Proposition 8

Tag 0D6A, permanent. Cite it as [Bourbaki, Tag 0D6A].

Commutative Algebra, chapter X, § 4. Anneaux réguliers

Let $\rho : A \to B$ be a homomorphism of noetherian rings making B a faithfully flat A-module.

a) For every finitely generated A-module M, one has $dp_A(M) = dp_B(B \otimes_A M)$ .

b) If the ring B is regular, the ring A is regular.

0 \to N \to L \to M \to 0

where the $A$ -module $L$ is free of finite type. One has $dp_A(N) = dp_A(M) - 1$ (A, X, p. 135, Cor. 2 of Prop. 1). Since $B$ is flat over $A$ , the sequence

0 \to B \otimes_A N \to B \otimes_A L \to B \otimes_A M \to 0

is exact and one has $dp_B(B \otimes_A N) = dp_B(B \otimes_A M) - 1$ . The induction hypothesis applied to $N$ enables one to conclude. This proves a); assertion b) follows from a) and Prop. 4 of No. 2.

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

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