Bourbaki

Example 2

Tag 04IQ, permanent. Cite it as [Bourbaki, Tag 04IQ].

Commutative Algebra, chapter III, § 5. Flatness properties of filtered modules

Every direct sum of ideally Hausdorff modules is an ideally Hausdorff module, by virtue of the relations

gn(a⊗A⨁λ∈LMλ)=gn⨁λ∈L(a⊗AMλ)=⨁λ∈Lgn(a⊗AMλ). \mathfrak{g}^n \left( a \otimes_A \bigoplus_{\lambda \in L} M_\lambda \right) = \mathfrak{g}^n \bigoplus_{\lambda \in L} (a \otimes_A M_\lambda) = \bigoplus_{\lambda \in L} \mathfrak{g}^n (a \otimes_A M_\lambda).

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

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Transcribed from the printed volume. Not translated.