# The core of a module and the adjoint of an ideal over a two dimensional regular local ring

## TL;DR

This paper generalizes a fundamental formula for the core of integrally closed ideals to modules over two-dimensional regular local rings, linking the core to the product of the module and the adjoint of an ideal.

## Contribution

It extends the core formula from ideals to modules, providing a new characterization involving the adjoint of an ideal in a two-dimensional regular local ring.

## Key findings

- Core of a module equals the product of the module and the adjoint of an ideal.
- Generalizes the core formula from ideals to modules.
- Shows inclusion of cores under certain module conditions.

## Abstract

The core of an module is the intersection of all its reductions. The main result asserts that the core of a finitely generated, torsion-free, integrally closed module over a two dimensional regular local ring is the product of the module and the adjoint of an ideal. This generalizes the fundamental formula for the core of an integrally closed ideal in a two-dimensional regular local ring due to Huneke and Swanson. As an application, we show that for integrally closed modules $M$ and $N$ over a two-dimensional regular local ring with $M\subset N$ and $M^{**}=N^{**}$, the core of $M$ is contained in the core of $N$.