Example 5

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

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

Let A be a graded ring of type $\mathbf{Z}$ and E a graded A-module of type $\mathbf{Z}$ ; let $A_{(i)}$ (resp. $E_{(i)}$ ) be the subgroup of homogeneous elements of degree i of A (resp. E). Let A and E be given the filtrations associated with their graduations (no. 1, Example 1) and let A' and E' denote the filtered ring and filtered A-module thus obtained. Then it is immediate that the Z-linear mapping $A \to \mathrm{gr}(A')$ which maps an element of $A_{(n)}$ to its canonical image in

\mathrm{gr}_n(A) = \left( \bigoplus_{i \geq n} A_{(i)} \right) / \left( \bigoplus_{i \geq n+1} A_{(i)} \right)

is a graded ring isomorphism. A canonical graded A-module isomorphism

E \to \mathrm{gr}(E')

is defined similarly.

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

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