A study of <italic>F</italic>-rationality and <italic>F</italic>-injectivity.

Abstract

This thesis deals with various aspects of <italic>F</italic>-rationality and <italic>F</italic>-injectivity, two concepts that are part of tight closure theory. It is studied whether <italic>F</italic>-rationality is preserved by flat maps <italic>R</italic> &rarr; <italic>S</italic> with geometrically <italic> F</italic>-rational fibers, with focus on the local situation. New tools are introduced, such as the Radu-Andre homomorphisms and rings. They are used to prove that <italic>F</italic>-rationality is preserved under flat local base change with <italic>F</italic>-rational generic fiber and geometrically <italic> F</italic>-injective closed fiber, under the mild assumption that the Radu-Andre rings are Noetherian. The corresponding problem for flat local purely inseparable extensions is also considered. Sufficient conditions for <italic> F</italic>-rationality to remain preserved under such base changes whenever the closed fiber is a point are given, together with a number of examples. The thesis also deals with the notion of cyclic covers and applications to tight closure theory. A special type of canonical cover, called the pseudocanonical cover, is defined and used to find sufficient conditions for strong <italic> F</italic>-regularity and <italic>F</italic>-purity. Examples showing how these results can be applied are presented. The Frobenius structure of the local cohomology is investigated as well. The set of <italic>F</italic>-stable primes and the <italic>F</italic>-stability number are introduced. Both are invariants of the ring and provide interesting information on the singularity of the ring. The final chapter discusses several open problems in these areas.