Proposition 16

Let $A$ be an integrally closed noetherian ring, $J$ an ideal of $A$ of height $\geqslant 2$, and $M$ a reflexive $A$-module of finite type. One has $\operatorname{prof}_A(J; M) \geqslant 2$.

Let us choose a finite-dimensional vector space $V$ over the field of fractions of $A$ and a lattice $N$ in $V$ isomorphic to $M$ (VII, § 4, No. 2, Remark 1). The prime ideals associated with $V/N$, being of height 1 (loc. cit., Theorem 2), do not contain $J$; after Remark 2 of No. 1, this implies $\operatorname{prof}_A(J; V/N) \geqslant 1$. On the other hand, the $A$-module $V$ is divisible and torsion-free, hence injective (A, X, p. 17, Corollary 2 of Proposition 10), which implies $\operatorname{prof}_A(J; V) = +\infty$. The inequality $\operatorname{prof}_A(J; N) \geqslant 2$ then follows from Proposition 1 of No. 1.