Note on Good Ideals in Gorenstein Local Rings

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 $$ I\in \Cal{X}_A(Q) \Longleftrightarrow IS_P \in \Cal{X}_{S_P}(Q_P). $$