𝐹-regularity, test elements, and smooth base change

This paper deals with tight closure theory in positive characteristic. After a good deal of preliminary work in the first five sections, including a treatment of F F -rationality and a treatment of F F -regularity for Gorenstein rings, a very widely applicable theory of test elements for tight closure is developed in § 6 6 and is then applied in § 7 7 to prove that both tight closure and F F -regularity commute with smooth base change under many circumstances (where "smooth" is used to mean flat with geometrically regular fibers). For example, it is shown in § 6 6 that for a reduced ring R R essentially of finite type over an excellent local ring of characteristic p p, if c c is not in any minimal prime of R R and R c Rc is regular, then c c has a power that is a test element. It is shown in § 7 7 that if S S is a flat R R -algebra with regular fibers and R R is F F </inline-f