Corollary 1

Tag 0CN7, permanent. Cite it as [Bourbaki, Tag 0CN7].

Commutative Algebra, chapter VIII, § 3. Dimension des anneaux noethériens

Let H be a nonempty hypersurface in V. The codimension of H in V is equal to 0 or 1.

One has codim(H, V) = 1 if and only if H contains no irreducible component of V. If this is so, all the irreducible components of H are of codimension 1 in V.

For every irreducible component Z of H, one has

0 \leq \operatorname{codim}(Z, V) \leq 1

by Proposition 4, b), and one has codim(Z, V) = 0 if and only if Z is an irreducible component of V. One has by definition

\operatorname{codim}(H, V) = \inf_Z \operatorname{codim}(Z, V)

where Z ranges over the set of irreducible components of H. Corollary 1 follows at once from these remarks.

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

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