On two conjectures of Murthy

Abstract

This article concerns two conjectures of M. P. Murthy. For Murthy's conjecture on complete intersections, the major breakthrough has still been the result proved by Mohan Kumar in 1978. In this article we improve "Mohan Kumar's bound" when the base field is $\overline{\mathbb F}_p$, and illustrate some applications of our result. Murthy's other conjecture is on a "splitting problem", which is roughly about finding the precise obstruction for a projective $R$-module $P$ of rank $\text{dim}(R)-1$ to split off a free summand of rank one, where $R$ is a smooth affine algebra over an algebraically closed field $k$. Asok-Fasel achieved the initial breakthrough, by settling it for $3$-folds and $4$-folds when $char(k)\neq 2$. For $k=\overline{\mathbb F}_p$ ($p\neq 2$) and $\text{dim}(R)\geq 5$ we define an obstruction group and an obstruction class for $P$ (whose determinant is trivial). As application we obtain: $P$ splits if and only if it maps onto a complete intersection ideal of height $\text{dim}(R)-1$.