First Coefficient Ideals and the S2-ification of a Rees Algebra

Let (A, , k) be a d-dimensional (d ≥ 1) quasi-unmixed analytically unramified local domain with infinite residue field. If I is an -primary ideal, Shah defined the first coefficient ideal of I to be the largest ideal I{1} containing I such that ei(I) = ei(I{1}) for i = 0, 1. Assume that A is (S2) and let = n Intn be the S2-ification of the extended Rees algebra S = A[It, t− 1]. We prove that In = (In){1} for every n ≥ 1. One of the consequences is a procedure of computing first coefficient ideals.