_k$
such that $I_k$ has a principal reduction, $I_k$ is 
minimally generated by $k + 3$ elements, and the 
fiber cone $F(I_k)$ has multiplicity 3 and is not Cohen--Macaulay.


\vskip 3ex

\ex
\label{\negative}
Let $(S, \n)$ be a two-dimensional regular local ring  and let 
$R= S[t^3, t^4, t^5]_{(\n, t^3, t^4, t^5)}$, where $t$ is an
indeterminate over $S$. 
For $k$ a fixed positive integer, let 
$I_k =(t^3, t^4, \n^k t^5)R$. Since $\n^k$ is minimally generated
by $k+1$ elements as an ideal in $S$, we see that $I_k$ is
minimally generated by $k + 3$ elements.
Notice that $t^3I_k \subsetneq I_k^2 = (t^6, t^7, t^8)$ and that 
$I_k^3 = t^3 I_k^2$. Thus $t^3$ is a principal reduction of $I_k$
and $I_k$ has reduction number 
$r(I_k) = 2$. Moreover, $I_k^2$ is minimally generated by
$t^6, t^7, t^8$, so the multiplicity of the
fiber cone $F(I_k)$ is $f_0 = 3$ and $f_1 = 6 - (1 + k + 3)$.
Since $k$ can be  arbitrarily large, there is no lower bound
on $f_1$ as we vary over ideals of $R$ having a principal reduction.

Since $t^3\n^k t^5 \subseteq \m I_k^2$, where $\m$ is the maximal
ideal of $R$, we see that the image of $t^3$ in $I_k/\m I_k = F_1$
in $F(I_k)$ is a zero-divisor. Therefore  
Remark \cmcheck \me as given below implies  that $F(I_k)$ is not
Cohen--Macaulay and thus has depth zero. 

\vskip 3ex

\section{The Cohen--Macaulay property of the fiber cone.}

\vskip 3ex

Interesting work on the Cohen--Macaulay property of the
fiber cone $F(I)$ has been done by K. Shah in
\cite{S}, by T. Cortadellas and S. Zarzuela in \cite{CZ}
and by C. D'Cruz, K. N. Raghavan and J. K. Verma in \cite{DRV}.
In particular, we use the following:

\vskip 2ex

\rmk
\label{\cmcheck}
As Shah notes in \cite{S},  the freeness lemma of
Hironaka \cite{N, (25.16)} implies that $F(I)$ is 
Cohen--Macaulay if and only if $F(I)$ is a free module
over one (or equivalently every) Noether normalization 
subring. If $x$ is a principal reduction of $I$
and $\overline x$ denotes the image of $x$ in 
$I/\m I = F_1$, then $K[\overline x]$ is a 
Noether normalization of $F(I)$. Since $K[\overline x]$ is
a principal ideal domain, $F(I)$ is a free $K[\overline x]$-module
if and only if it is a torsionfree $K[\overline x]$-module.
Since $F(I)$ is graded and $\overline x$ is homogeneous, we
see that $F(I)$ is torsionfree as a $K[\overline x]$-module if 
and only if $\overline x$ is a regular element of $F(I)$.
Therefore $F(I)$ is Cohen--Macaulay if and only if 
$\overline x$ is a regular element of $F(I)$.

\vskip 2ex
It follows from Remark \cmcheck \me that for the ideal 
$I_n=(t^a, t^b, t^{c_3}, \dots, t^{c_n})$
of the ring $R_n=K[\![t^s:s\in S_n ]\!]$ given in  Example \FPnoIneq, \me
the fiber cone $F(I_n)$ is not Cohen-Macaulay:
the relation $(R_1)$ of Example \FPnoIneq \me 
implies that $t^at^{c_3} \in I_n^2\m$. Therefore 
the image in $F(I_n)$ of $t^a$
(which is a principal reduction of $I_n$) is a zerodivisor in $F(I_n)$.     
The fact that $F(I_n)$ is not Cohen-Macaulay also follows from  
Proposition \FPnoeq.

