In this article, we solve Grothendieck's localization problem for a certain class of rings that arises from tight closure theory. It is quite essential to use the relative version of the Frobenius map (Raclu-Andre morphism) to investigate the fibers of a flat ring homomorphism of characteristic p > 0. We also give some interesting geometric applications of the localization problem. (C) 2009 Published by Elsevier B.V.