# An Example in Complete Intersections and an Erratum

## TL;DR

This paper discusses the Complete Intersection conjecture, highlights recent counterexamples to a claimed proof, and clarifies inconsistencies in the literature related to ideals in polynomial rings over fields.

## Contribution

It provides clarification on the inconsistencies in the literature regarding the Complete Intersection conjecture and discusses recent counterexamples to a claimed proof.

## Key findings

- Counterexamples to the stronger claim about the conjecture
- Inconsistencies in the literature on the conjecture
- Discussion on the implications of these counterexamples

## Abstract

This is essentially an erratum, with some example to indicate inconsistencies. Suppose $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$. The Complete Intersection conjecture states that, for any ideal $I$ in $A$, $\mu(I)=\mu(I/I^2)$, where $\mu$ denotes the minimal number of generators. When $k$ is an infinite field, with $1/2\in k$, a proof of this conjecture was claimed recently, which was a consequence of a stronger claim. A counter example of this stronger claim surfaced recently. This note discusses such examples and attempts to provide some clarity to the inconsistencies in the literature.

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Source: https://tomesphere.com/paper/1702.00087