\vskip 2ex

Proposition \FPnoeq \me as given below
is a consequence of results of Cortadellas and Zarzuela \cite{CZ}
and D'Cruz, Raghavan and Verma \cite{DRV} on the structure of 
the Hilbert function of a Cohen--Macaulay fiber cone $F(I)$. 
One can use  \cite{BH, Proposition 4.1.12} to see the 
asserted relation between the fiber postulation number and
reduction number from their results. We give a direct elementary 
proof of the result.

\vskip 2ex



\prop
\label{\FPnoeq}
Suppose $(R,\m)$ is a Noetherian
local ring and $I$ is a nonprincipal regular ideal of $R$ having
a principal reduction $xR$. If $F(I)$ is 
Cohen--Macaulay, then $fp(I) = r(I) - 1$.
\endb

\proof
Let $\overline x$ denote the image of $x$ in 
$I/\m I = F_1$. As noted in Remark \cmcheck, \me
$F(I)$ is Cohen--Macaulay if and only if $\overline{x}$ is 
a regular element of $F(I)$. By
Proposition \FPnoRedno, \me $fp(I) \le r(I) - 1$, and 
by definition of 
$r(I) = r$, we have $xI^{r-1} \subsetneq I^r$.  

Assume, by way of contradiction, that $fp(I) < r - 1$. This means 
that $\dim F_{r-1} = f_0 = \dim F_r$. Since 
$\overline x$ is a regular element of $F(I)$,
$f_0 = \dim \overline{x}F_{r-1}$. Since 
$\overline{x}F_{r-1} \subseteq F_r$, this 
implies $\overline{x}F_{r-1} = F_r$. However, 
this implies $xI^{r-1} = I^r$,
a contradiction. \qed

\vskip 3ex

\rmk
\label{\converse}
The converse of Proposition \FPnoeq \me is not true in general.
For example, let $R = K[t^4, t^5, t^{11}]_{(t^4, t^5, t^{11})}$,
where $t$ is an indeterminate over the field $K$. 
Let $I = \m$, the maximal ideal of $R$. Since the image of 
$t^4 \in I/I^2$ is a zero-divisor, $F(I) = G(I)$ is not
Cohen--Macaulay. On the other hand, $fp(I) = n(I) = 2$
and $r(I) = 3$, so $fp(I) = r(I) - 1$. 

Other examples illustrating the failure of the converse of
Proposition \FPnoeq \me are the ideals $I_k$ of 
Example \negative. \me  In these examples one
has $fp(I_k) = 1$ and $r(I_k) = 2$.



\vskip 3ex

\dis
\label{\fibercm} 
Shah in \cite{S} proves several interesting
results on the Cohen--Macaulay property of $F(I)$. In \cite{S, Theorem 1}
he proves that if $I$ is an ideal of a Noetherian local ring $(R,\m)$ and 
if $I$ is integral over an ideal generated by a regular sequence
$\underline x$ such that $I^2 = (\underline{x})I$, then $F(I)$ is 
Cohen--Macaulay. As Shah notes, it was proved earlier by C. Huneke
and J. Sally in \cite{HS, Proposition 3.3} that if $(R,\m)$ is a
Cohen--Macaulay local ring and $I$ is an $\m$-primary ideal
such that $I^2 = (\underline{x})I$, where $\underline x$ is a
regular sequence, then $F(I)$ is Cohen--Macaulay.
It is immediate from these results that, if $I$ is a regular
ideal having a principal reduction $xR$ and if $r(I) \le 1$,
then the fiber cone $F(I)$ is Cohen--Macaulay. In \cite{S, Theorem 2},
Shah proves that if $I$ is an ideal of a Noetherian local ring $(R,\m)$ and 
if $I$ is integral over an ideal generated by a regular sequence
$\underline x$ such that $I^3 = (\underline{x})I^2$, 
$I^2 \cap (\underline{x}) = I(\underline{x})$, and 
$I^2\m = I(\underline{x})\m$, then $F(I)$ is Cohen--Macaulay.
In \cite{CZ; Theorem 3.2}, Cortadellas and Zarzuela generalize Shah's
result and show  that if $I$ is an ideal and $J$ a minimal reduction of $I$
generated  by a regular sequence with $J \cap I^n = J I^{n-1}$ for all 
$1 \le n \le r_J (I)$, then $F(I)$ is Cohen--Macaulay if and only if
$J \cap \m I^n = J \m I^{n-1}$ for all 
$1 \le n \le r_J (I)$.

