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This paper is concerned with tight closure in a commutative Noetherian ring$R$ of prime characteristic $p$, and is motivated by an argument of K. E. Smithand I. Swanson that shows that, if the sequence of Frobenius powers of a properideal $I$ of $R$ has linear growth of primary decompositions, then tightclosure (of $I$) commutes with localization at the powers of a single element.It is shown in this paper that, provided $R$ has a weak test element, lineargrowth of primary decompositions for other sequences of ideals of $R$ thatapproximate, in a certain sense, the sequence of Frobenius powers of $I$ wouldnot only be just as good in this context, but, in the presence of a certainadditional finiteness property, would actually imply that tight closure (of$I$) commutes with localization at an arbitrary multiplicatively closed subsetof $R$.  Work of M. Katzman on the localization problem for tight closure raised thequestion as to whether the union of the associated primes of the tight closuresof the Frobenius powers of $I$ has only finitely many maximal members. Thispaper develops, through a careful analysis of the ideal theory of the perfectclosure of $R$, strategies for showing that tight closure (of a specified ideal$I$ of $R$) commutes with localization at an arbitrary multiplicatively closedsubset of $R$ and for showing that the union of the associated primes of thetight closures of the Frobenius powers of $I$ is actually a finite set. Severalapplications of the strategies are presented; in most of them it was alreadyknown that tight closure commutes with localization, but the resultingaffirmative answers to Katzman's question in the various situations consideredare believed to be new.

Ideals in a perfect closure, linear growth of primary decompositions, and tight closure