The structure of the core of ideals

The core of an $R$-ideal $I$ is the intersection of all reductions of $I$. This object was introduced by D. Rees and J. Sally and later studied by C. Huneke and I. Swanson, who showed in particular its connection to J. Lipman’s notion of adjoint of an ideal. Being an a priori infinite intersection of ideals, the core is difficult to describe explicitly. We prove in a broad setting that: $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(I)$ is a finite intersection of minimal reductions; $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(I)$ is a finite intersection of general minimal reductions; $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(I)$ is the contraction to $R$ of a `universal’ ideal; $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(I)$ behaves well under flat extensions. The proofs are based on general multiplicity estimates for certain modules.