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.

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.