On the localization theorem for FF-pure rings

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 (Radu–Andrè morphism) to investigate the fibers of a flat ring homomorphism of characteristic p>0p>0. We also give some interesting geometric applications of the localization problem.