\vskip 2ex
\prop
\label{\twogen}
Suppose $I$ is a regular ideal having a principal reduction $(x)$ 
in a Noetherian local ring $(R,\m)$. If 
$I$ is 2-generated, then $F(I)$ is Cohen--Macaulay.
\endb

\proof
Consider a presentation $\phi$ of $F(I)$ as a 
graded $K$-algebra homomorphic
image of the polynomial ring $K[X,Y]$, where 
$\phi(X)$ and $\phi(Y)$ are the images in $I/\m I = F_1$ 
of two generators of $I$. To show $F(I)$ is Cohen--Macaulay
it suffices to show that $\ker \phi$ is principal.
By a result of P. Eakin and A. Sathaye \cite{ES, page 440}, 
if $I^n$ is generated by less than $n+1$ elements, then
$xI^n = I^{n+1}$. Hence if $r = r(I)$ is the reduction
number of $I$, then for each positive integer $n \le r$
the ideal $I^n$ is minimally generated by $n+1$
elements. Thus  the multiplicity of $F(I)$ is $r+1$ and
$\dim F_n = r+1$ for each $n \ge r$. 
Therefore  the minimal degree of a form
$f \in \ker \phi$ is $r+1$ and there exists a form 
$f \in \ker \phi$ with $\deg f = r+1$. It 
follows that $\ker \phi$ is generated by $f$. 
Therefore $F(I)$ is a complete intersection and 
hence Cohen--Macaulay.
\qed

\vskip 2ex
\cor
\label{\mult2}
Suppose $(R, \m)$ is a one-dimensonal Cohen--Macaulay local ring
of multiplicity 2. 
For each $\m$-primary ideal $I$ of $R$, 
the fiber cone $F(I)$ is Cohen--Macaulay.  
\endb   
\proof
Since $R$ has multiplicity 2, the integral closure of $R$
has at most 2 maximal ideals and  each $\m$-primary ideal $I$
of $R$ is 2-generated. Therefore $I$ 
has a principal reduction and by Proposition~\twogen, \me
$F(I)$ is Cohen--Macaulay. \qed


\vskip 2ex

\ques
For which one-dimensional Cohen--Macaulay local rings
$(R, \m)$ is it true that $F(I)$ is Cohen--Macaulay 
for every $\m$-primary ideal $I$? Is this true if 
$R$ has multiplicity 3 ? 

\vskip 2ex

\ex
\label{\all}
Let $R =  K[\![t^3, t^4, t^5]\!]$. Then $K[\![t]\!]$ 
is the integral closure $\overline R$ of $R$
and the maximal ideal $\m = (t^3, t^4, t^5)R$ is the
conductor of $\overline R$ into $R$. 
If $I$ is an $\m$ primary ideal of  $R$, then either
(i) $I$ is principal, or (ii) $I$ is minimally 2-generated,
or (iii) $I$ is minimally 3-generated and $I = I\overline R$. 
In this third case, $I = t^s\overline R$ for some integer
$s \ge 3$. It follows that $t^sI = I^2$ and $I$ has reduction
number one as an ideal of $R$. Therefore the fiber cone
$F(I)$ is Cohen--Macaulay for each $\m$-primary ideal $I$ 
of $R$. 

\vskip 2ex

\rmk
For $R$ as in Example \all,  \me there exist $\m$-primary ideals
$I$ of $R$ such that the associated graded ring $G(I)$ is not
Cohen--Macaulay. For example, $I = (t^3, t^4)$ is not a 
Ratliff-Rush ideal since $I \subsetneq \m$ and $I^2 = \m^2$.
Therefore $I$ is an example of an ideal 
for which $F(I)$ is Cohen--Macaulay and $G(I)$ is not
Cohen--Macaulay.

