Formal curves are set theoretic complete intersections

TL;DR

This paper proves that in a formal power series ring over any field, every one-dimensional prime ideal can be expressed as the radical of exactly d-1 elements, showing a set-theoretic complete intersection property.

Contribution

It establishes that one-dimensional prime ideals in formal power series rings are set-theoretic complete intersections with a minimal number of generators.

Findings

One-dimensional prime ideals are radicals of d-1 elements.

The result holds over any field and in formal power series rings.

Provides a constructive approach to generating prime ideals as set-theoretic intersections.

Abstract

Let $k$ be any field. Let $R=k[[X_1,\ldots,X_d]]$ and $\frak p$ a $1$-dimensional prime ideal. In this note we present $d-1$ elements such as $f_1,\ldots,f_{d-1}$ in $\frak p$ such that $\mathfrak{p}=\text{rad}(f_1,\ldots,f_{d-1})$.