  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 $\text{dim}(R)-1$.
