Note on good ideals in Gorenstein local rings
Bull. Korean Math. Soc. 2002 Vol. 39, No. 3, 479-484
Printed September 1, 2002
Mee-Kyoung Kim
Sungkyunkwan University
Abstract : Let $I$ be an ideal in a Gorenstein local ring $A$ with
the maximal ideal $\frak{m}$ and $d=\dim A$. Then we say that $I$
is a $good$ $ideal$ in $A$, if $I$ contains a reduction
$Q=(a_1,a_2,\cdots,a_d)$ generated by $d$ elements in $A$ and
$\roman{G}(I)=\oplus_{n\geq 0}I^n/I^{n+1}$ of $I$ is a Gorenstein
ring with $\roman{a}(\roman{G}(I))=1-d$, where
$\roman{a}(\roman{G}(I))$ denotes the $\roman{a}$-invariant of
$\roman{G}(I)$. Let $S=A[Q/a_1]$ and $P=\frak{m}S$. In this paper,
we show that the following conditions are equivalent. \roster
\item"{$(1)$}" $I^2=QI$ and $I=Q : I$.
\item"{$(2)$}" $I^2S=a_1IS$ and $IS=a_1S :_{S} IS$.
\item"{$(3)$}" $I^2S_P=a_1IS_P$ and $IS_P=a_1S_P :_{S_P} IS_P$.
\endroster
We denote by $\Cal{X}_A(Q)$ the set of good ideals $I$ in
$\Cal{X}_A$ such that $I$ contains $Q$ as a reduction. As a
Corollary of this result, we show that $$ I\in \Cal{X}_A(Q)
\Longleftrightarrow IS_P \in \Cal{X}_{S_P}(Q_P). $$
Keywords : Rees algebra, associated graded ring, Cohen-Macaulay ring, Gorenstein ring, $\roman{a}$-invariant
MSC numbers : Primary 13A30; Secondary 13H10
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