FILTRATION, ASYMPTOTIC σ-PRIME DIVISORS AND SUPERFICIAL ELEMENTS

Let (a,m) be a noetherian local ring with infinite residue field a/m and i be a m-primary ideal of a. let f = (in)n∈n be a good filtration on a such that i1 containing i. let σ be a semi-prime operation in the set of ideals of a. let l ≥ 1 be an integer and (f (l) )σ = σ(in+l) : σ(in) for all large integers n and ρ f σ (a) = min n ∈ n | σ(il) = (f (l) )σ, for all l ≥ n . here we show that, if i contains an σ(f)-superficial element, then σ(il+1) : i1 = σ(il) for all l ≥ ρ f σ (a). we suppose that p is a prime ideal of a and there exists a semi-prime operation σbp in the set of ideals of ap such that σbp (jap ) = σ(j)ap , for all ideal j of a. hence assa a/σ(il)
⊆ assa a/σ(il+1)
, for all l ≥ ρ f σ (a