On the complete intersection conjecture of Murthy

TL;DR

This paper proves Murthy's conjecture on the minimal number of generators of ideals in polynomial rings over infinite fields, extending recent results to broader classes of rings.

Contribution

It extends Murthy's conjecture proof to polynomial rings over infinite fields and related algebraic structures, generalizing previous work by Fasel.

Findings

Murthy's conjecture holds for polynomial rings over infinite fields.

The results apply to ideals containing monic polynomials in polynomial rings.

The paper generalizes the conjecture to regular rings over perfect fields.

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 $k$.