An Example in Complete Intersections and an Erratum

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.