Proposition 1

Tag 0D2H, permanent. Cite it as [Bourbaki, Tag 0D2H].

Commutative Algebra, chapter X, § 2. Modules et anneaux macaulayens

Let $A$ be a noetherian ring, $M$ a finitely generated $A$ -module, and $p$ and $q$ prime ideals of $\mathrm{Supp}(M)$ such that $p \subset q$ . Suppose that $\dim_{A_q}(M_q) = \mathrm{prof}_{A_q}(M_q)$ . Then one has $\dim_{A_p}(M_p) = \mathrm{prof}_{A_p}(M_p)$ and

\dim_{A_q}(M_q) = \dim_{A_p}(M_p) + \dim(A_q/\mathfrak{p}A_q) .

This follows directly from cor. 1 of prop. 13 of § 1, No. 7.

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

Cited by

Nothing.

Cites

Nothing.