On the complete intersection conjecture of Murthy

Abstract

Suppose $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$ and $I$ is an ideal in $A$. Then M. P. Murthy conjectured that $\mu(I)=\mu(I/I^2)$, where $\mu$ denotes the minimal number of generators. Recently, Fasel \cite{F} settled this conjecture, affirmatively, when $k$ is an infinite perfect field, with $1/2\in k$ {\rm (always)}. We are able to do the same, when $k$ is an infinite field. In fact, we prove similar results for ideals $I$ in a polynomial ring $A=R[X]$, that contains a monic polynomial and $R$ is essentially finite type smooth algebra over an infinite field $k$, or $R$ is a regular ring over a perfect field $k$.