Theorem 2

Tag 04JO, permanent. Cite it as [Bourbaki, Tag 04JO].

Commutative Algebra, chapter IV, § 1. Prime ideals associated with a module

Let M be a finitely generated module over a Noetherian ring A and $(M_i)_{0 \leq i \leq n}$ a composition series of M such that, for $0 \leq i \leq n - 1$ , $M_i / M_{i+1}$ is isomorphic to $A / \mathfrak{p}_i$ where $\mathfrak{p}_i$ is a prime ideal of A. Then

(4)

\operatorname{Ass}(M) \subset \{\mathfrak{p}_0, \ldots, \mathfrak{p}_{n-1}\} \subset \operatorname{Supp}(M);

the minimal elements of these three sets are the same and coincide with the minimal elements of the set of prime ideals containing Ann(M).

The inclusion Ass(M) $\subset \{p_0, \ldots, p_{n-1}\}$ follows immediately from Propositions 1 and 3 of no. 1. For $0 \leq i \leq n - 1$ ,

p_i \in \operatorname{Supp}(A/p_i) = \operatorname{Supp}(M_i/M_{i+1})

(Chapter II, § 4, no. 4, Example), whence $p_i \in \operatorname{Supp}(M_i) \subset \operatorname{Supp}(M)$ (Chapter II, § 4, no. 4, Proposition 16), which shows the inclusion

\{p_0, \ldots, p_{n-1}\} \subset \operatorname{Supp}(M).

Corollary 1 to Proposition 7 of no. 3 shows that Ass(M) and Supp(M) have the same minimal elements and (4) shows that these are just the minimal elements of $\{p_0, \ldots, p_{n-1}\}$ . The last assertion then follows from Chapter II, § 4, no. 4, Proposition 17.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Theorem 2

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