Corollary 1

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

Commutative Algebra, chapter VIII, § 5. Anneaux locaux réguliers

Let $A$ be a noetherian local ring, and $t$ an element of $m_A$ . The following conditions are equivalent:

(i) $A$ is regular, and $t$ does not belong to $m_A^2$ ; (ii) $A/tA$ is regular and $\dim(A/tA) < \dim(A)$ ; (iii) $A/tA$ is regular, and $t$ is not a divisor of 0 in $A$ .

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

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