Proposition 4

Let A be a noetherian ring, x_1, ..., x_r elements of A, x the ideal which they generate, and M a finitely generated A-module such that M/xM is nonzero and of finite length. a) One has d_x(M) \leq r. b) If d_x(M) = r, then e_x(M) \leq \operatorname{long}_A(M/xM). c) If the sequence (x_1, ..., x_r) is completely secant for M (A, X, p. 157), then d_x(M) = r and e_x(M) = \operatorname{long}_A(M/xM). The converse is true if the x_i belong to the radical of A.

Let H be the polynomial ring (A/x)[X_1, ..., X_r]. Let us endow G = \bigoplus_n x^nM/x^{n+1}M with the structure of a graded H-module for which (X_i)_G is multiplication by the class of x_i modulo x^2. With the notation P_G, Q_G, c_G of No. 2, one has H_{M,x} = P_G, hence (1 - T)^{-d_x(M)}R = (1 - T)^{-r}Q_G, where R(1) = e_x(M) > 0 and Q_G(1) = c_G.

One therefore has either d_x(M) < r and c_G = 0, or d_x(M) = r and c_G = e_x(M). Moreover, by Prop. 1 of No. 2, one has c_G \leq \operatorname{long}(M/xM), and there is equality if and only if the canonical homomorphism A/x[X_1, ..., X_r] \otimes_{A/x} M/xM \to \bigoplus_n x^nM/x^{n+1}M is bijective. This entails the proposition, in view of A, X, p. 160, th. 1.