  The aim of this paper is to extend the main result of C. Huneke and G.
Lyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are
images of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian
domain of positive characteristic that is an image of a Cohen-Macaulay local
ring. We prove that all local cohomology of $R$ (below the dimension) maps to
zero in a finite extension of the ring. As a direct consequence we obtain that
the absolute integral closure of $R$ is a big Cohen-Macaulay algebra. Since
every excellent local ring is an image of a Cohen-Macaulay local ring, this
result is a generalization of the main result of M. Hochster and Huneke in
[Ann. of Math. 135 (1992), 45--79] with a simpler proof.
