On the vanishing of local cohomology of the absolute integral closure in positive characteristic

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.