Integral Closures of Ideals and Coefficient Ideals of Monomial Ideals

Abstract

The integral closure $\overline{I}$ of an ideal $I$ in a ring $R$ consists of all elements $x \in R$ that are integral over $I$. If $R$ is an algebra over an infinite field $k$, one can define general elements of $I=\left( x_{1},\ldots,x_{n}\right)$ as $x_{\alpha}=\sum_{i=1}^{n}\alpha_{i}x_{i}$ with $(\alpha_{1},\ldots,\alpha_{n})$ belonging to a Zariski-open subset of $k^{n}$. We prove that for any ideal $I$ of height at least $2$ in a local, equidimensional excellent algebra over a field of characteristic zero, the integral closure specializes with respect to a general element of $I$. That is, we show that $\overline{I}/(x)=\overline{I/(x)}$. In a Noetherian local ring $(R,m)$ of dimension $d$, one has a sequence of ideals approximating the integral closure of $I$ for $I$ an $m$-primary ideal. The ideals \[ I \subseteq I_{\{d\}} \subseteq \cdots \subseteq I_{\{1\}} \subseteq I_{\{0\}}=\overline{I}\]are the coefficient ideals of $I$. The $i^{\text{th}}$ coefficient ideal $I_{\{i\}}$ of $I$ is the largest ideal containing $I$ and integral over $I$ for which the first $i+1$ Hilbert coefficients of $I$ and $I_{\{i\}}$ coincide. With a goal of understanding how coefficient ideals behave under specialization by general elements, we turn to the case of monomial ideals in polynomial rings over a field. A consequence of the specialization of the integral closure is that the $i^{\text{th}}$ coefficient ideal specializes when the $i^{\text{th}}$ coefficient ideal coincides with the integral closure. To this end, we give a formula for first coefficient ideals of $m$-primary monomial ideals generated in one degree in $2$ variables in order to describe when $I_{\{1\}}=\overline{I}$. In the $2$-dimensional case, we characterize the behavior of all coefficient ideals with respect to specialization by general elements. In the $d$-dimensional case for $d \geq 3$, we give a characterization of when $I_{\{1\}}=\overline{I}$ for $m$-primary monomial ideals generated in one degree. In the final chapter, we give an application to the core, by characterizing when $\core(I)=\adj(I^{d})$ for such ideals.