Proposition 7

Let $0 \to M \to N \to Q \to 0$ be an exact sequence of $A$-modules. Suppose that $N$ is finitely generated and torsion-free. (i) If $M$ is reflexive, then $\operatorname{Ass}(Q) \subset P \cup \{\{0\}\}$ (in other words, every ideal associated with $Q$, is, either $(0)$, or of height 1 ). (ii) Conversely, if $N$ is reflexive and $\operatorname{Ass}(Q) \subset P \cup \{\{0\}\}$, then $M$ is reflexive.

As $A$ is Noetherian, $M$ is also finitely generated; if we write $V = M_{(K)}$, $W = N_{(K)}$, $M$ (resp. $N$) is canonically identified with a lattice of $V$ (resp. $W$) (no. 1, Proposition 1). Consider the two exact sequences:

(i) We deduce (Chapter IV, § 1, no. 1, Proposition 3) that: If $M$ is reflexive, then $\operatorname{Ass}(V/M) \subset P$ (Theorem 2); on the other hand, clearly $\operatorname{Ass}(W/V)$ is, either empty, or reduced to $\{0\}$; whence (i). (ii) Similarly: The hypotheses therefore imply that $\operatorname{Ass}(V/M) \subset P \cup \{\{0\}\}$. But $V/M$ is a torsion $A$-module and hence $\{0\} \notin \operatorname{Ass}(V/M)$; Theorem 2 then shows that $M$ is reflexive.