Primary Decompositions of Powers of Ideals

Let R be a Noetherian ring and I an ideal. We prove that there exists an integer k such that for all n ≥ 1 there exists an irredundant primary decomposition I = q1 ∩ · · · ∩ ql such that √ qi nk ⊆ qi whenever ht (qi/I) ≤ 1. In particular, if R is a local ring with maximal ideal m and I is a prime ideal of dimension 1, then mI ⊆ I, where I denotes the n’th symbolic power of I . We study some asymptotic properties of primary decompositions of powers of ideals in a Noetherian ring. In particular, we consider the following question: Let (R,m) be a regular local ring and P a prime ideal of dimension 1. Then for some cn ∈ N , mnP (n) ⊆ P. How does this cn depend on n? (cf. [2]) The main theorem 1 says that cn is bounded linearly, i.e. there exists an integer k such that mP (n) ⊆ P for all n ≥ 0. Note that if P = P ∩Jn is a primary decomposition of P with m ⊆ √ Jn and mn ⊆ Jn, then mnP (n) ⊆ P. So we tackle the question via selected irreducible primary components of powers of ideals. Hence we consider more generally: if I is an ideal in a Noetherian ring and P ∈ ∪n=1Ass (R/I), does there exist an irredundant primary decomposition of I = qn1 ∩ · · · ∩ qnkn such that if √ qni = P , then the least integer cn for which P cn ⊆ qni is bounded linearly with respect to n? We would also like to know whether there are good primary decompositions of ideals of the form (xq1, . . . , x q n), q ranging over powers of the characteristic of R. A positive answer to this question would solve the open question whether tight closure commutes with localization, at least for ideals generated by elements x1, . . . , xn for which ∪qAss (R/(xq1, . . . , xn)) is a finite set. There may be infinitely many primes associated to such ideals if q is allowed to vary over all positive integers. An example of Hochster is the following: let R = Z[X,Y ], a 1Primary 13H99 2This paper is in final form and no version of it will be submitted for publication elsewhere.