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

Abstract:

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. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal ${\mathfrak {a}}$ of $R$ has linear growth of primary decompositions, then tight closure (of ${\mathfrak {a}}$) ‘commutes with localization at the powers of a single element’. It is shown in this paper that, provided $R$ has a weak test element, linear growth of primary decompositions for other sequences of ideals of $R$ that approximate, in a certain sense, the sequence of Frobenius powers of ${\mathfrak {a}}$ would not only be just as good in this context, but, in the presence of a certain additional finiteness property, would actually imply that tight closure (of ${\mathfrak {a}}$) commutes with localization at an arbitrary multiplicatively closed subset of $R$. Work of M. Katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the Frobenius powers of ${\mathfrak {a}}$ has only finitely many maximal members. This paper develops, through a careful analysis of the ideal theory of the perfect closure of $R$, strategies for showing that tight closure (of a specified ideal ${\mathfrak {a}}$ of $R$) commutes with localization at an arbitrary multiplicatively closed subset of $R$ and for showing that the union of the associated primes of the tight closures of the Frobenius powers of ${\mathfrak {a}}$ is actually a finite set. Several applications of the strategies are presented; in most of them it was already known that tight closure commutes with localization, but the resulting affirmative answers to Katzman’s question in the various situations considered are believed to be new.