Example 1

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

Algebra, chapter X, § 6. Utilisation de résolutions non canoniques

Let $a$ be an element of $A$ such that the mapping $\varphi : x \mapsto xa$ of $A$ into itself is injective (" $a$ is not a right zero divisor "). Using the resolution

0 \to A_s \xrightarrow{\varphi} A_s \to A/Aa \to 0

one sees that for every right $A$ -module $M$ , one has

\operatorname{Tor}_i^A (M, A/Aa) = 0 \quad \text{for } i > 1

and that the $k$ -module $\operatorname{Tor}_1^A (M, A/Aa)$ is isomorphic to $\operatorname{Ker} (a_M)$ .

Analogously, for every left $A$ -module $M$ , one has

\operatorname{Ext}_A^i (A/Aa, M) = 0 \quad \text{for } i > 1

and the $k$ -module $\operatorname{Ext}_A^1 (A/Aa, M)$ is isomorphic to $M/aM$ .

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

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