Corollary

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

Let $A$ be a local noetherian ring and $q$ an ideal of definition of $A$ . Then one has $\dim(A) = \dim(\operatorname{gr}_q(A))$ .

Applying proposition 5 to the case $M = H = \operatorname{gr}_q(A)$ , one obtains the relation

\sum_{n \geq 0} \operatorname{long}_{A/q}(q^n/q^{n+1}) \cdot T^n = (1 - T)^{-d} Q(T)

with $d = \dim(\operatorname{gr}_q(A))$ and $Q(1) \neq 0$ . One has $d = \dim(A)$ by Theorem 3 of § 4, No. 4, whence the corollary.

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

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