Lemma 2

Tag 0E1A, permanent. Cite it as [Bourbaki, Tag 0E1A].

Algebra, chapter X, § 8. Dimension homologique

Let $\rho : A\to A'$ be a ring homomorphism, M an a-module, M′ an a′-module. If the a-module $A'_d$ is flat, one has $\mathrm{dp}_A(A'\otimes_A M)\leq\mathrm{dp}_A(M)$ . If the a-module $A'_s$ is projective, one has $\mathrm{dp}_A(M')\leq\mathrm{dp}_{A'}(M')$ .

The first assertion is clear if $\mathrm{dp}_A(M) = \pm \infty$ ; if $\mathrm{dp}_A(M) = n \in \mathbf{N}$ , there exists an exact sequence of $A$ -modules

0 \to P_n \to P_{n-1} \to \ldots \to P_0 \to M \to 0

where the $P_i$ are projective ; the sequence of $A'$ -modules

0 \to A' \otimes_A P_n \to A' \otimes_A P_{n-1} \to \ldots \to A' \otimes_A P_0 \to A' \otimes_A M \to 0

is exact, since $A'_d$ is flat, and the $A'$ -modules $A' \otimes_A P_i$ are projective (II, p. 89, Corollary) ; therefore $\mathrm{dp}_{A'}(A' \otimes_A M) \leq n = \mathrm{dp}_A(M)$ . The second assertion is clear if $\mathrm{dp}_{A'}(M') = \pm \infty$ ; if $\mathrm{dp}_{A'}(M') = m \in \mathbf{N}$ , there exists an exact sequence of $A'$ -modules

0 \to P'_m \to P'_{m-1} \to \ldots \to P'_0 \to M' \to 0 ,

where the $P'_i$ are projective ; the underlying sequence of $A$ -modules is exact. Moreover each $P'_i$ is a sub- $A'$ -module direct factor of a module ${A'_s}^{(1)}$ , hence is a projective $A$ -module ; therefore one has $\mathrm{dp}_A(M') \leq m = \mathrm{dp}_{A'}(M')$ .

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

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