Corollary

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

Commutative Algebra, chapter VIII, § 7. Multiplicités

Suppose that B is flat over A and that ρ(mA) B = m_B. Then*

e_{q_B}^B(M \otimes_A B) = e_q^A(M).

This applies in particular when B is the completion \* or the henselisation \* of A relative to an ideal distinct from A, \* or a blowing-up of A, for example a strict henselisation of A. \*

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

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