\vskip 2ex

\rmk
\label{\numerical}
Suppose $(R,\m)$ is a one-dimensional Cohen--Macaulay local
ring and $I$ is an $\m$-primary ideal having a principal
reduction $xR = J$ with reduction number $r$. 
As a special case of \cite{DRV, Theorem 2.1}
it follows that the fiber cone $F(I)$ is Cohen--Macaulay
if and only if the Hilbert series 
$P(t) = \sum_{n=0}^\infty \lambda(I^n/\m I^n)t^n$ has the
form $h(t)/(1-t)$, where 
$h(t) = \sum_{i=0}^r\lambda(I^i/(JI^{i-1} + \m I^i)$. It is
not true, however, that the Cohen--Macaulay property of
the fiber cone $F(I)$ is determined by its Hilbert series
$P(t)$. For example, if $R=K[\![t^6,t^{11},t^{15}, t^{31} ]\!]$
and $I=(t^6, t^{11}, t^{31})$ are as  
in Example \FPnoIneq, where $R=K[\![t^s  : s \in S_3]\!]$
and $I$ corresponds to the semigroup ideal $I_3$ of $S_3$,
 \me  $F(I)$ is not Cohen--Macaulay
and the Hilbert series for $I$  is $P(t) = (1+2t)/(1-t)$.  
This is also the Hilbert series of a Cohen--Macaulay 
fiber cone; for example it is the Hilbert series of 
the maximal ideal of $R = K[\![ x,y,z ]\!]$, where $y^2 = yz = z^2 = 0$.




\vskip 3ex

\section{The multiplicity of the fiber cone. }

\vskip 3ex

Suppose   $I$ is a regular ideal of $R$ having a principal
reduction $xR$. In this situation, the blowing up 
scheme $\Proj R[It]$ of $\Spec R$ with respect to $I$ 
is affine and has the form $\Spec R[I/x]$.
Therefore we refer to the ring $R^I : = R[I/x]$ as
the  {\it blowing--up ring} 
of $I$. It is simple to prove that 
$R^I := R[I/x] = \bigcup_{n\ge 0} I^n/x^n = I^r / x^r$
where $r = r(I)$. Moreover, the blowing up ring $R^I$ of $I$
is also the blowing up ring of each power $I^n$ of $I$.

\vskip 3ex

\lemma
\label{\mult}
The multiplicity $f_0$ of $F(I)$ is the
minimal number of generators of $R^I$ as an $R$-module. 
Thus $f_0 = \la(R^I/\m R^I)$.
\endb

\proof
We know that $f_0 = \la(I^r/ \m I^r)$. In other words the multiplicity of $F(I)$
is given by the cardinality of a minimal set of generators for $I^r$ 
as an $R$-module. Moreover, the fact that $x$ is a regular 
element of $R$ implies that if  $\{a_1, \ldots, a_{f_0}\}$ 
is a minimal set of generators for $I^r$, then 
$\{a_1/x^r, \ldots, a_{f_0}/x^r\}$ is  a minimal set 
of generators of  $I^r/x^r$ as an $R$-module. \qed

\vskip 2ex

\rmk
\label{\tangent} 
Suppose   $(R,\m)$ is a one-dimensional Cohen--Macaulay local
ring and $I$ is an $\m$-primary ideal having a principal
reduction $xR$. The associated graded ring $G(I)$ is 
Cohen--Macaulay if and only if $I^n = \widetilde{I^n}$ for
each $n \in \N$. Moreover, $\widetilde{I^n} = I^nR^I \cap R$,
where $R^I = R[I/x]$ is the blowing up ring of $I$. We use this
to observe that $G(I)$ is Cohen--Macaulay if 
$R=K[\![t^6,t^{11},t^{15},t^{31}]\!]$ and $I=(t^6,t^{11},t^{31})$
are as in Example \FPnoIneq,  \me where $R = K[\![t^s : s \in S_3]\!]$
and $I$ corresponds to the semigroup ideal $I_3$ of $S_3$.
Since $r(I) = 2$, it follows from \cite{DGH, Proposition 2.2}
that the asymptotic Ratliff-Rush number  $h(I) \le 2$.
Thus $\widetilde{I^n} = I^n$ for each $n \ge 2$. Hence it 
suffices to show that $I = IR^I \cap R$. Observe that 
$R^I = R[t^5] = K[\![t^5, t^6]\!]$, $IR^I = t^6R^I$, 
 and $\lambda(R/I) = 2$.
Thus it suffices to show that $t^{15} \not\in t^6K[\![t^5, t^6]\!]$,
and this is clear since $t^9 \not\in K[\![t^5, t^6]\!]$. 

