# NOTE ON GOOD IDEALS IN GORENSTEIN LOCAL RINGS

• Published : 2002.08.01
• 28 3

#### 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 $Q=(a_1,a_2,...,a_d)$ generated by d elements in A and $G(I)=\bigoplus_{n\geq0}I^n/I^{n+1}$ 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$_1$] and P = mS. In this paper, we show that the following conditions are equivalent. (1) $I^2$ = QI and I = Q:I. (2) $I^2S$ = $a_1$IS and IS = $a_1$S：sIS. (3) $I^2$Sp = $a_1$ISp and ISp = $a_1$Sp ：sp ISp. We denote by $X_A(Q)$ the set of good ideals I in $X_A(Q)$ such that I contains Q as a reduction. As a Corollary of this result, we show that $I\inX_A(Q)\Leftrightarrow\IS_P\inX_{SP}(Qp)$.

#### Keywords

Rees algebra;associated graded ring;Cohen-Macaulay ring, Gorenstein ring;a-invariant

#### References

1. Interscience Local rings M. Nagata
2. Trans. Amer. Math. Soc. v.353 Good ideals in Gorenstein local rings S. Goto;S. Iai;K. Watanabe https://doi.org/10.1090/S0002-9947-00-02694-5
3. Pac. J. Math. v.20 Ideals of the Principal Class, R-Sequences and a Certain Monoidal Transformation E. D. Davis https://doi.org/10.2140/pjm.1967.20.197
4. Cambridge studies in advanced mathematics v.39 Cohen-Macaulay rings W. Bruns;J. Herzog
5. Commutative ring theory H. Matsumura
6. J. Math. Soc. Japan v.30 On graded rings, Ⅰ S. Goto;K. Watanabe https://doi.org/10.2969/jmsj/03020179