Commutative Algebra Seminar

Title

Elliptic ideals in 2 dimensional normal local rings

Abstract

This is a joint work in progress with Tomohiro Okuma and Ken-ichi Yoshida.

Let (A, m) be a 2-dimensional excellent normal local ring. Let I be an integrally closed m-primary ideal, and let f : X -> Spec(A) be a resolution of singularity such that I O_X = O_X(-Z) is invertible. Also let Q be a minimal reduction of I. We define

\bar{r}(I) = min {r | \bar{I^{n+1}} = Q bar{I^n} for all n \ge r},

where \bar{I^n} is the integral closure of I^n.

If A is a rational singularity, then H^1(X, O_X(-Z)) = 0 and \bar{I^2} = QI for every integrally closed ideal I, \bar{r}(I) = 1. We call I an elliptic ideal if \bar{r}(I) = 2. This naming comes from the fact that if A is an elliptic singularity, then \bar{r}(I) \le 2 for every integrally closed ideal.

Today we discuss about the following topics:

1. Let \bar{G}(I) denote the associated graded ring of the filtration {\bar{I^n}}. For what I, is \bar{G}(I) Gorenstein? If I is an elliptic ideal, then \bar{G}(I) is Cohen-Macaulay but Gorenstein in very limited cases. We discuss about the condition for \bar{G}(I) to be Gorenstein.

2. We give a formula for Core(I) for elliptic ideals.