We remark that for this ring $R = K[\![t^6, t^{11}, t^{15}, t^{31}]\!]$,
the associated graded ring $G(\m) = F(\m)$ of the maximal ideal 
$\m$ is not Cohen--Macaulay. This is readily seen from the fact
that the image of $t^6$ in $\m/\m^2$ in $G(\m)$ is a zero divisor
of $G(\m)$ since $t^{31} \not\in \m^2$ and $t^6t^{31} = 
t^{15}(t^{11})^2 \in \m^3$. 

\vskip 3ex


\dis
\label{\fibermult} 
Suppose $(R,\m)$ is a one-dimensional Cohen--Macaulay local
ring of multiplicity $e$. It is well known that every
$\m$-primary ideal $I$ of $R$ can be generated by $e$ 
elements \cite{SV, Theorem 1.1}.
Therefore $F(I)$ has multiplicity at most $e$ for each
$\m$-primary ideal $I$ of $R$. Moreover,
high powers of $\m$ require $e$ generators and
$F(\m) = G(\m)$ has multiplicity $e$. Also
there exist $\m$-primary principal 
ideals of $R$ and it is clear that if $I$ is a principal
$\m$-primary ideal, then $F(I)$ has multiplicity one. 
Thus the integers $1$ and $e$ are the multiplicity of 
$F(I)$ for $\m$-primary ideals $I$ of $R$.
It is natural to ask what integers $f$ with 
$1 \le f \le e$ are realized as the
multiplicity of $F(I)$ for some 
$\m$-primary ideal $I$  of $R$. We exhibit in 
Example \onlytwo \me the existence,  for each positive 
integer $e \ge 2$, of a one-dimensional 
Cohen--Macaulay local domain $(R,\m)$ having 
multiplicity $e$ such that  for each $\m$-primary
ideal $I$ of $R$, either $I$ is principal and $F(I)$ has
multiplicity one, or else $F(I)$ has multiplicity $e$. 
On the other hand, we observe in Example \every \me
the existence of a one-dimensional Cohen--Macaulay 
local domain $(R, \m)$ of multiplicity $e$ 
such that for every integer $f$ with $1 \le f \le e$
there exists an $\m$-primary ideal $I$ with 
$F(I)$ having multiplicity $f$.

\vskip 2ex

\ex
\label{\onlytwo}
Fix a positive integer $e \ge 2$ and let $K/E$ be an algebraic
extension of fields such that $[K:E] = e$ and such that
there are no fields properly between $E$ and $K$. Let
$t$ be an indeterminate over $K$,  let $S = K[t]_{(t)}$,
and $R = E + tS$. Then $S$ is the
integral closure of $R$ and $S$ is the only subring of $S$
that properly contains $R$. Hence if $I$ is a nonprincipal
$\m$-primary ideal of $R$, then $R^I = S$, so $F(I)$ has
multiplicity $e$. 


