Good Ideals in Gorenstein Local Rings

Let I be an m-primary ideal in a Gorenstein local ring (A,m) with dimA = d, and assume that I contains a parameter ideal Q in A as a reduction. We say that I is a good ideal in A if G = ∑ n≥0 I n/In+1 is a Gorenstein ring with a(G) = 1−d. The associated graded ring G of I is a Gorenstein ring with a(G) = −d if and only if I = Q. Hence good ideals in our sense are good ones next to the parameter ideals Q in A. A basic theory of good ideals is developed in this paper. We have that I is a good ideal in A if and only if I2 = QI and I = Q : I. First a criterion for finite-dimensional Gorenstein graded algebras A over fields k to have nonempty sets XA of good ideals will be given. Second in the case where d = 1 we will give a correspondence theorem between the set XA and the set YA of certain overrings of A. A characterization of good ideals in the case where d = 2 will be given in terms of the goodness in their powers. Thanks to Kato’s Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set XA of good ideals in A heavily depends on d = dimA. The set XA may be empty if d ≤ 2, while XA is necessarily infinite if d ≥ 3 and A contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring k[X1,X2,X3] in three variables over a field k. Examples are given to illustrate the theorems.