NOTE ON GOOD IDEALS IN GORENSTEIN LOCAL RINGS

Title & Authors
NOTE ON GOOD IDEALS IN GORENSTEIN LOCAL RINGS
Kim, Mee-Kyoung;

Abstract
Let I be an ideal in a Gorenstein local ring A with the maximal ideal m and d = dim A. Then we say that I is a good ideal in A, if I contains a reduction $\small{Q=(a_1,a_2,...,a_d)}$ generated by d elements in A and $\small{G(I)=\bigoplus_{n\geq0}I^n/I^{n+1}}$ of I is a Gorenstein ring with a(G(I)) = 1-d, where a(G(I)) denotes the a-invariant of G(I). Let S = A[Q/a$\small{_1}$] and P = mS. In this paper, we show that the following conditions are equivalent. (1) $\small{I^2}$ = QI and I = Q:I. (2) $\small{I^2S}$ = $\small{a_1}$IS and IS = $\small{a_1}$S：sIS. (3) $\small{I^2}$Sp = $\small{a_1}$ISp and ISp = $\small{a_1}$Sp ：sp ISp. We denote by $\small{X_A(Q)}$ the set of good ideals I in $\small{X_A(Q)}$ such that I contains Q as a reduction. As a Corollary of this result, we show that $\small{I\inX_A(Q)\Leftrightarrow\IS_P\inX_{SP}(Qp)}$.
Keywords
Rees algebra;associated graded ring;Cohen-Macaulay ring, Gorenstein ring;a-invariant;
Language
English
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