Corollary 5

Tag 04EM, permanent. Cite it as [Bourbaki, Tag 04EM].

Commutative Algebra, chapter III, § 2. General results on filtered rings and modules

Let A be a commutative ring and m an ideal of A such that $A/m$ is Noetherian, $m/m^2$ is ajinitely generated $(A/m)$ -module and A is Hausdorff and complete with the m-adic topology. Then gr(A) and A are Noetherian.

gr(A) is an $(A/m)$ -module generated by $m/m^2$ (no. 3, Example 3) and hence the ring gr(A) is Noetherian by Corollary 3. From this we deduce that A itself is Noetherian (no. 9, Corollary 2 to Proposition 12).

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

Cited by

Corollary 2, § 2
Corollary 6, § 2
Proposition 8, § 3
Exercise 7, § 5

Cites

no. 3, Example 3
Proposition 12
no. 9, Corollary 2
Corollary 3