Bourbaki

Exercise 1, § 8

Algebra, X, p. 202

Let u:A→Bu : A \to B be a homomorphism of rings, M a B-module. Prove the inequality

dpA(M)⩽dpB(M)+dpA(Bs) \mathrm{dp}_A(M) \leqslant \mathrm{dp}_B(M) + \mathrm{dp}_A(B_s)

(Reason by induction on dpB(M)\mathrm{dp}_B(M) or use the spectral sequence of Exercise 6, p. 188.)

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.