On the Arithmetic of Tight Closure

This paper deals with a question regarding tight closure in characteristic zero which we now review. Let R be a commutative ring of prime characteristic p and let I ⊆ R be an ideal. Recall that for e ≥ 0, the e-th Frobenius power of I, denoted I [p e], is the ideal of R generated by all p-th powers of elements in I. We say that f ∈ I∗, the tight closure of I, if there exists a c not in any minimal prime of R with the property that cf e ∈ I [p] for all large e ≥ 0. This notion, due to M. Hochster and C. Huneke, is now an important tool in commutative algebra and algebraic geometry, particularly since it gives a systematic framework for reduction to positive characteristic. We refer the reader to [17] for the basic properties of tight closure in characteristic p. How does the containment f ∈ I∗ depend on the prime characteristic? To make sense of this question suppose that RZ is a finitely generated ring extension of Z and that I ⊆ RZ is an ideal, f ∈ RZ. Then we may consider for every prime number p the specialization RZ/(p) = RZ ⊗Z Z/(p) of characteristic p together with the extended ideal Ip ⊆ RZ/(p), and one may ask whether fp ∈ I∗ p holds or not. We refer to this question about the dependence on the prime numbers as the “arithmetic of tight closure”. Many properties in commutative algebra exhibit an arithmetically nice behaviour: for example, RQ is smooth (normal, Cohen-Macaulay, Gorenstein) if and only if RZ/(p) is smooth (normal, Cohen-Macaulay, Gorenstein) for almost all prime numbers (i.e., for all except for at most finitely many). In a similar way we have for an ideal I ⊆ RZ that IQ = IRQ is a parameter ideal or a primary ideal if and only if this is true for almost all specializations Ip. Furthermore, f ∈ I if and only if fp ∈ Ip holds for almost all prime characteristics: see [16, Chapter 2.1] and appendix 1 in [18] for these kinds of results. When R is a finitely generated Q-algebra, Hochster and Huneke define the tight closure of an ideal I ⊆ R, in the same spirit as the examples above, with the help of a Z-algebra RZ where R = RZ ⊗Z Q, as the set of all f ∈ R for which fp ∈ (Ip) holds for almost all p. This definition is independent of the chosen model RZ. The reader should consult [16] for properties of tight closure in characteristic zero. This definition works well, because the most important features from tight closure theory in positive characteristic, like F -regularity of regular rings, colon capturing,