Lemma 1

— Let A be a noetherian local ring, M a finitely generated and pure A-module, and x an element of mA. The following conditions are equivalent:

(i) one has dim(M/xM) = dim(M) − 1 ; (ii) the homothety xM is injective.

One may suppose M nonzero. Assertion (i) is equivalent to the fact that x belongs to none of the minimal elements p of Supp(M) such that dim(A/p) = dim(M) (VIII, § 3, No. 2, Prop. 3), and assertion (ii) is equivalent to the fact that x belongs to none of the prime ideals associated with M (IV, § 1, No. 1, Cor. 2 to Prop. 2). Since M is pure, its associated prime ideals are the minimal elements of Supp(M); therefore (i) and (ii) are equivalent.

Let A be a noetherian local ring and M a nonzero finitely generated A-module. Recall (VIII, § 3, No. 2) that a sequence (x₁, ..., x_r) of elements of mA is said to be secant for M if one has dim(M/(x₁M + ... + x_rM)) = dim(M) − r.