Remark 1

Suppose the ideal J completely secant; let (x_1, ..., x_r) be a sequence of elements of J such that for every maximal ideal m ∈ V(J) the canonical images of the x_i in J/mJ form a basis of this A/m-vector space. Then the $A/J$-module $J/J^2$ is free and the canonical images of the x_i in J/J^2 form a basis of it: it is enough in fact to verify that the images of the x_i form a basis of the $A_m/J_m$-module $J_m/J_m^2$ for every m ∈ V(J) (II, § 3, No. 3, Th. 1), which results from loc. cit., No. 2, Prop. 5 and Cor. 2, since the $A_m/J_m$-module $J_m/J_m^2$ is projective (Th. 1).