Example 2

Tag 0D2E, permanent. Cite it as [Bourbaki, Tag 0D2E].

Commutative Algebra, chapter X, § 2. Modules et anneaux macaulayens

Let $M'$ be a direct factor submodule of a finitely generated Macaulay $A$ -module $M$ . Then $M'$ is Macaulay; in fact, for every maximal ideal $m$ of $A$ , the $A_m$ -module $M'_m$ is a direct factor of $M_m$ and one consequently has

\mathrm{prof}_{A_m}(M'_m) \geq \mathrm{prof}_{A_m}(M_m) \geq \dim_{A_m}(M_m) \geq \dim_{A_m}(M'_m),

by Remark 4 of

§ 1

, No. 1 and VIII,

§ 1

, No. 4, Proposition 9 c).

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

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