Multiplicities and reductions of adjacent integrally closed ideals in two-dimensional regular local rings

Sunsook Noh, Purdue University

Abstract

Let (R,m) be a 2-dimensional regular local ring with algebraically closed residue field. For two adjacent integrally closed ideals $I \supseteq J$, the multiplicity difference $e(J)$ $-$ $e(I)$ is studied. We obtain a formula for $e(J)$ $-$ $e(I)$ in terms of the Rees variations of J which are not Rees valuations of I. We obtain a sharper upper bound for $e(J) - e(I)$ in certain cases. When J is simple associated to the prime divisor w of the second kind, we show that $w(I) = w(J) - 1, e(J) - e(I) = 1$, and I is a w-ideal. Furthermore, it is shown that I is the unique integrally closed ideal right above J. We observe that a generalization of this result to $J\sp{n}$ for $n \geq 2$ is not possible even though $w(J\sp{n}) - w(I) = 1$ for an arbitrary integrally closed ideal I right above $J\sp{n}$. In particular, we show that there are infinitely many integrally closed ideals right above $J\sp{n}$. We also show that there is more than one, and sometimes infinitely many integrally closed ideals right above J in certain other cases where J is not simple. It is shown that the value semigroup $w(R)$ on R for a prime divisor w of the second kind of R is symmetric. We also show that the value semigroups of v and w on R which are associated to two adjacent simple integrally closed ideals are equal. For a prime divisor w, the sequence of w-ideals in R is discussed and the lengths between some of these w-ideals are measured. The intersection multiplicity $(I$ $\cdot$ $J)$ of two integrally closed ideals I and J is obtained in terms of the Rees valuations of either I or J. And this is used for measuring the lengths between certain w-ideals. Minimal reductions of adjacent integrally closed ideals are studied.

Degree

Ph.D.

Advisors

Heinzer, Purdue University.

Subject Area

Mathematics

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