\vskip 2ex

\ex
\label{\every}
Fix a positive integer $e \ge 2$ and let $K$ be a field.
Let $t$ be an indeterminate over $K$ and let 
$R = K[t^e, t^{e+1}, \ldots, t^{2e-1}]_{(t^e, t^{e+1}, \ldots,t^{2e-1})}$.
Fix an integer $k$ with $1 \le k < e$. Let 
$I = (t^e, t^{2e-k}, t^{2e-k+1}, \ldots, t^{2e-1})$. 
Then $R^I$ as an $R$-module is minimally generated by 
the $k+1$ elements $1, t^{e-k}, t^{e-k+1}, \ldots, t^{e-1}$. 
Therefore by Lemma \mult \me $F(I)$ has multiplicity $f := k+1$


\vskip 2ex



In Example \highmult \me we exhibit a 3-dimensional
Cohen--Macaulay local domain $(R,\m)$ of multiplicity 2
with the following property. For
each positive integer $k$, the ring $R$ has  a 
stable  ideal $I_k$ having  a principal reduction
such that the fiber cone $F(I_k)$ 
of $I_k$ is Cohen--Macaulay and has multiplicity $k + 2$. 
Therefore, in contrast with the case where $\dim R = 1$, for
$R$ of higher dimension there may exist no upper
bound on the multiplicity of $F(I)$ as $I$ varies 
over the ideals of $R$ that have a principal reduction.
\vskip 3ex

\ex
\label{\highmult}
Let $(S, \n)$ be a two-dimensional regular local ring  and let 
$R= S[t^2, t^3]_{(\n, t^2, t^3)}$, where $t$ is an
indeterminate over $S$. 
Fix a positive integer $k$, and let 
$I=(t^2, \n^k t^3)R$. Since $\n^k$ is minimally generated
by $k+1$ elements as an ideal in $S$, we see that $I$ is
minimally generated by $k + 2$ elements. Moreover,
$I^2 = (t^4, \n^kt^5) = t^2I$. Thus $I$ has a principal
reduction and is a stable ideal, i.e., $r(I) = 1$. It 
follows that $F(I)$ is Cohen--Macaulay and has multiplicity
$f_0 = k + 2$.

\vskip 2ex

\section{A bound for the reduction number.}

\vskip 2ex

Suppose $(R,\m)$ is a one-dimensional Cohen--Macaulay
local ring of multiplicity $e$. It is well known from
results of Sally-Vasconcelos \cite{SV} and 
Eakin-Sathaye \cite{ES} that $e-1$ is then an upper
bound for the reduction number $r(I)$ of $\m$-primary
ideals of $R$ having a principal reduction. In a 
higher dimensional situation, S. Huckaba proves in
\cite{Hu1, Theorem 2.5} that if $(R,\m)$ is a quasi-unmixed
analytically unramified local ring containing an infinite
field of characteristic $\ne 2$ and if the multiplicity of
$R$ is 2, then 
each regular ideal $I$ of $R$ of analytic spread one
has  reduction number $r(I) \le 1$. Now the two-dimensional
regular local domain $S$ of Example \highmult \me can
be chosen so that $R$ as in Example \highmult \me 
satisfies all these properties. It then follows that
for ideals $I$ of $R$ having a principal reduction there
is no upper bound on the multiplicity of $F(I)$, but the 
reduction number $r(I) \le 1$ for each $I$.


Huckaba  \cite{Hu1, Question 2.6} raises the interesting
question of  
whether an analogous result to \cite{Hu1, Theorem 2.5} 
holds in the case where  $R$ has 
multiplicity $e > 2$. To illustrate a special case of
this question, suppose $(R,\m)$ is a complete
Noetherian local domain of dimension $d$ containing 
an infinite  coefficient field $K$. Then there
exist $x_1, \dots x_d \in \m$ that form a system 
of parameters for $R$ and generate a reduction 
of $\m$. Since $R$ is complete $A := K[[x_1, \dots ,x_d]]$
is a $d$-dimensional regular local subring of $R$ 
and $R$ is a finitely generated $A$-module.  Moreover,
the multiplicity $e$ of $R$ is precisely the
degree of the fraction field extension $[Q(R):Q(A)]$
\cite{ZS, Corollary 2, page 300}. In this situation,
$\lambda(R/(x_1, \dots,x_d)R) \ge e$ is the number 
of generators for $R$ as an $A$-module and $R$ is
Cohen--Macaulay if and only if $R$ is a free 
$A$-module if and only if $\lambda(R/(x_1,\dots,x_d)R) = e$.
Since $R$ is complete, the integral closure $\overline R$ 
of $R$ is again local and is a finitely generated $R$-module
and thus also a finitely generated $A$-module.
The freeness lemma of Hironaka \cite{N, (25.16)} 
implies that  $\overline R$ is a free $A$-module if and only
if $\overline R$ is Cohen--Macaulay if and only if
$\lambda(\overline R/(x_1,\dots,x_d)\overline R) = e$. If
$\overline R$ has residue field $K$, then $e$ is also the
multiplicity of $\overline R$, but if the residue field 
of $\overline R$ is a proper extension of the residue 
field $K$ of $R$ with respect to the canonical inclusion map 
of $R$ into $\overline R$, then $\overline R$ has multiplicity
less than $e$.



