# EQUIMULTIPLE GOOD IDEALS WITH HEIGHT 1

• Published : 2002.01.01

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m. Then we say that I is an equimultiple good ideal in A, if I contains a reduction Q = ( $a_1$, $a_2$,ㆍㆍㆍ, $a_{s}$ ) generated by s elements in A and G(I) =(equation omitted)$_{n 0}$ $I^{n}$ / $I^{n＋1}$ of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = h $t_{A}$ I and a(G(I)) denotes the a-invariant of G(I). Let $X_{A}$$^{s} denote the set of equimultiple good ideals I in A with h t_{A} I = s, R(I) = A [It] be the Rees algebra of I, and K_{R(I)} denote the canonical module of R(I). Let a I such that I^{n＋l} = a I^{n} for some n\geq0 and \mu$$_{A}$(I)$\geq$2, where $\mu$$_{A}(I) denotes the number of elements in a minimal system of generators of I. Assume that A/I is a Cohen-Macaulay ring. We show that the following conditions are equivalent. (1) K_{R(I)}(equation omitted)R(I)＋as graded R(I)-modules. (2) I^2 = aI and aA : I\in X^1$$_{A}$._{A}$./. #### Keywords #### File #### References 1. W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge studies in advanced mathematics, vol. 39, Cambridge University, Cambridge-New York-Port Chester-Sydney, 1993 2. S. Goto, S. Iai, and M. Kim, Good ideals in Gorenstein local rings obtained by idealization, Proc. Amer. Math. Soc. (to appear) 3. S. Gato, S. Iai, and K. Watanabe, Good ideals in Gorenstein local rings, Trans. Amer. Math. Soc. (to appear) 4. S. Gato and M. Kim, Equimultiple good ideals, J. Math. Kyoto Univ. (to appear) 5. S. Goto and K. Watanabe, On graded rings, I, J. Math. Soc. Japan 30 (1978), 179-213 https://doi.org/10.2969/jmsj/03020179 6. J. Herzog and E. Kunz (eds.), Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics, vol. 238, Springer-Verlag, Berlin${\cdot}$Heidelberg${\cdot}$New York${\cdot}$Tokyo, 1971. 7. H. Matsumura, Commutative ring theory, Cambridge University, Cambridge${\cdot}$London${\cdot}\$Sydney, 1986.
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