Lemma 1

Tag 0D6C, permanent. Cite it as [Bourbaki, Tag 0D6C].

Commutative Algebra, chapter X, § 4. Anneaux réguliers

One has

\delta(B) + [\mathrm{Ker}(d\rho) : \kappa_B] = \delta(A) + \delta(\kappa_A \otimes_A B) + (\dim(A) - \dim(B) + \dim(\kappa_A \otimes_A B)) .

Let $C$ denote the local ring $\kappa_A \otimes_A B$ . Consider the exact sequence of $B$ -modules

B \otimes_A \mathfrak{m}_A \to \mathfrak{m}_B \to \mathfrak{m}_C \to 0 ;

by tensor product with $\kappa_B$ , one obtains an exact sequence of $\kappa_{13}$ -vector spaces

\kappa_B \otimes_{\kappa_A} (\mathfrak{m}_A/\mathfrak{m}_A^2) \xrightarrow{d\rho} \mathfrak{m}_B/\mathfrak{m}_B^2 \longrightarrow \mathfrak{m}_C/\mathfrak{m}_C^2 \to 0,

whence one deduces the equality

[\mathfrak{m}_B/\mathfrak{m}_B^2 : \kappa_B] + [\mathrm{Ker}(d\rho) : \kappa_B] = [\mathfrak{m}_A/\mathfrak{m}_A^2 : \kappa_A] + [\mathfrak{m}_C/\mathfrak{m}_C^2 : \kappa_C],

which implies the lemma.

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

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