The following interesting observation shown to us by Craig 
Huneke proves the existence of a global bound
on the reduction number $r(I)$ of ideals having a 
principal reduction in certain rings $R$.



\vskip 2ex

\prop
\label{\huneke}
Suppose $I$ is a regular ideal of a ring $R$ and that
$xR$ is a principal reduction of $I$. If there
exists an $n$-generated faithful $R$-module 
$M = <m_1, \ldots, m_n>$ such that $IM = xM$, then
$I$ has reduction number $r(I) \le n-1$.
\endb

\proof 
Let $u_1, \ldots, u_n$ be (not necessarily distinct) elements
of $I$. Since $IM = xM$, we obtain $n$ equations
$u_i m_i = x\sum_{j=1}^n a_{ij} m_j$, where the $a_{ij} \in R$.
Rearranging to make the system of equations homogeneous,
we obtain a coefficient matrix  $A$ 
that applied to the column vector $(m_1, \ldots, m_n)^T$ gives the zero vector. 
By multiplying $A$ by its adjoint, one sees that $\det(A) \in R$ is  
in $(0:_R M) = (0)$. Thus $\det(A) = 0$.
The explicit expression of $\det(A)$
shows that $u_1 \cdots u_n \in x I^{n-1}$. In conclusion 
$I^n = xI^{n-1}$ and $r(I) \le n - 1$. \qed

\vskip 2ex
In general, if $\overline R$ is the integral closure
of an integral domain $R$, then principal ideals of $\overline R$ are
integrally closed. Thus if $I$ is an ideal of $R$ 
having a principal reduction $xR$, then 
$x\overline R  = I\overline R$. Thus
the following corollary to Proposition \huneke \me is immediate.

\vskip 2ex

\cor
\label{\globalbound}
Suppose  $R$ is an integral  domain and that the integral closure 
$\overline R$ of $R$ in its fraction field is $n$-generated as
an $R$-module. Then each ideal $I$ of $R$ having a 
principal reduction has reduction number $r(I) \le n - 1$
\endb

We record several other results that follow from
Proposition \huneke. 
\vskip 2ex

\cor
\label{\multbound}
Suppose $(R,\m)$ is a Noetherian local ring and $I$ is 
a regular ideal of $R$ having a principal reduction 
$xR$. Let $R^I = R[I/x]$ denote the blowing up ring of $I$.
Then the reduction number $r(I) \le \lambda(R^I/\m R^I) - 1$,
so $r(I) \le f_0 - 1$, where $f_0$ is the multiplicity of 
the fiber ring $F(I)$.
\endb
\proof
Since $R^I$ is a faithful $R$-module and $IR^I = xR^I$,
Proposition \huneke \me implies the first assertion.  
The second assertion follows from Lemma \mult. \qed


\vskip 2ex

\cor
\label{\cohenint}
Suppose  $(R,\m)$ is a complete local 
domain of multiplicity $e$ containing an infinite  field.
If the integral closure $\overline R$ of $R$ is Cohen--Macaulay,
then every ideal $I$ of $R$ of analytic spread one has
reduction number $r(I) \le e - 1$.
\endb

