Proposition 2

Tag 0D40, permanent. Cite it as [Bourbaki, Tag 0D40].

Commutative Algebra, chapter X, § 3. Profondeur et dimension homologique

Let $A$ be a ring, $M$ and $N$ $A$ -modules, $i$ an integer and $S$ a multiplicative subset of $A$ . One has a canonical isomorphism of $S^{-1}A$ -modules

S^{-1} \mathrm{Tor}_i^A(M, N) \longrightarrow \mathrm{Tor}_i^{S^{-1}A}(S^{-1}M, S^{-1}N) .

If the ring $A$ is noetherian and the $A$ -module $M$ finitely generated, one has a canonical isomorphism of $S^{-1}A$ -modules

S^{-1} \mathrm{Ext}_A^i(M, N) \longrightarrow \mathrm{Ext}_{S^{-1}A}^i(S^{-1}M, S^{-1}N) .

Since the $A$ -module $S^{-1}A$ is flat, this follows from A, X, p. 110, Prop. 9 and p. 111, Prop. 10.

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

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Cites

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