Corollary 5
Tag 04LB, permanent. Cite it as [Bourbaki, Tag 04LB].
Commutative Algebra, chapter IV, § 2. Primary decomposition
In a complete semi-local Noetherian ring every decreasing sequence ideals whose intersection is 0 is a filter base which converges to 0.
It is sufficient to apply Corollary 4 to the A-module .
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.