Tight closure and base change.

We first introduce a very general method of reducing questions about the tight closure of submodules to questions about the tight closure of ideals. Next we examine when the weak F-regularity of the associated graded ring forces the ring itself to be weakly F-regular. This is then used to give a partial answer to the deformation question; under certain circumstances, we show that if $R/(f)$ is F-regular, then so is R. In Chapter 5 we show that the tight closure and Frobenius closure of binomial ideals are again binomial. Situations where the test ideal and characteristic ideals in binomial rings are binomial are also discussed. A new result of Karen E. Smith is also presented showing that tight closure commutes with localization in rings defined by binomial ideals. Given a domain A and an A-algebra $R\sb{A},$ we also examine when information on the tight closure of ideals in the generic fiber of $R\sb{A}$ passes down to almost all fibers. This is followed by a discussion of how tight closure behaves under completion.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/130542/2/9732142.pd