Proposition 2

Tag 05PR, permanent. Cite it as [Bourbaki, Tag 05PR].

Algebra, chapter II, § 4. Relations between tensor products and homomorphism modules

(i) When $F$ is a projective (resp. finitely generated projective) B-module, the canonical homomorphism (7) is injective (resp. bijective).

(ii) When $E$ is a finitely generated projective $A$ -module, the canonical homomorphism (7) is bijective.

(i) Fixing $E$ and $G$ , for every left $B$ -module $F$ , we write

T(F) = \mathrm{Hom}_A(E, G) \otimes_B F, \quad T'(F) = \mathrm{Hom}_A(E, G \otimes_B F);

for every left $B$ -module homomorphism $u : F \to F'$ , we write $T(u) = 1 \otimes u$ (1 here denoting the identity mapping of $\mathrm{Hom}_A(E, G)$ ),

T'(u) = \mathrm{Hom}(1_E, 1_G \otimes u);

on the other hand we write $\nu_F$ instead of $\nu$ . Then we have the following lemmas:

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

Cited by

Proposition 10, § 2
Corollary, § 4
Proposition 8, § 5
Proposition 16, § 7
Lemma 1, § 12
Proposition 1, § 13
Proposition 4, § 3

Cites

Nothing.