Corollary 1

Tag 0D6Z, permanent. Cite it as [Bourbaki, Tag 0D6Z].

Commutative Algebra, chapter X, § 5. Intersections complètes

Let $\hat{A}$ be the separated completion of A for the J-adic topology and $\hat{J} = J\hat{A}$ the separated completion of J. In order that the ideal $\hat{J}$ of $\hat{A}$ be completely secant, it is necessary and sufficient that the ideal J of A be completely secant.

In fact, the canonical mapping gr_J(A) → gr_J(\hat{A}) is an isomorphism of graded rings; it is therefore enough to apply criterion (iv).

More generally:

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

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