Corollary 2

If one has dp_{A_m}(M_m) < +\infty for every maximal ideal m of Supp(M) , one has dp_A(M) < +\infty .

In fact, the subspace X of Supp(M) formed by the maximal ideals is quasi-compact (II, § 4, No. 2, Prop. 8 and 9 and No. 3, Cor. 7 of Prop. 11) ; the mapping m \mapsto dp_{A_m}(M_m) of X into \overline{\mathbf{R}} is upper semi-continuous (Prop. 3), hence bounded (TG, IV, p. 30, Cor. of Th. 3).