Example

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

Algebra, chapter X, § 4. Produits de torsion

Let $a$ be an ideal of $A$ . The exact sequence

0 \to a \to A_s \to A/a \to 0

of left a-modules, gives rise to an exact sequence of torsion products, in which the terms Tor $_i^A$ (M, A) are zero for $i > 0$ . From this one deduces isomorphisms

\operatorname{Tor}_{i+1}^A(M, A/\alpha) \to \operatorname{Tor}_i^A(M, \alpha), \quad i > 0

and an exact sequence

0 \to \operatorname{Tor}_1^A(M, A\alpha) \to M \otimes_A \alpha \to M \otimes_A A \to M \otimes A \alpha \to 0 :

it follows that $\operatorname{Tor}_1^A(M, A/\alpha)$ is identified with the kernel of the canonical homomorphism $M \otimes_A \alpha \to M$ .

For example, taking for M a module of the form $A_d/b$ , where b is a right ideal of A, one obtains an isomorphism of $\operatorname{Tor}_1^A(A/b, A/\alpha)$ onto $(\alpha \cap b)/ba$ .

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

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