Asymptotic Behaviour of Standard Bases

We prove that the elements of any standard basis of In, where I is an ideal of a Noetherian local ring and n is a positive integer, have order bounded by a linear function in n. We deduce from this that the elements of any standard basis of In in the sense of Grauert-Hironaka, where I is an ideal of the ring of power series, have order bounded by a polynomial function in n. The aim of this paper is to study the growth of the orders of the elements of a standard basis of I, where I is an ideal of a Noetherian local ring. Here we show that the maximal order of an element of a standard basis of I is bounded by a linear function in n. For this we prove a linear version of the strong Artin-Rees lemma for ideals in a Noetherian ring. The main result of this paper is Theorem 3. First we prove the following proposition inspired by Corollary 3.3 of [4]: Proposition 1. Let A be a Noetherian ring and let I and J be ideals of A. There exists an integer λ ≥ 0 such that ∀x ∈ A, ∀n,m ∈ N, n ≥ λm, (x) ∩ (J + I) = ((x) ∩ (J + Im))(xn−λm). Proof. Let B := A/J . By Theorem 3.4 of [5], there exists λ such that for any m ≥ 1, there exists an irredundant primary decomposition I = Q 1 ∩ · · · ∩Q (m) r such that if P (m) i := √ Q (m) i , then (P (m) i ) λm ⊂ Q i for 1 ≤ i ≤ m. We denote by Q (m) i the image of Q (m) i in A/(J + I ) for 1 ≤ i ≤ r. We denote by P i the inverse image of P (m) i in A, for 1 ≤ i ≤ r. Let x ∈ A. If x ∈ P i , then x ∈ (P (m) i ) n and (Qi (m) : x) = A/(J + I) for any n ≥ λm. If x / ∈ P i , then x / ∈ (P (m) i ) n and (Q (m) i : x ) = Q (m) i for any n ≥ λm. Thus, for any n ≥ λm, ( 0A/(J+Im) : x n ) = (⋂