Corollary 3

Tag 04E6, permanent. Cite it as [Bourbaki, Tag 04E6].

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

Suppose that X and Y are Hausdorff, the filtrations of X and Y exhaustive and X complete. Then, if gr(u) is bijective, u is bijective.

Let A be a local ring, m its maximal ideal and M an A-module; let A and M be given the m-adic filtrations and let gr(A) and gr(M) be the graded ring and the graded gr(A)-module associated with A and M. We have seen (no. 3, Example 3) that the canonical mapping (11) is always surjective; we are going to consider the following property of M: (GR) The canonical mapping

\gamma_M : \operatorname{gr}(A) \otimes_{\operatorname{gr}_0(A)} \operatorname{gr}_0(M) \to \operatorname{gr}(M)

is bijective.

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

Cited by

Nothing.

Cites

no. 3, Example 3