Remark 2

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

Algebra, chapter II, § 3. Tensor products

Under the hypotheses of Proposition 7, let $E'_\lambda$ be a submodule of $E_\lambda$ , $F'_\mu$ a submodule of $F_\mu$ and let $E' = \bigoplus_{\lambda \in L} E'_\lambda$ , $F' = \bigoplus_{\mu \in M} F'_\mu$ . Let Im $(E' \otimes_A F')$ (resp. Im(E'_λ ⊗_A F'_μ)) denote the image of E' ⊗_A F' (resp. E'_λ ⊗_A F'_μ) in E ⊗_A F (resp. E_λ ⊗_A F_μ) under the canonical mapping; then the isomorphism (23) identifies the sub- $\mathbf{Z}$ -modules

\operatorname{Im}(E' \otimes_A F') \quad \text{and} \quad \bigoplus_{(\lambda, \mu) \in I \times M} \operatorname{Im}(E'_\lambda \otimes_A F'_\mu);

this follows immediately from the commutativity of the diagram

\begin{array}{ccc} \left( \bigoplus_\lambda E'_\lambda \right) \otimes_A \left( \bigoplus_\mu F'_\mu \right) & \longrightarrow & \left( \bigoplus_\lambda E_\lambda \right) \otimes_A \left( \bigoplus_\mu F_\mu \right) \\ \downarrow & & \downarrow \\ \bigoplus_{\lambda, \mu} (E'_\lambda \otimes_A F'_\mu) & \longrightarrow & \bigoplus_{\lambda, \mu} (E_\lambda \otimes_A F_\mu) \end{array}

where the vertical arrows are the canonical isomorphisms.

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

Cited by

Nothing.

Cites

Proposition 7