On the limit closure of a sequence of elements in local rings

TL;DR

This paper systematically studies the limit closure of sequences in local rings, characterizing systems of parameters, and providing explicit computations and examples, with implications for unmixed rings.

Contribution

It offers new characterizations of systems of parameters and a topological description of unmixed local rings, extending previous results.

Findings

Identifies elements always in the limit closure of a system of parameters.

Provides a topological characterization of unmixed local rings.

Explicitly computes the limit closure in two-dimensional cases.

Abstract

We present a systematic study for the limit closure $(\underline{x})^{\lim}$ of a sequence of elements $\underline{x}$ (eg. a system of of parameters) in a local ring. Firstly, we answer the question which elements are always contained in the limit closure of a system of parameters. Then we apply this result to give a characterization of systems of parameters which is a generalization of previous results of Dutta and Roberts in \cite{DR} and of Fouli and Huneke in \cite{FH}. We also prove a topological characterization of unmixed local rings. In two dimensional case, we compute explicitly the limit closure of a system of parameters. Some interesting examples are given.