Remark 3

Tag 0E35, permanent. Cite it as [Bourbaki, Tag 0E35].

Algebra, chapter X, § 9. Complexes de Koszul

If the family $x$ is completely transverse for $A$ , the complex $K.(x, A)$ defines a free resolution of the $A$ -module $A/x$ , with $x = \sum_{i \in I} x_i A$ ; therefore for every integer $i \geqslant 0$ and for every $A$ -module $M$ there are isomorphisms

\text{Ext}_A^{n-i}(A/x, M) \to H^{n-i}(x, M) \to H_i(x, M) \to \operatorname{Tor}_i^A(A/x, M) .

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

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