Asymptotic prime divisors and analytic spreads

Abstract

Abstract. Let / be an ideal in a Noetherian domain R, and let / be the integral closure of /. Let A*(I) = A^s(R/I") for n large (it being known that for large n this set does not vary with n). Suppose that R satisfies the altitude formula. Then it is shown that P e Â*(I) if and only if height P ■> l(IP), the analytic spread of /,,.

Asymptotic growth of algebras associated to powers of ideals

“…Here ℓ(H) denotes the analytic spread of an ideal H. For ordinary symbolic powers a related result was proven by Katz and Ratliff [20,Theorem A and Corollary 1]. One direction of Theorem 1.5 follows from part (a) of Theorem 1.1, which is inspired by a result of McAdam [23], where we give a short direct proof of the fact that under the above conditions on R and J, S J (I) is a graded subalgebra of the integral closure of the Rees algebra of I, provided that ℓ(I P ) < dim R P for all P ∈ V (J). This result can also be deduced from Theorem 4.1 of Katz's paper [K] and Theorem 5.6 of Schenzel's paper [S].…”

“…The multiplicity of a finitely generated R-module M will be denoted by e(M ). Theorem 1.1 below is inspired by results of MacAdam [23], Ratliff [25], Katz [19], Schenzel [27] and others on the asymptotic associated primes of ideals of small analytic spread. Part (a) of Theorem 1.1 follows from Theorem 4.1 of Katz's paper [19] and Theorem 5.6 of Schenzel's paper [27].…”

A note on asymptotic prime sequences

“…As T is quasiunmixed (and therefore satisfies the altitude formula). Theorem 3 in [4] implies that a((I,xx,_xk)T) = dim T. Since the images of xx.xk in Fform an asymptotic prime sequence over IT(by 1.8), 1.6 implies that a(IT) -dim T -k. Beginning with yx, 1.6 applied k times in T shows that a((I, V|,... ,yk)T) = dim T. Again Theorem 3 in [4] implies that MT E A* ((I, v,,. ..,yk)T) so by 1.8, k -s. Finally the argument given indicates clearly that k 3= i(7), and may be repeated to provide a contradiction if k > s(I).…”

Symbolic powers of prime ideals and their topology

“…Because P satisfies the altitude formula it follows that a(I) < dim P by virtue of S. McAdam [6,Theorem 3]. In order to prove (iii) =* (i) we first note that it is enough to show that D(IR) is finitely generated over R(IR).…”