Localization and Test Exponents for Tight Closure

We introduce the notion of a test exponent for tight closure, and explore its relationship with the problem of showing that tight closure commutes with localization, a longstanding open question. Roughly speaking, test exponents exist if and only if tight closure commutes with localization: mild conditions on the ring are needed to prove this. We give other, independent, conditions that are necessary and sufficient for tight closure to commute with localization in the general case, in terms of behavior of certain associated primes and behavior of exponents needed to annihilate local cohomology. While certain related conditions (the ones given here are weaker) were previously known to be sufficient, these are the first conditions of this type that are actually equivalent. The difficult calculation of §4 uses associativity of multiplicities and many other tools to show that sufficient conditions for localization to commute with tight closure can be given in which asymptotic statements about lengths of modules defined using the iterates of the Frobenius endomorphism replace the finiteness conditions on sets of primes introduced in §3. The result is local and requires special conditions on the rings: one is that countable prime avoidance holds. This is not a very restrictive condition however: it suffices, for example, for the ring to contain an uncountable field. Countable prime avoidance also holds in any complete local ring. But we also need the existence of a strong test ideal (see