Corollary

Tag 0CT4, permanent. Cite it as [Bourbaki, Tag 0CT4].

Commutative Algebra, chapter VIII, § 7. Multiplicités

Suppose that $A$ is local, with infinite residue field, and set $d = \dim_A(M)$ . There exists a sequence $x_1, ..., x_d$ of elements of $q$ such that, on setting $x = Ax_1 + \cdots + Ax_d$ , one has

e_q(M) = e_x(M) \leq \operatorname{long}(M/xM) < + \infty .

This follows at once from the theorem and Remark 4.

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

Cited by

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Cites

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