Proposition 14

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

Given an exact sequence

(23)

0 \to E' \to E \to E'' \to 0

of right vector spaces over a field K and linear mappings and a left vector space F over K, the corresponding sequence of $\mathbf{Z}$ -linear mappings

0 \to E' \otimes_K F \to E \otimes_K F \to E'' \otimes_K F \to 0

is exact and splits.

As the sequence (23) splits, this is a particular case of § 3, no. 7, Corollary 5 to Proposition 7 and § 3, no. 6, Proposition 5.

Because of Proposition 14, when $E'$ is a vector subspace of $E$ , $j : E' \to E$ the canonical injection, $E' \otimes_K F$ is usually identified with a sub- $\mathbf{Z}$ -module of $E \otimes_K F$ by means of the injection $j \otimes 1_F$ . With this convention:

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

Cited by

Corollary, § 7
Remark, § 7
Proposition 16, § 7
Proposition 18, § 7
Proposition 19, § 7
§ 4. Tensor products of algebras
Proposition 10, § 10
Lemma 2, § 11
§ 21. Linear Representations of Finite Groups

Cites

§ 3, no. 6, Proposition 5
§ 3, no. 7, Corollary 5 to Proposition 7