
\titleofpresentation{Ratliff-Rush Monomial Ideals}

\presenter{Veronica Crispin Qui{\~n}onez} 

Let $R$ be a Noetherian ring and let an ideal $I$ in it be regular,
that is, let $I$ contain a nonzerodivisor. Then the ideals
$(I^{l+1}:I^l),\; l\geq 1,$ increase with $l$. The union $\tilde
I=\bigcup_{l\geq 1}^{\infty} (I^{l+1}:I^l)$ was first studied by
Ratliff and Rush in \cite{crispinRR}. They show that $(\tilde I)^l=I^l$ for
sufficiently large $l$ and that $\tilde I$ is the largest
ideal with this property. Hence, $\tilde{\tilde I} =\tilde
I$. Moreover, they show that $\widetilde{I^l}=I^l$ for sufficiently
large $l$. We call $\tilde I$ the Ratliff-Rush ideal associated to
$I$, and an ideal such that $\tilde I=I$ a Ratliff-Rush ideal. The
Ratliff-Rush reduction number of $I$ is defined as $r(I)=\min\,\{l\in\mathbb Z_{\geq
  0}\:\vert\:\tilde I=(I^{l+1}:I^l)\}$. The operation $\:\tilde{}\:$ cannot be considered as a closure
operation in the usual sense, since $J\subseteq I$ does not generally
imply $\tilde J\subseteq\tilde I$. An example from \cite{crispinRS} shows this: let
$J=\langle y^4,xy^3,x^3y,x^4\rangle\subset  I=\langle
y^3,x^3\rangle\subset k[x,y]$, then $I$ is Ratliff-Rush but
$x^2y^2\in\tilde J\backslash\tilde I$.

One of the reasons to study Ratliff-Rush ideals is the following. Let
$I$ be a regular $\mathfrak m$-primary ideal in a local ring
$(R,\mathfrak m,k)$. The Hilbert function $H_I(l)={\rm
  dim}_k (R/I^l)$ is a polynomial $P_I(l)$ called the Hilbert polynomial
of $I$ for all large $l$. Then $\tilde I$ can be defined as the unique
largest ideal containing $I$ and having the same Hilbert polynomial as
$I$.

Ratliff-Rush ideals associated to monomial ideals are monomial by
definition, which makes the computations easier. There is always a
positive integer $L$ such that $\tilde I=I^{L+1}:I^L$, but it is not
clear how big that $L$ is (see Example~1.8 in \cite{crispinRS}). If $I$ is a monomial
ideal and $m$ is some monomial, then for all $l\geq 0$ we have \begin{equation} \label{mIekv}
  (mI)^{l+1}:(mI)^l=(m^{l+1}I^{l+1}):(m^lI^l)=m(I^{l+1}:I^l).
\end{equation} Principal ideals are trivially Ratliff-Rush. Any
non-principal monomial ideal $J$ in the rings $k[x,y]$ and
$k[[x,y]]$ can be written as $J=mI$, where $m$ is a monomial and $I$
is an $\langle x,y\rangle$-primary ideal; hence it suffices
to consider $\langle x,y\rangle$-primary monomial
ideals. Moreover, (\ref{mIekv}) shows that the Ratliff-Rush reduction
numbers of $I$ and $mI$ are the same.

In this talk we show how to compute the Ratliff-Rush ideal
associated to a monomial ideal in a certain class in the rings $k[x,y]$ and $k[[x,y]]$ and find an upper bound
for the Ratliff-Rash reduction number for such an ideal. We start by giving some results about numerical semigroups that are crucial
for our later work. We conclude by duscussing several useful examples.


\begin{ourbiblio}{4}  

%\bibitem{HJLS} W. Heinzer, B. Johnston, D. Lantz and K. Shah, \textit{Coefficient Ideals in and Blowups of a Commutative Noetherian Domain}, J. of Algebra {\bf 162} (1993), 355-391.

\bibitem{crispinRR} L. J. Ratliff, Jr and D. E. Rush, \textit{Two Notes on
    Reductions of Ideals}, Indiana Univ. Math. J. {\bf 27} (1978),
  no. 6, 929-934.

\bibitem{crispinRS} M. E. Rossi and I. Swanson, \textit{Notes on the
  Behavior of
  the Ratliff-Rush Filtration}, Commutative Algebra
    (Grenoble/Lyon, 2001), 313-328, Contemp. Math., 331, Amer. Math. Soc., Providence, RI, 2003.

\end{ourbiblio}

\presenteraddress{KTH Matematik, Royal Institute of Technology, Stockholm, Sweden}

\email{vcrispin@kth.se} 

\web{www.math.kth.se/~vcrispin}
