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• W1539571809 abstract "Let A be a locally analytically unramified local ring and J1, . . . , Jk, I ideals such that Ji ⊆ √ I for all i, the ideal I is not nilpotent, and ⋂ k I = (0). Let C = C(J1, . . . , Jk; I) ⊆ R be the cone generated by {(m1, . . . , mk, n) ∈ N | J1 1 . . . Jk k ⊆ I}. We prove that the topological closure of C is a rational polyhedral cone. This generalizes results by Samuel, Nagata, and Rees. Introduction In this note we continue the study of the asymptotic properties of powers of ideals initiated by Samuel in [8]. Let A be a commutative noetherian ring with identity and I, J ideals in A with J ⊆ √ I. Also, assume that the ideal I is not nilpotent and ⋂ k I k = (0). Then for each positive integer m one can define vI(J,m) to be the largest integer n such that J m ⊆ I. Similarly, wJ(I, n) is defined to be the smallest integer m such that J m ⊆ I. Under the above assumptions, Samuel proved that the sequences {vI(J,m)/m}m and {wJ (I, n)/n}n have limits lI(J) and LJ(I), respectively, and lI(J)LJ(I) = 1 [8, Theorem 1]. It is also observed that these limits are actually the supremum and infimum of the respective sequences. One of the questions raised in Samuel’s paper is whether lI(J) is always rational. This has been positively answered by Nagata [4] and Rees [5]. The approach used by Rees is described in the next section of this paper. We consider the following generalization of the problem described above. Let J1, . . . , Jk, I be ideals in a locally analytically unramified ring A such that Ji ⊆ √ I for all i, I is not nilpotent, and ⋂ k I k = (0), and let C = C(J1, . . . , Jk; I) ⊆ Rk+1 be the cone generated by {(m1, . . . ,mk, n) ∈ Nk+1 | J1 1 . . . J mk k ⊆ In}. We prove that the topological closure of C is a rational polyhedral cone; i.e., a polyhedral cone bounded by hyperplanes whose equations have rational coefficients. Note that the case k = 1 follows from the results proved by Samuel, Nagata, and Rees; the cone C is the intersection of the half-planes given by n ≥ 0 and n ≤ lI(J)m1. In Section 3 we look at the periodicity of the rate of change of the sequence {vI(J,m)}m, more precisely, the periodicity of the sequence {vI(J,m+1)− vI(J,m)}m. The last part of the paper describes a method of computing the limits studied by Samuel in the case of monomial ideals. 1. The Rees valuations of an ideal In this section we give a brief description of the Rees valuations associated to an ideal. For a noetherian ring A which is not necessarily an integral domain, a discrete valuation on A is defined as follows. 1991 Mathematics Subject Classification. Primary 13A15. Secondary 13A18. The second author gratefully acknowledges partial financial support from the National Science Foundation, CCF-0515010 and Georgia State University, Research Initiation Grant. 1 2 CĂTĂLIN CIUPERCĂ, FLORIAN ENESCU, AND SANDRA SPIROFF Definition 1.1. Let A be a noetherian ring. We say that v : A → Z ∪ {∞} is a discrete valuation on A if {x ∈ A | v(x) = ∞} is a prime ideal P , v factors through A → A/P → Z ∪ {∞}, and the induced function on A/P is a rank one discrete valuation on A/P . If I is an ideal in A, then we denote v(I) := min{v(x) | x ∈ I}. If R is a noetherian ring, we denote by R the integral closure of R in its total quotient ring Q(R). Definition 1.2. Let I be an ideal in a noetherian ring A. An element x ∈ A is said to be integral over I if x satisfies an equation x + a1x n−1 + . . . + an = 0 with ai ∈ I. The set of all elements in A that are integral over I is an ideal I, and the ideal I is called integrally closed if I = I. If all the powers I are integrally closed, then I is said to be normal. Given an ideal I in a noetherian ring A, for each x ∈ A let vI(x) = sup{n ∈ N | x ∈ In}. Rees [5] proved that for each x ∈ A one can define vI(x) = lim k→∞ vI(x ) k , and for each integer n one has vI(x) ≥ n if and only if x ∈ In. Moreover, there exist discrete valuations v1, . . . , vh on A in the sense defined above, and positive integers e1, . . . , eh such that, for each x ∈ A, (1.1) vI(x) = min {vi(x) ei | i = 1, . . . , h } . We briefly describe a construction of the Rees valuations v1, . . . , vh. Let p1, . . . , pg be the minimal prime ideals p in A such that p+I 6= A, and letRi(I) be the Rees ring (A/pi)[It, t−1]. Denote by Wi1, . . . ,Wihi the rank one discrete valuation rings obtained by localizing the rings Ri(I) at the minimal primes over tRi(I), let wij (i = 1, . . . , g, 1 ≤ j ≤ hi) be the corresponding discrete valuations, and let Vij = Wij ∩ Q(A/pi) (i = 1, . . . , g). Then define vij(x) := wij(x+pi) and eij := wij(t −1)(= vij(I)) for all i, and for simplicity, renumber them as e1, . . . , eh and v1, . . . , vh, respectively. Rees [5] proved that v1, . . . , vh are valuations satisfying (1.1). We refer the reader to the original article [5] for more details on this construction. Remark 1.3. With the notation established above, for every positive integer n we have" @default.
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• W1539571809 title "Asymptotic growth of powers of ideals" @default.
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