Remark 3

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

Commutative Algebra, chapter VIII, § 1. Dimension de Krull d’un anneau

Under the conditions of prop. 9, c), one has $\mathrm{codim}(\mathrm{Supp}(M), \mathrm{Spec}(A)) = \inf(\mathrm{codim}(\mathrm{Supp}(M'), \mathrm{Spec}(A)), \mathrm{codim}(\mathrm{Supp}(M/M'), \mathrm{Spec}(A)))$ . This follows from the formula $\mathrm{Supp}(M) = \mathrm{Supp}(M') \cup \mathrm{Supp}(M/M')$ and from remark 4 of n° 2.

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

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