Good ideals in Gorenstein local rings

Abstract:

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