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\begin{center}
{\large\bf
Almost complete intersection
lattice ideals
}
\end{center}

\vspace{5mm}

\begin{center}
Kazufumi Eto
\end{center}

\vspace*{10mm}


\noindent
\centerline {Department of Mathematics,}\newline
\centerline {Nippon Institute of Technology,}\newline
\centerline {Saitama 345-8501, Japan} 

\vspace{5mm}


In this talk, 
we will investigate minimal generating systems of 
almost complete intersection lattice ideals.

First, we introduce some definitions and notations.
Let $N>r>0$ be natural numbers,
\Z\ the ring of integers,
$A=k[X_1, \dots, X_N]$
a polynomial ring over a field $k$
and $B=k[X_1^{\pm1}, \dots, X_N^{\pm1}]$.
For $v\in\Z^N$, we write
$v^-$ (resp. $v^+$)
for the negative part (resp. the positive part) of $v$,
hence $v=v^-+v^+$.
We also write
$X^v$ in place of $\prod_{i=1}^N X_i^{v_i}$
where $v_i$ is the $i$-th entry of $v$
and $F(v)=X^{-v^-}-X^{v^+}\in A$.
For a submodule $V$ in $\Z^N$ of rank $r$,
put $I(V)$ the ideal $(1-X^v)_{v\in V}\cap A$ in $A$,
called a \textit{lattice ideal} of $V$.
Then $I(V)$ is an ideal generated by binomials 
of the form $F(v)$
of height $r$
and any $X_i$ is non zero divisor on $A/I(V)$.

We always assume that $V$ is contained in 
the kernel of a map $\Z^N\to\Z$
defined by natural numbers $n_1, \dots, n_N$.
Then $I(V)$ is a homogeneous ideal
when we put $\deg X_i=n_i$ for each $i$.
We use the degree in this sence.

About complete intersection lattice ideals,
following theorem holds.

\begin{thm}[{\cite[Theorem 2.4]{tokyo}}]
Let $V\subset\Z^N$ a submodule of rank $r$.
For $v_1, \dots, v_r\in V$,
$I(V)=(F(v_1), \dots, F(v_r))$
if and only if following two conditions are satisfied;
\begin{enumerate}
\item
$V=\sum_{j=1}^r\Z v_j$,
\item
For any $S\subset[1, N]$
and for any $T\subset[1, r]$ with $|S|=|T|$,
there is $j\in T$ with $F(v_j)\notin (X_i)_{i\in S}$.
\end{enumerate}
\end{thm}

We want to know whether the same type theorem as before
holds for almost complete intersection case.
In fact, following holds.

\begin{thm}
Let $V\subset\Z^n$ a submodule of rank $r$.
Assume that $I(V)$ is an almost complete intersection
and generated by $F(v_1), \dots, F(v_{r+1})$.
And further assume $v_1+v_2+\cdots+v_{r+1}=0$.
If there are proper subsets $S\subset[1, N]$
and $T\subset[1, r+1]$ with $|S|=|T|$
satisfying $F(v_j)\in(X_i)_{i\in S}$ for each $j\in T$,
then the ideal $(F(v_j))_{j\notin T}$
is a complete intersection lattice ideal $I(W)$
where $W=\sum_{j\notin T}\Z v_j$.
\end{thm}

To use this theorem,
we will study minimal generating systems of
almost complete intersection lattice ideals
in general case.

\begin{thebibliography}{99}

\bibitem{comm1}
K.~Eto,
\newblock {Almost complete intersection monomial curves in A$^4$},
\newblock {\it Comm. in Algebra}, {\bf 22} (1994) 5325-5342.

\bibitem{tokyo}
K.~Eto,
\newblock {{D}efining ideals of complete intersection monoid rings},
\newblock {\it Tokyo J. Math.}, {\bf 18} (1995) 185-191.

\bibitem{comm2}
K.~Eto,
\newblock {A free rersolution of a binomial ideal},
\newblock {\it Comm. in Algebra}, {\bf 27} (1999) 3459-3472.

\bibitem{fs1961}
K.~Fischer and J.~Shapiro.
\newblock Mixed matrices and binomial ideals.
\newblock {\em J. Pure and Appl. Alg.}, {\bf 113}:39--54, (1996).

\bibitem{kunz}
E.~Kunz,
\newblock {\it Introduction to Commutative Algebra and Algebraic Geometry},
\newblock  Birkh\"auser, Boston, 1985.

\end{thebibliography}

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