Example 2

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

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

Suppose that $A$ is an integral domain; let $K$ be the field of fractions of $A$ and $M$ an $A$ -module. Using the flat resolution

0 \to A \to K \to K/A \to 0

(X, p. 9, example 5), we see that $\operatorname{Tor}_i^A (K/A, M) = 0$ for $i > 1$ ; moreover, in view of II, p. 116, prop. 26, (ii), the $A$ -module $\operatorname{Tor}_1^A (K, A, M)$ is isomorphic to the torsion submodule of $M$ .

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

Cited by

Nothing.

Cites

Nothing.