Corollary 2

Let ρ : A → B be a homomorphism of noetherian rings making B into a flat A-module, and J an ideal of A.

a) If J is completely secant, the ideal JB of B is completely secant.

b) Suppose that the ideal JB of B is completely secant and that every maximal ideal m ∈ V(J) is the inverse image of a maximal ideal of B. Then the ideal J is completely secant. This is the case for example if B is a faithfully flat A-module.

First observe that, since the A-module B is flat, $J^n \otimes_A B$ is identified with $J^nB$ and $(J^n/J^{n+1}) \otimes_{A/J} (B/JB)$ with $J^nB/J^{n+1}B$ for every integer $n \geq 0$. Assertion a) then results from criterion (iii). Under the hypotheses of b), the $A/J$-module B/JB is faithfully flat (I, § 2, No. 7, Cor. 2 of Prop. 8 and § 3, No. 5, Prop. 9) and J is completely secant by criterion (iv), taking account of I, § 3, No. 1, Prop. 2 and No. 6, Prop. 12. The last assertion results from Prop. 8 of I, § 3, No. 5.