Bourbaki

Exercise 9, § 1

Commutative Algebra, VIII, p. 82

Let AA be a ring, pp a prime ideal of AA and MM a finitely generated a-module. Prove that one has:

dim⁡Ap(Mp)+dim⁡A/p(M/pM)≤dim⁡A(M). \dim_{A_p}(M_p) + \dim_{A/p}(M/pM) \leq \dim_A(M).

Solution

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Machine translation of the English, by gpt-5-6-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.