Good ideals in Gorenstein local rings

Abstract

Let I I be an m \mathfrak {m} -primary ideal in a Gorenstein local ring ( A A , m \mathfrak {m} ) with dim ⁡ A = d \dim A = d , and assume that I I contains a parameter ideal Q Q in A A as a reduction. We say that I I is a good ideal in A A if G = ∑ n ≥ 0 I n / I n + 1 G = \sum _{n \geq 0} I^{n}/I^{n+1} is a Gorenstein ring with a ( G ) = 1 − d \mathrm {a} (G) = 1 - d . The associated graded ring G G of I I is a Gorenstein ring with a ( G ) = − d \mathrm {a}(G) = -d if and only if I = Q I = Q . Hence good ideals in our sense are good ones next to the parameter ideals Q Q in A A . A basic theory of good ideals is developed in this paper. We have that I I is a good ideal in A A if and only if I 2 = Q I I^{2} = QI and I = Q : I I = Q : I . First a criterion for finite-dimensional Gorenstein graded algebras A A over fields k k to have nonempty sets X A \mathcal {X}_{A} of good ideals will be given. Second in the case where d = 1 d = 1 we will give a correspondence theorem between the set X A \mathcal {X}_{A} and the set Y A \mathcal {Y}_{A} of certain overrings of A A . A characterization of good ideals in the case where d = 2 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 X A \mathcal {X}_{A} of good ideals in A A heavily depends on d = dim ⁡ A d = \dim A . The set X A \mathcal {X}_{A} may be empty if d ≤ 2 d \leq 2 , while X A \mathcal {X}_{A} is necessarily infinite if d ≥ 3 d \geq 3 and A A contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring k [ X 1 , X 2 , X 3 ] k[X_{1},X_{2},X_{3}] in three variables over a field k k . Examples are given to illustrate the theorems.