Proposition 8

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

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

a) If $N$ is flat over $A$ and over $B$ , homomorphism (11) is bijective.

b) If $N$ is flat over $A$ and projective over $B$ , homomorphism (12) is bijective.

Indeed $N \otimes_A L(Q)$ is isomorphic to a direct sum of copies of $N$ , hence is a flat (resp. projective) $B$ -module when the $B$ -module $N$ is flat (resp. projective); one then applies Theorem 1 (X, p. 100).

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

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