Remark 5

Tag 0CT5, permanent. Cite it as [Bourbaki, Tag 0CT5].

Commutative Algebra, chapter VIII, § 7. Multiplicités

In the situation of the preceding corollary, one has

e_q(M) = e_x(M) \leq \operatorname{long}(M/xM)

and $\operatorname{long}(M/qM) \leq \operatorname{long}(M/xM)$ ; the three cases

e_q(M) < \operatorname{long}(M/qM) , \quad e_q(M) = \operatorname{long}(M/qM) , \quad e_q(M) > \operatorname{long}(M/qM)

are possible (p. 106, Exerc. 16 and 17).

Exercises

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

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