\proof
Since $R$ is complete, $R$ contains a coefficient field $K$.
If $(x_1, \ldots,x_d)$ is a minimal reduction of $\m$, then 
$A := K[[x_1, \ldots, x_d]]$ is a $d$-dimensional regular
local subring of $R$, $R$ is a finitely generated $A$-module
and $[Q(R): Q(A)] = e$. Since $\overline R$ is Cohen--Macaulay,
$\overline R$ is a free $A$-module on $e$ generators. 
Therefore $\overline R$ is $e$-generated as 
an $R$-module. By Proposition \huneke,  \me  
the reduction number $r(I) \le e-1$ 
for each ideal $I$ of $R$ having a principal reduction,
or equivalently in our setting, each ideal of analytic
spread one. \qed

\vskip 2ex

\rmk
\label{\analyticirr}
There are other situations to which 
Proposition \huneke \me applies.
Suppose $(R,\m)$ is a Noetherian local ring and let
$\widehat R$ denote the $\m$-adic completion of $R$.
If $I$ is a regular ideal of $R$ having a principal
reduction $xR$, then $x\widehat R$ is a principal 
reduction of $I\widehat R$. Moreover, 
$xI^n = I^{n+1}$ if and only if $xI^n\widehat R =
I^{n+1}\widehat R$. Thus $r(I) = r(I\widehat R)$.
Also $R$ and $\widehat R$ have the same multiplicity.
Thus if $(R,\m)$ is an analytically irreducible 
Noetherian local domain of multiplicity $e$ and if
its completion $\widehat R$ contains an infinite 
field and has a Cohen--Macaulay normalization,
then each ideal $I$ of $R$ of analytic spread one
has reduction number $r(I) \le e-1$.
 

\vskip 2ex

\cor
\label{\twodim}
Suppose  $(R,\m)$ is a $2$-dimensional complete local 
domain of multiplicity $e$ containing an infinite  field.
Then every ideal $I$ of $R$ of analytic spread one has
reduction number $r(I) \le e - 1$.
\endb

\proof 
Since a $2$-dimensional integrally closed Noetherian
domain is Cohen--Macaulay, this follows from 
Corollary \cohenint. \me \qed

\vskip 2ex

\cor
\label{\analytictwodim} 
Suppose $(R,\m)$ is a 2-dimensional  Noetherian analytically 
irreducible local domain containing an infinite field.
If $R$ has multiplicity $e$, then 
every ideal $I$ of $R$ of analytic spread one has
reduction number $r(I) \le e - 1$. 
\endb

\proof 
This follows from Remark \analyticirr \me and Corollary \twodim.



\vskip 2ex



\rmk
\label{\serre}
It is known that a complete local ring that contains
a field and satisfies Serre's condition $S_n$ is 
Cohen--Macaulay if it has multiplicity $\le n$ \cite{H2}.
Thus Corollary \cohenint \me yields in a special case
the result of Huckaba \cite{Hu1, Theorem 2.5} mentioned
above.





\vskip 3ex

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\endRefs

\vskip 4ex
\centerline{\sl Universit\`a di Catania - Dipartimento di Matematica,}
\centerline{\sl Viale Andrea Doria, 6 - 95125 Catania, Italy}
\centerline{\sl E-mail: {mdanna\@dipmat.unict.it}}
\vskip 2ex
\centerline{\sl Universit\`a di L'Aquila - Dipartimento di Matematica,}
\centerline{\sl Via Vetoio, Loc. Coppito -  67100 L'Aquila, Italy}
\centerline{\sl E-mail: {guerran\@univaq.it}}
\vskip 2ex
\centerline{\sl Purdue University - Department of Mathematics,}
\centerline{\sl West Lafayette, Indiana 47907, USA}
\centerline{\sl E-mail: {heinzer\@math.purdue.edu}}


\footnote{ }{{\footitfont 1991 Mathematics Subject Classification.}
13A30, 13C05, 13E05, 13H15}

\footnote{ }{{\footitfont Key words and phrases.}
\footfont principal reduction, fiber cone, fiber postulation number, 
 multiplicity, Cohen--Macaulay.}




\enddocument

