All given rings in this paper are commutative, associative with identity, and Noetherian.Recently, L. Ein, R. Lazarsfeld, and K. Smith [ELS] discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: if h is the largest height of an associated prime of I, then I (hn) ⊆ I n for all n ≥ 0. Here, if W is the complement of the union of the associated primes of I, I (t) denotes the contraction of I t R W to R, where R W is the localization of R at the multiplicative system W . Their proof depends on the theory of multiplier ideals, including an asymptotic version, and, in particular, requires resolution of singularities as well as vanishing theorems. We want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist.Our objective here is to give stronger results that can be proved by methods that are, in some ways, more elementary. Our results are valid in both equal characteristic 0 and in positive prime characteristic p, but depend on reduction to characteristic p. We use tight closure methods and, in consequence, we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic. For the most basic form of the result, all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring, every ideal is tightly closed. We note that the main argumentThe authors were supported in part by grants from the National Science Foundation. Version of July 25, 2001.
MELVIN HOCHSTER AND CRAIG HUNEKEhere is closely related to a proof given in [Hu, p. 45] that regular local rings in characteristic p are UFDs, which proceeds by showing that Frobenius powers of height one primes are symbolic powers. Our main results in all characteristics are summarized in the following theorem. Note that I * denotes the tight closure of the ideal I. The characteristic zero notion of tight closure used in this paper is the equational tight closure of [HH6] (see, in particular Definition (3.4.3) and the remarks in (3.4.4) of [HH6]). This is the smallest of the characteristic zero notions of tight closure, and therefore gives the strongest result. See §3.1 for a discussion of the Jacobian ideal J (R/K) utilized in part (c). Theorem 1.1. Let R be a Noetherian ring containing a field. Let I be any ideal of R, and let h be the largest height 1 of any associated prime of I. (a) If R is regular, I (hn+kn) ⊆ (I (k+1) ) n for all positive n and nonnegative k. In particular, I (hn) ⊆ I n for all positive integers n. (b) If I has finite projective dimension then I (hn) ⊆ (I n ) * for all positive integers n.(c) If R is finitely generated, geometrically reduced (in characteristic 0, this simply means that R is reduced) and equidimensional over a field K, and locally I is either 0 or contains a nonzerodivisor (this is automatic if R is a domain), then, with J = J (R/K), for every nonnegative integer k and positive integer n, we have that...
